## High Performance and Parallel Computing for Materials Defects and Multiphase Flows (01 Jan - 31 Mar 2015)

Jointly organized with Department of Mathematics, NUS

## ~ Abstracts ~

An innovative phase transition modeling for reproducing cavitation
Rémi Abgrall, Universität Zürich, Switzerland

This work is devoted to model the phase transition for two-phase flows with a mechanical equilibrium model. First, a five-equation model is obtained by means of an asymptotic development starting from a non-equilibrium model (seven-equation model), by assuming a single-velocity and a single pressure between the two phases, and by using the Discrete Equation Method (DEM) for the model discretization. Then, a splitting method is applied for solving the complete system with heat and mass transfer, i.e., the solution of the model without heat and mass transfer terms is computed and, then, updated by supposing a heat and mass exchange between the two phases. Heat and mass transfer is modeled by applying a thermo-chemical relaxation procedure allowing to deal with metastable states.

The interest of the proposed approach is to preserve the positivity of the solution, and to reduce at the same time the computational cost. Moreover, it is very flexible since, as it is shown in this paper, it can be extended easily to six (single velocity) and seven-equation models (non-equilibrium model).

Several numerical test-cases are presented, i.e. a shock-tube and an expansion tube problems, by using the five equation model coupled with the cavitation model. This enables us to demonstrate, using the standard cases for assessing algorithms for phase transition, that our method is robust, efficient and accurate, and provides results at a lower CPU cost than existing methods. The influence of heat and mass transfer is assessed and we validate the results by comparison with experimental data and to the existing state-of-art methods for cavitation simulations.

Joint work with M.G. Rodio (INRIA Bordeaux Sud Ouest)

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Computational studies of novel solar energy materials: defects and inverse sign
Juan Ruiz Álvarez, Universidad Alcalá, Spain

With the rapid development of the modern computational techniques, computational studies on the condensed matter, including understanding physical mechanism, simulating specific dynamics processes and designing desired materials, have played a more and more important role. A direct design of materials with desired properties, so-called inverse deign, has been a long-standing dream in computational materials science.

In this talk, I will show our newly developed algorithm for the inverse deign of materials (IM2ODE: Inverse Design of Materials by Multi-Objective Differential Evolution). As an example, I will demonstrate how one can obtain a direct-gap carbon or silicon phase that is proper for the solar absorbers with this package. Furthermore, I will present some our recent results on the halide perovskite ABX3, stability, defects, especially a new algorith for band offset calculation, with the resuts for the band alignment of inogranic halide pervoke will also be discussed.

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Complex flows with the Lattice Boltzmann method
Regina Ammer, Universität Erlangen-Nürnberg, Germany

First, the lattice Boltzmann method (LBM) is introduced that can be used for simulating complex 3D flows including hydrodynamic and thermodynamic phenomena. Basic ideas of the algorithm are provided and specific modelling aspects for simulating flows with phase-change, heat transfer and condensation-evaporation problems are discussed as extensions of the classical LBM.

Furthermore, it is shown that the lattice Boltzmann method plays an important role in simulations involving free surfaces. A volume-of-fluid based free surface treatment (FSLBM) has been applied successfully for various applications in the past but so far an analysis of this method has been missing.

We show that this original FSLBM boundary condition is of first order in spatial accuracy with a Chapman-Enskog analysis. Following the same ansatz, an improved second order accurate free surface boundary condition is proposed.

Finally, numerous applications using the FSLBM are proposed: simulations for additive manufacturing processes, protein foams, and interaction of moving obstacles with the fluid flow.

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Large-scale stencil and particle applications and their performances on a GPU supercomputer
Takayuki Aoki, Tokyo Institute of Technology, Japan

The GPU (Graphics Processing Unit) has been used in many world top-class supercomputers as an accelerator with high computational performance and wide memory bandwidth. We have developed several large-scale stencil-based PDE and particle-based applications well suitable for GPU supercomputers. We demonstrate turbulent flow simulations with a Lattice Boltzmann Method for air currents over a 10 km x 10 km central part of metropolitan Tokyo with 1m resolution in association with a sophisticated LES model and a phase-field simulation for the dendritic solidification of a binary alloy with 0.3 trillion cells. We also show granular and fluid simulations based on short-range particle interactions, in which a dynamic load balance has to be introduced. For most of the applications, the performance scalabilities based on the "roofline model" are shown.

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Asymptotic and numerical methods for rarefied gas flows based on kinetic theory
Kazuo Aoki, Kyoto University, Japan

Part 1:
The first half of this part is devoted to a brief introduction to kinetic theory of gases, which contains a summary of the Boltzmann equation and its basic properties, the boundary conditions, etc. Then, we consider the free-molecular gas (or the Knudsen gas), i.e., a gas which is so rarefied that the collisions between gas molecules can be neglected (that is, the mean free path of the gas molecules is infinitely long compared with the characteristic length of the system). We present an exact solution that describes the effect of boundary temperature in a quite general situation and its applications to some concrete problems.

Part 2:
In this part, we consider the near continuum regime (or near the fluid-dynamic limit), i.e., the case where the mean free path is small compared with the characteristic length. We show the outline of the formal asymptotic analysis of the steady boundary-value problem of the Boltzmann equation that provides the fluid-dynamic type equations, their boundary conditions of slip type, and the kinetic correction to fluid-dynamic solutions in the vicinity of the boundary (the Knudsen layer) systematically. A special emphasis is put on the case in which the fluid-dynamic limit thus obtained is not covered by the conventional fluid dynamics (the ghost effect).

Part 3:
In this part, we consider the general case where the mean free path is of the same order of magnitude as the characteristic length (the transition regime). In this case, the basic tool for solving the Boltzmann equation is a numerical method. Roughly speaking, there are two kinds of approach: One is stochastic (Monte Carlo methods) and the other is deterministic (discrete-velocity methods, finite-volume methods, etc.). Here, we discuss some deterministic methods with special interest in capturing the propagation of discontinuities in the velocity distribution function induced by a convex body or discontinuous boundary data.

Part 4:
The last part is devoted to the numerical analysis of moving boundary problems. We first discuss unsteady motion of a body in a free-molecular gas. Then, we consider unsteady flows of a (collisional) rarefied gas caused by the oscillation of a body. To be more specific, restricting ourselves to simple one-dimensional problems (i.e., gas flows induced by an infinitely wide plate oscillating in its normal direction), we investigate the unsteady behavior of the gas on the basis of a model Boltzmann equation. The aim of this part is to show properties and difficulties inherent to moving boundary problems in kinetic theory using a simple one-dimensional setting.

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Motion of an array of plates in a rarefied gas caused by radiometric force
Kazuo Aoki, Kyoto University, Japan

We consider an infinitely long two-dimensional channel containing a rarefied gas. Inside the channel, there is an array of infinitely many two-dimensional plates (with finite width and without thickness) perpendicular to the axis of the channel, and the array can move freely along the channel. If one side of each plate of the array is heated and the other side is not, it is subject to a force because of the temperature difference on its two sides (radiometric force). In consequence, the array starts moving, and finally its motion reaches the steady motion with a constant velocity, where the radiometric force and the drag force acting on each plate counterbalance. We investigate the behavior of the gas in this final steady motion and obtain the speed of the array numerically on the basis of kinetic theory, for a wide range of the Knudsen number. The present problem can be regarded as a simplified model for the moving vanes of the Crookes radiometer. This is a joint work with Satoshi Taguchi.

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Dynamic rupture earthquake simulations on petascale heterogeneous supercomputers
Michael Bader, Technische Universität München, Germany

SeisSol is a software suite for dynamic rupture earthquake simulations based on the Arbitrary high order DERivative Discontinuous Galerkin (ADER-DG) method. SeisSol uses unstructured tetrahedral meshes to approximate complicated geometries in realistic geological models. Multiphysics simulations of the dynamic rupture process and the resulting seismic wave propagation lead to unique model complexity. SeisSol has recently been optimized for heterogeneous (Xeon-Phi-based) supercomputers. I will discuss optimization techniques, present scalability studies on the currently largest Xeon-Phi supercomputers and show results from a petascale simulation with nearly 100 billion degrees of freedom executed on SuperMUC.

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Variational principles and computation of linear response eigenvalue problems with application to excited state calculations
Zhaojun Bai, University of California, USA

Linear response eigenvalue problems (LREPs) arise from time-dependent density functional theory in the linear response formalism. LREPs are challenging due to their unconventional doubly-structured (non-Hermitian) representation. In this talk, we will first present recent theoretical results on variational principles of LREPs. These results are the extension of well-known variational principles for Hermitian eigenvalue problems. Then we will present the application of these principles in the development of a conjuguate-gradient like method for large-scale LREPs. Numerical results for the calculations of multiple low-lying excitation energies of molecules will be presented. This is a joint work with Ren-cang Li, Dario Rocca and Giulia Galli.

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Mathematical models and numerical methods for Bose-Einstein condensation
Weizhu Bao, National University of Singapore

The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. In this talk, I will present a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.

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Modeling, analysis & simulation for quantized vortices in superfluidity and superconductivity
Weizhu Bao, National University of Singapore

Quantized vortices have been experimentally observed in type-II superconductors, superfluids, nonlinear optics, etc. In this tutorial, I will review different mathematical equations for modeling quantized vortices in superfluidity and superconductivity, including the nonlinear Schrodinger/Gross-Pitaevskii equation, Ginzburg-Landau equation, nonlinear wave equation, etc. Asymptotic approximations on single quantized vortex state and the reduced dynamic laws for quantized vortex interaction are reviewed and solved analytically in several cases. Efficient and accurate numerical methods will be presented for computing quantized vortex lattices and ther dynamics. Direct numerical simulation results from different PDE models are reported for quantized vortex dynamics and they are compared with those from the reduced dynamics laws. Some open problems and emerging applications will be discussed.

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Geometric graph-based methods for high dimensional data
Andrea Bertozzi, University of California Los Angeles, USA

We present new methods for segmentation of large datasets with graph based structure. The method combines ideas from classical nonlinear PDE-based image segmentation with fast and accessible linear algebra methods for computing information about the spectrum of the graph Laplacian. The goal of the algorithms is to solve semi-supervised and unsupervised graph cut optimization problems. I will present results for image processing applications such as image labeling and hyperspectral video segmentation, and results from machine learning and community detection in social networks, including modularity optimization posed as a graph total variation minimization problem.

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Particle laden thin films: theory and experiment
Andrea Bertozzi, University of California Los Angeles, USA

Modeling of particle laden flow, especially in the case of higher particle concentrations, does not readily allow for first principles models. Rather, semi-empirical models of the bulk dynamics require careful comparision with experiments. At UCLA we have developed this theory for the geometry of viscous thin film flow with non-neutrally buoyant particles. We have found that for these slower flows, that diffusive flux models, involving a balance between shear-induced migration and hindered settling, can provide reasonably accurate predictive models. I will discuss the current state of this work including recent extensions to bidensity slurries and the relevant mathematics needed to understand the dynamics. Lubrication theory can be derived for this problem and results in a coupled system of conservation laws including regular shock dynamics and singular shocks. I will also briefly discuss relevant applications such as spiral separators.

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Multigrid methods for structured matrices on large-scale supercomputers
Matthias Bolten, University of Wuppertal, Germany

Matrices often posses a lot of structure. This structure allows for a rigorous analysis of the numerical methods that are applied to these matrices, e.g., to obtain the solution of linear systems. Often these results can be transferred to similar problems, e.g., to partial differential equations discretized on structured grids.

For many problems multigrid methods are optimal solvers. By optimality we mean that the convergence rate is bounded from above independently from the system size and that the number of arithmetic operations grows linear with the system size. Originally multigrid methods have been developed especially for the solution of linear systems that arise when partial differential equations are discretized, later this approach has been extended to general algebraic multigrid methods. Based on these developments multigrid methods for structured matrices have been developed.

Structured problems often arise in physics or engineering when structured discretizations of the continuous problems have been used. In many cases the system matrices belong to well-analyzed matrix classes, so the applications can directly benefit from the theoretical results. If this is not the case, the results can often be used by asymptotic arguments, as well. In any case the structure can be used for an efficient - and in some cases hardware-aware - implementation.

In the talk multigrid methods for structured matrices and their application on large-scale supercomputers are presented.

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On adaptive methods in heterogeneous media
Jed Brown, Argonne National Laboratory, USA

Conventional adaptive methods are ill-suited to problems with heterogeneous materials throughout the domain because coarse discretizations cannot resolve the macroscopic properties in the absence of scale separation. We propose $\tau$ adaptivity, a multigrid technique that provides most of the benefits of adaptive mesh refinement for problems with localized nonlinearities in heterogeneous media. The method is based on the Full Approximation Scheme (FAS) and takes advantage of the $\tau$ correction (influence of fine scales on coarser scales) changing much more slowly than the localized nonlinearities. In parallel implementations, $\tau$ adaptivity can be combined with modest overlap to eliminate most horizontal data dependencies, enabling local decisions about adaptivity and lower overhead load balancing. This talk will introduce these methods, explore the class of problem structure that can be exploited, and discuss open questions and practical concerns.

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Sparse grids: higher dimensionalities and HPC aspects
Hans-Joachim Bungartz, Technische Universität München, Germany

There are a lot of challenges Computational Science and Engineering and High-Performance Computing are confronted with, many of them being of "Multi-X" type: multi-physics problems, multi-scale models, multi-level algorithms, or multi-core systems are just some prominent representatives. Moreover, multi-dimensionality has become an issue, since multi-dimensional problems have become accessible to numerical approaches, where formerly Monte Carlo methods or informatics-based techniques such as neural networks were the dominant males. High-dimensional numerical quadrature, stochastic moments, computational finance, optimization, parameter identification, and classification provide nice examples for the increasing relevance of higher dimensionalities in a computational context.

The crucial roadblock to overcome for higher dimensionalities is the so-called curse of dimensionality, expressing the exponential increase of degrees of freedom with growing dimensionality (O(Nd) characteristics). Sparse grids are one of the success stories in discretization beyond a continuum mechanics horizon. The talk will provide a brief overview of sparse grid methods and current application scenarios and then focus on recent results concerning higher dimensionalities and promising HPC-related features.

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Multiscale modeling and simulation of gas-liquid flows
Vivek Buwa, Indian Institute of Technology-Delhi, India

Several engineering processes involve various gas-liquid flows e.g. in chemical processing, oil and gas, power generation technologies, biochemical operations, etc. Gas-liquid flows occurring in these processes are complex and often involve simultaneous heat/mass transport and chemical reactions. The inherently unsteady gas-liquid flows, with widely varying length- and time-scales, pose severe challenges to their experimental characterization and numerical simulations. The speaker will introduce various length- and time-scales associated with gas-liquid flows. The continuum methods used to simulate large-scale gas-liquid flows and their limitations will be discussed with the help of examples. Further, the computational methods used to simulate microscopic (small-scale) gas-liquid flows with moving and deforming surfaces and their applications to simulate the rise behavior of single/multiple bubbles, segmented gas-liquid/gas-liquid-liquid flows in micro-channels and drop spreading on solid surfaces will be presented.

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Parallel domain decomposition methods and multi-physics applications
Xiao-Chuan Cai, University of Colorado Boulder, USA

Domain decomposition is a class of algorithms for solving partial differential equations on parallel computers. In this talk, we first discuss some classical results concerning overlapping domain decomposition methods and then focus on some recent work on domain decomposition methods for multi-physics problems arising in solid mechanics, fluid mechanics, and optimization problems, etc. For linear problems we present some Schwarz preconditioned Krylov subspace methods, and for nonlinear problems, we discuss some Schwarz preconditioned inexact Newton methods.

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Parallel solvers for nonlinear PDEs
Xiao-Chuan Cai, University of Colorado Boulder, USA

In this talk we discuss a general purpose parallel solution framework for solving nonlinear algebraic system of equations arising from the discretization of nonlinear partial differential equations or optimization problems constrained by partial differential equations. The framework includes algorithmic components like inexact and preconditioned Newton method, Krylov subspace method and domain decomposition methods. High performance measured using strong or weak scalability or the total compute time can be achieved only when the algorithmic components are carefully selected. The focus of the talk is about the interplay and customization of these algorithmic components. As examples, we discuss applications in solving compressible Euler equations for atmospheric flows, incompressible Navier-Stokes equations for blood flows, Cahn-Hilliard-Cook equations for spinodal decomposition of materials, and some PDE constrained optimization problems.

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Density functional theory
Eric Cances, Ecole des Ponts ParisTech and INRIA, France

First-principle molecular simulation has become an essential tool in chemistry, condensed matter physics, molecular biology, materials science, and nanosciences. It is also an inexhaustible source of exciting mathematical and numerical problems. In this tutorial, I will focus on Density Functional Theory and the Kohn-Sham model, which is to date the most widely used approach in first-principle molecular simulation, as it provides the best compromise between accuracy and computational efficiency. The outline of the tutorial is as follows:

1 - Basics of quantum mechanics and first-principle modeling of molecular systems

2 - Density functional theory (DFT) and the Kohn-Sham model

3 - The supercell model for condensed matter simulations

4 - DFT for perfect crystals, crystals with a single point defect, and crystals with a random distribution of point defects.

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Parallel spectral-element direction splitting method for incompressible Navier-Stokes equations
Lizhen Chen, Beijing Computational Science Research Center, China

An efficient parallel algorithm for the time dependent incompressible Navier-Stokes equations is developed. The time discretization is based on a direction splitting method which only requires solving a sequence of one-dimensional Poisson type equations at each time step. Then, a spectral-element method is used to approximate these one-dimensional problems. A Schur-complement approach is used to decouple the computation of interface nodes from that of interior nodes, allowing an efficient parallel implementation. The unconditional stability of the full discretized scheme is rigorously proved for the two-dimensional case. Numerical results are presented to show that this algorithm retains the same order of accuracy as a usual spectral-element projection type schemes but it is much more efficient, particularly on massively parallel computers.

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A second-order cell-centered Lagrangian scheme for two-dimensional elastic-plastic problems
Jun Bo Cheng, Beijing Institute of Applied Physics and Computational Mathematics, China

We introduce a high-order cell-centered Lagrangian scheme for two dimensional elastic-plastic problems with the hypo-elastic constitutive model and the von Mises yield criterion. There are many numerical methods for these problems, including staggered Lagrangian schemes, Eulerian Godunov schemes and cell-centered Lagrangian schemes. In these methods, however, the wave structure is not exploited in the construction of the methods. Because the hypo-elastic constitutive model is a non-conservative equation, it is difficult to build an approximate Riemann solver in the normal direction of cell edge for the governing equations with the hypo-elastic constitutive model. In this paper, we analyze the wave structure of the Riemann problem for elastic-plastic materials and then develop a four-rarefaction-wave Riemann solver with elastic waves (TRRSE). Based on the developed TRRSE, we proposed a second-order cell-centered Lagrangian scheme for two-dimensional elastic-plastic solid problems. Moreover, we use a second-order scheme to discretize the constitutive equation based on geometry conservation law in order to keep high accuracy. Several numerical experiments are carried out, and the numerical results are compared with the exact solution and the results obtained by other authors. The comparison shows that the current scheme is convergent, stable and essentially non-oscillatory.

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Partitioning and load balancing in parallel quantum chemistry
Edmond Chow, Georgia Institute of Technology, USA

This talk focuses on the parallelization of the most fundamental method in quantum chemistry, the Hartree-Fock method. This method has a complicated data access pattern, making it challenging to find a partitioning of the work that reduces communication. Load balance is also difficult due to sparsity and irregularly sized tasks. We first present a basic introduction to the partitioning and load balance problem for Hartree-Fock. We then present a new optimized and scalable distributed implementation of Hartree-Fock. Calculations are shown for up to 8100 nodes of the Tianhe-2 supercomputer, simultaneously using both CPUs and Intel Xeon Phi coprocessors.

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A parallel orbital-updating approach for electronic structure calculations
Xiaoying Dai, Chinese Academy of Sciences, China

In this talk, we will talk about an orbital iteration based parallel approach for electronic structure calculations. This approach is based on our understanding for the single-particle equations of independent particles that move in an effective potential. With this new approach, the solution of the single-particle equation is reduced to some solutions of independent linear algebraic systems and a small scale algebraic problem. It is demonstrated by our numerical experiments that this new approach is quite efficient for electronic structure calculations for a class of molecular systems. This presentation is based on some joint works with Xingao Gong, Aihui Zhou, and Jinwei Zhu.

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Towards high resolution 3D computations of quantum turbulence in Bose-Einsten condensates
Ionut Danaila, Université de Rouen, France

Bose-Einstein condensates (BEC) are ideal superfluid systems to realize quantum turbulence (QT): vortex cores in BECs are larger than in superfluid Helium, making easier their observation. Recent experimental and numerical studies reported that vortex states in BEC can evolve towards a turbulent regime when an oscillatory excitation is applied. We discuss in this work how to accurately prepare initial states with vortices before running numerical simulations of QT based on the Gross-Pitaevskii equation (GPE). As in classical fluid turbulence, the numerical and physical accuracy of the initial condition could be crucial in computing properties of numerically generated QT. We first focus on stationary solutions with vortices. Different cases are presented, from a dense Abrikosov lattice in a fast rotating BEC to giant vortices. High resolution numerical simulations using parallel computing are used to accurately capture physically important features of the vortices (vortex radius, inter-vortex spacing, vortex density profile). The stationary vortex lattice will be then used as initial condition in the unsteady (real-time) GPE and submitted to external oscillatory excitations to explore routes to QT.

A new numerical system solving the Gross-Pitevskii equation was designed and implemented using MPI-OpenMP parallel programming. The new code is able to solve both real-time and imaginary-time Gross-Pitaevskii equations. It was used to accurately simulate steady configurations with a large number of vortices, and afterwards their dynamics. For the space discretization we use either Fourier pseudo-spectral methods (with periodic boundary conditions) or compact sixth order finite difference schemes (with periodic or homogeneous Dirichlet boundary conditions). The numerical code performed with very good scaling properties on parallel supercomputers up to 16,000 cores. The highest grid resolution used in computations was 1024^3.

Joint work with Ionut Danaila, Université de Rouen, France and Ph. Parnaudeau, A. Suzuki, Université Pierre et Marie Curie, Paris, France and J.-M Sac-Epée, Université de Lorraine, Metz, France.

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Fast semi-Lagrangian schemes for kinetic equations
Giacomo Dimarco, Université Paul Sabatier, France

In this talk we will introduce a new class of semi-Lagrangian schemes for solving kinetic equations. Following a discrete velocity approach the velocity space is discretized into a set of fixed velocities. Consequently the original kinetic equation is replaced by a set of linear transport equations plus a coupling interaction term corresponding to the collision operator. The purpose of the method is to drastically reduce the cost of the transport part of the discrete velocity model using a semi-Lagrangian technique, allowing to exactly solve the transport part on the entire domain at a negligible cost : the shape of the distribution function is fixed once for all in the space domain and no reconstructions of the distribution function are needed anymore to find the feet of the characteristic. Hence, the cost of the solution of the original kinetic equation is almost entirely due to the projection of the solution onto the grid to compute the collision operator. We then demonstrate the capability of the scheme to deal with parallel architectures. We will present its behaviors on a classical architecture using OpenMP and then on GPU architecture using CUDA.

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Incompressible N-phase flows: physical formulation, numerical algorithm, and parallelism
Steven Suchuan Dong, Purdue University, USA

This talk focuses on simulating the motion of a mixture of N (N>=2) immiscible incompressible fluids with given densities, dynamic viscosities and pairwise surface tensions. We present an N-phase formulation within the phase field framework that is thermodynamically consistent, in the sense that the formulation satisfies the conservations of mass/momentum, the second law of thermodynamics and Galilean invariance. In addition, we also present an efficient algorithm for numerically simulating the N-phase system that has overcome the issues caused by the variable mixture density/viscosity and the couplings among the (N-1) phase field variables and the flow variables. Issues of parallelization in the context of a spectral element implementation will also be discussed. We compare simulation results with the Langmuir-de Gennes theory to demonstrate that the presented method produces physically accurate results for multiple fluid phases. Numerical experiments will be presented for several problems involving multiple fluid phases, large density contrasts and large viscosity contrasts to demonstrate the capabilities of the method for studying the interactions among multiple types of fluid interfaces.

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Solving eigenvalue problems in Kohn-Sham DFT theory
Weiguo Gao, Fudan University, China

Kohn-Sham Density Functional Theory (KS-DFT) is one of the most promising techniques for studying electronic structures of materials. This talk summarizes our work on the nonlinear eigenvalue problems arising from KS-DFT. First, we introduce the nonlinear eigenvalue problem through a simple 3D model and summarize some known results including the existence of the minima, the convergence of the self-consistent field iteration. We then present our new method, LOBPCG, which has no steps of orthogonalization but maintains the same convergence speed. Moreover, we examine the Newton's method and a modification of the orbital minimization method. Finally we demonstrate our recent numerical method of the DFT plane wave pseudo-potential calculation on GPU clusters.

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Scalable multilevel stokes solver for Mantle convection problems
Björn Gmeiner, Universität Erlangen-Nürnberg, Germany

At the beginning of the presentation, we will introduce the concept of hierarchical hybrid grid to design scalable and fast multigrid solvers for incompressible Stokes flow that lead to saddle point formulations. Based on a hierarchy of block-wise uniformly refined grids, a semi-structured mesh is obtained that constitutes the starting point for the design and implementation of a massively parallel Stokes solver.

Weak and strong scaling properties are presented for recent supercomputers and show the applicability of our solver in case of extreme resolutions as occurring in Earth mantle convection simulations. We demonstrate that a co-design of models, discretization, algorithms, and their parallel implementation based on a systematic performance engineering methodology can lead to numerical solvers that are not just asymptotically optimal but also fast as measured by time-to-solution.

Currently, we are working on the implementation of parallel multi-level Monte-Carlo methods on top of the multigrid solver. Here, some aspects in combining the high-performance multigrid solver and multilevel uncertainty quantification are addressed.

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Dimension-adaptive sparse grid simulations of multiscale viscoelastic flows
Michael Griebel, Institut für Numerische Simulation, Germany

Polymeric viscoelastic fluids can be modelled by using the Navier-Stokes equations on the macroscopic scale with an additional stress tensor and a higher-dimensional Fokker-Plank equation or a corresponding stochastic PDE on the microscopic scale. Here, the dimension of the microscopic problem is 3N where N+1 is the number of beads in the underlying spring bead model for viscoelasticity. For the numerical treatment of the overall system, we couple the the stochastic Brownian configuration field method with our fully parallelized three-dimensional Navier-Stokes solver NaSt3DGPF. But due to the microscopic problem, we directly encounter the curse of dimensionality. To this end, we suggest the so-called dimension-adaptive sparse grid approach. It allows to deal with moderate-sized subproblems in an adaptive fashion. Furthermore, all arising subproblems can be treated fully in parallel. This way, reliable multiscale simulations of viscoelastic flow problems for microscopic models with N >1 get possible for the first time.
This is joint work with Alexander Rüttgers from Bonn.

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Coarse grained density functional theory for the study of defects in crystalline materials
Xiao Hao, Nanyang Technological University,

Defects determine critical properties of crystalline materials even though they occur at relatively low concentrations. They can interact over long distances through slowly decaying fields whose strength depends on the electronic structure of the core. Thus the study of defects requires electronic resolutions with continuum range. This talk will describe a sub-linear scaling method for computing the electronic structure of solids at continuum scales with no a priori ansatz or ad hoc patches, and illustrate it with selected examples.

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Coupling molecular dynamics and first-principle electronic structure modeling of disordered heterostructures
Olle Heinonen, Argonne National Laboratory, USA

Metal/insulator/metal heterostructures are of considerable interest because of their complex behavior with important potential applications. For example, many structures in which the insulator is an oxide, such as Ti-oxide or Hf-oxide, exhibit resistive switching with applications in low-power scalable memories. The conducting properties depend sensitively on the structure at the insulator-metal interface. While the conductance can be modeled using non-equilibrium Green's function (NEGF) methods, it is very expensive, and it is also very time-consuming, to generate different disordered structures. Here, we will report on a software workflow that combines molecular dynamics simulations with NEGF calculations using the Swift parallel scripting language. Many parallel instances of MD simulations using LAMMPS of quenched disorder of interfaces are fed directly and seamlessly to the smeagol NEGF code. Swift then launches in parallel conductance calculations for different values of applied bias voltage to obtain I-V curves, in addition to other electronic structure information, such as densities-of-states. While smeagol itself is limited to about 1,000 atoms the calculation of each bias point may require up to 64k cores. Using Swift , we can effortlessly parallelize several voltage calculations for different atomic configurations concurrently, leading to massively parallel execution on several 100k cores.
Joint work with O. Heinonen, K. Maheshwari, B. Narayanan, X. Zhong, D. Karpeyev, S. Sankaranayarayanan. M. Wilde, P. Zapol, and I. Rungger.

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The parareal method: challenges and opportunities
Jan Hesthaven, Ecole Polytechnique Federale de Lausanne, Switzerland

During the last decade there has been an intense research activity devoted to the development of parallel-in-time methods to further push the boundaries of performance of large scale simulations and improve scaling at the large scale. This development has been very successful for range of problems, including molecular dynamics and diffusion dominated problems.

We shall discuss a particular family of such methods, known as the parareal methods, and discuss their behavior of general problems. While the succes of such methods for solving diffusion dominated problems is well established, the progress for convection dominated problems has been considerably slower with the classic methods being plagued by stability problems or slow performance.

In this talk we shall discuss these issues to expose the cause of the stability problems. This understanding allows us to further explore different strategies to resolve these issues. Time permitting we shall also discuss resilience of the parareal method in an extreme scale computing environment.

This work is done in collaboration with Y Maday (UPMC, Paris), A. Nielsen (EPFL), F. Chen (CUNY Baruch)

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Mathematical models and simulations for solid-state dewetting problems
Wei Jiang, Wuhan University, China

The Solid-state dewetting problem of thin films is a hot research topic which is attracting increasing attention from a lot of scientists. Studies on the solid-state dewetting problem involve in the multidisciplinary research, such as material sciences, physics, applied mathematics, scientific computing and so on. Different from the traditional "liquid-wetting-solid" type problem, to some extent, the solid-state dewetting problem belongs to the "solid wetting-solid" type problem. It includes two important research subjects, named as the surface diffusion flow and the moving contact line, and furthermore, the surface energy anisotropy plays important roles in the problem. In the talk, I will discuss several mathematical models which are used to simulate the solid-state dewetting problems, such as the phase field model and sharp-interface model.

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An asymptotic preserving unified gas kinetic scheme for grey radiative transfer equations
Song Jiang, Institute of Applied Physics and Computational Mathematics, China

The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon's mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction among radiation and material. In the optical thin limit, he photon free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the grey radiative transfer equations, where the radiation transport equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics

[K. Xu and J.C. Huang, J. Comput. Phys. 229 (2010), 7747-7764], and is an extension of a recent work [L. Mieussens, J. Comput. Phys. 253(2013), 138-156] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional grey radiative system. The newly developed scheme has the asymptotic preserving (AP) property n the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon's mean free path and time step being less than the photon collision time. Besides the diffusion limit, the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method. Due to the direct modeling for the time evolution solution of the interface radiative intensity, a smooth transition of the transport physics from optical thin to optical thick can be accurately recovered. Many numerical examples are included to validate the current approach.
(joint work with Wenjun Sun and Kun Xu)

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Parallel, adaptive, multilevel solution of nonlinear systems arising in phase-field problems
Peter Jimack, University of Leeds, UK

We describe the parallel implementation of adaptive mesh refinement, implicit time stepping, and a nonlinear multigrid solver in order to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt which is substantially cooled below its freezing point at the microscale, and is of physical importance in material science for improved understanding of the formation of solid metal alloys. Hitherto, this fully coupled thermal-solute problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques and parallel implementation this problem is shown here to be tractable at appropriate resolution with relatively moderate computational resources.

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Existence, stability and dynamics of solitary waves, vortices and vortex rings in Bose-Einstein condensates: from theory to experiments
Panayotis Kevrekidis, University of Massachusetts, USA

In this talk, we will present an overview of some of our recent theoretical, numerical and experimental efforts concerning the static, stability, bifurcation and dynamic properties of coherent structures that can emerge in one- and higher-dimensional settings within Bose-Einstein condensates. We will discuss how this ultracold setting can be approximated at a mean-field level by a deterministic PDE of the nonlinear Schrodinger type and what the fundamental nonlinear waves of the latter are, such as dark solitons, vortices and vortex rings. Then, we will try to go to a further layer of simplified description via nonlinear ODEs encompassing the dynamics of the waves within the traps that confine them, and the interactions between them. Finally, we will attempt to compare the analytical and numerical implementation of these reduced descriptions to recent experimental results and speculate towards a number of interesting possibilities for the future.

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David E. Keyes, King Abdullah University of Science and Technology, Saudi Arabia

Algorithmic adaptations are required to use anticipated exascale hardware near its potential, since the code base has been engineered to squeeze out flops. Instead, algorithms must now squeeze out synchronizations, memory, and transfers, while extra flops on locally cacheable data represent small costs in time and energy. Today's scalable solvers, in particular, exploit frequent global synchronizations. After decades of programming model stability with bulk synchronous processing (BSP), new programming models and new algorithmic capabilities (to take advantage, e.g., of forays in data assimilation, inverse problems, and uncertainty quantification) must be co-designed with the hardware. We briefly recap the architectural constraints, highlight some work at KAUST, and outline future directions.

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Numerical investigation of unsteady flow supercavitation and associated acoustic wave propagation
Boo Cheong Khoo, National University of Singapore

Cavitation/supercavitation usually occurs as a result of flow acceleration along underwater body surface and plays an important role in a variety of engineering problems. The present study is focused on the numerical investigation of pressure wave propagation through the cavitating compressible liquid flow, its interaction with cavitation bubble and the resulting unsteady cavitation/supercavitation evolution. The compressibility effects of liquid water are taken into account and the cavitating flow is governed by one-fluid cavitation models which are based on the compressible Euler equations with the assumption that the cavitation is the homogeneous mixture of liquid and vapour which are locally under both kinetic and thermodynamic equilibrium. Two kinds of homogeneous cavitation models are employed and compared and the unsteady features of cavitating flow due to the external perturbation, such as the cavitation deformation and collapse and consequent pressure increase are resolved numerically and discussed in detail. In addition, axisymmetric boundary element solver was developed to study acoustic wave propagation in a moving supercavitating vehicle.

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Software design for highly scalable numerical algorithms
Harald Köstler, Universität Erlangen-Nürnberg, Germany

In the last years it was possible to deal with more and more complex models for the various applications in computational science and engineering like fluid or other physical simulations. These models became feasible mostly because of the progress made in parallel computer architectures like found in GPUs and modern multi-core CPUs. However, the implementation effort on large HPC clusters increases, if one wants to achieve good performance. A solution to this problem is to formulate the numerical algorithms in an abstract way in a domain-specific language (DSL) and then automatically generate efficient C++ or CUDA code. A multi-layered approach is sketched that allows users to describe applications in a natural way from the mathematical model down to the program specification. As an example, we show scaling results for a generated multigrid solver on a large HPC cluster.

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The Cumulant LBM for turbulent flow and coupled multiphase problems on massively parallel CPU- and GPGPU architectures
Manfred Krafczyk, Technische Universität Braunschweig, Germany

M. Krafczyk, M. Geier, M. Schönherr, K. Kucher, A. Pasquali, Y. Wang We report on recent progress in modeling turbulent and multiphase / multicomponent flows on massively parallel CPU and GPGPU architectures. After presenting some technical aspects of the first GPGPU implementation of grid refinement for the Lattice Boltzmann method (LBM) on a GPGPU using a D3Q27 stencil we show how the Cumulant LBM can be utilized to successfully simulate turbulent implicit LES flows on a desktop system equipped with multiple GPGPUs for an automotive related drag computation. Another example of the capabilities for our approach is a multi-scale evaporation simulation over a realistic soil surface using thousands of CPU-cores including a hierarchical tree-type data structure for grid refinement.

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Numerical simulations for a leaky dielectric drop under DC electrical field
Ming-Chih Lai, National Chiao Tung University, Taiwan

In this talk, we present a hybrid immersed boundary (IB) and immersed interface method (IIM) to simulate the dynamics of a drop under an electric field in Navier-Stokes flows. Within the leaky dielectric framework with piecewise constant electric properties in each fluid, the electric stress can be treated as an interfacial force on the drop interface. Thus, both the electric and capillary forces can be formulated in a unified immersed boundary framework. The electric potential satisfies a Laplace equation which is solved numerically by an augmented immersed interface method which incorporates the jump conditions naturally along the normal direction. The incompressible Navier-Stokes equations for the fluids are solved using a projection method on a staggered MAC grid and the potential is solved at the cell center. The interface is tracked in a Lagrangian manner with mesh control by adding an artificial tangential velocity to transport the Lagrangian markers to ensure that the spacing between markers is uniform throughout the computations. A series of numerical tests for the present scheme have been conducted to illustrate the accuracy and applicability of the method.

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A rescaling scheme and its applications in moving boundary problems
Shuwang Li, Illinois Institute of Technology, USA

In this talk, I will first give a brief introduction on moving boundary problems and related numerical challenges. From a computation point of view, the complex morphology due to interface instabilities and the intrinsic slow/fast growth make long-time simulations extremely expensive/difficult. Here I present a time and space rescaling scheme, which can significantly reduces/extends the computation time (especially for slow/fast growth), and enables one to accurately compute the very long-time dynamics of moving interfaces. I will then show a few examples and demonstrate the efficiency of the method.

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Finite element methods for the dipolar Bose-Einstein condensates
Xiang-Gui Li, Beijing Information Science and Technology University, China

In this talk, finite element approximation is considered for computing the dipolar Bose-Einstein condensates with nonlocal nonlinear convolution terms in one- or multi-dimensional space. Following the idea of the imaginary time method, we compute the ground state solution of the 1-D Bose-Einstein condensate by directly applying the standard finite element method to solve a nonlinear parabolic differential-integral equation. Theoretical analysis is given to show the existence and convergence of this finite element method for one-dimensional problem. For the multi-dimensional dipolar BEC, it can be described a Gross-Pitaevskii equation coupled with a first-order velocity system. Then a combined discontinuous Galerkin method, which is a hybridized mixed discontinuous Galerkin method combining with the direct discontinuous Galerkin method, is proposed to compute ground states and dynamics. This combined DG method can keep the conservation of the particle number and avoid evaluating integrals with high singularity. Numerical examples are presented to demonstrate the accuracy and capability of the proposed method.

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Parallel treecode for boundary integral method in growth of elastically stressed precipitates
Xiaofan Li, Illinois Institute of Technology, USA

The evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method. We present a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions. We also give a space-time rescaling scheme for computing the long time evolution of multiple precipitates. We demonstrate the effectiveness of the parallel treecode and the rescaling scheme by studying the effect of elasticity in long-time morphological evolution of precipitates.

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New progress on energy-conserved S-FDTD methods in computational electromagnetics

Maxwell's equations, which are a set of partial differential equations describing the relation of electric and magnetic fields, have been widely used in computational electromagnetics. They have been playing an important role in many applications such as the radio-frequency, microwave, antennas, aircraft radar, integrated optical circuits, wireless engineering, and the design of CPU in microelectronic, etc. During the propagation of electromagnetic wave in lossless media without sources, the electromagnetic energy keeps constant for all time, which explains the physical feature of conservation of electromagnetic energy in long term behavior. It is significantly important to preserve this invariance in time, for developing efficient numerical schemes for Maxwell's equations and specially for a long term computation of electromagnetic fields. In this talk, we will present our new progress on energy-conserved S-FDTD methods, including high-order energy-conserved schemes for Maxwell's equations and new energy-conserved schemes in metamaterials. We will talk theoretical results on energy conservation, stability and convergence and will give numerical examples to show their performance. Applications in metamaterials will also be introduced in the talk.

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A second-order stable explicit interface advancing scheme for FSI with both rigid and elastic structures
Jie Liu, National University of Singapore

In this paper, we present a temporally second-order numerical scheme for fluid-structure interaction (FSI) problems in which the structure may be rigid or elastic. The explicit treatment of the interface motion and the semi-implicit treatment of other terms make our scheme very efficient. We prove the stability of our scheme which indicates that the explicit treatment may not damage the stability. An exact solution for FSI is derived. We use it to numerically check that our scheme converges at a rate of $O(\Delta t^2 + h^{m+1})$ when we use $P_m/P_{m-1}/P_m$ finite elements for the fluid velocity, fluid pressure and structure velocity. We also confirm the stability and accuracy of our scheme using the benchmark solution of flow past a cylinder-elastic bar. As the new fluid-structure system can deal with both "active motion" and "passive deformation" of structures, we use our scheme to study the locomotion of articulated structures in viscous fluid. Our rigid-elastic fish model obey all the local balance laws at the deforming interfaces. It can faithfully capture the vorticity generation and thrust generation at these deforming interfaces. Our computation shows that a planar three-link fish can propel itself in a viscous fluid by periodically change its shape variables.

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Modified ghost fluid method applied to treat compressible and incompressible flow coupling
Tiegang Liu, Beihang University, China

The Modified Ghost Fluid Method (MGFM) has provided us a robust way of treating immiscible material interfaces in compressible mutlimedium/multiphase flow with large density ratio, and it overcomes the difficulties encountered by the original ghost fluid method proposed by Fedkiw et al.

This work is devoted to extending the modified ghost fluid method (MGFM) to treat compressible and incompressible fluid coupling. By solving shock relationship in the compressible medium and a derived equation of interfacial pressure continuity together to predict the ghost fluid states, this approach not only ensures numerical stability and maintains the advantages of simplicity and high efficiency, but also provides a more accurate interface boundary condition. Specific applications to underwater bubble collapse are presented.

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Redesigning algorithms for extreme scale
Hatem Ltaief, King Abdullah University of Science and Technology, Saudi Arabia

Multicore and accelerator-based systems are now ubiquitous in the hardware landscape. These architectures present many challenges to the scientific end-users: including high concurrency, non-uniform memory access, programming language interoperability. Abstracting these issues from scientific software development is indispensable, especially in the exascale era, less than a decade away, where system complexity will rise significantly.

This talk will highlight numerical algorithms and code optimization techniques (e.g., mixed precision techniques, hierarchical algorithms, data motion reduction, asynchronous execution) and present scheduling frameworks to run numerical applications on many-core systems with high productivity in mind. There is no free lunch. It is still the responsibility of the user, for instance, to expose the fine-grain parallelism so that a dynamic runtime system can then distribute the work among available computing resources. Demonstrations will be shown for critical dense and sparse numerical operations (e.g., dense factorizations and eigensolvers, stencil computations, sparse matrix-vector multiply), which are inner kernels for many scientific and engineering applications. Latest performance results will be reported against existing high performance numerical libraries using the state-of-the-art available hardware based on commodity x86 and accelerators.

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The parareal in time algorithm: basic aspects and new developments
Yvon Maday, Université Pierre et Marie Curie, France

The need for faster numerical simulations of complex phenomena, and the definition in this context of what a complex phenomenon is, is evolving in line with the improvement of the platforms that are available for High Performance Computing. Indeed, what used to require hours or days of numerical simulations on large computers can now be run in fractions of seconds on laptops. Nevertheless the understanding of real phenomena, the control and optimization of processes and the monitoring of industrial problems propose new challenges where i) better accuracy, ii) use of more involved mathematical models, iii) simulations on bigger object or iv) on longer period of time for unsteady phenomena are required. The evolution of the computing platforms helps in addressing bigger problems but is not sufficient.

Exascale systems which are achievable in the next 5-10 years will contain millions of cores. In order to make efficient use of these systems, high-performance applications must have sufficient parallelism to support parallel execution across millions of threads of execution. The development of more efficient and more highly parallel scalable solvers is therefore at the forefront of Exascale applications research and development, in particular, the domain decomposition methods or task partitioning approaches reach their limits in their ability to use the entire computational resource with the same efficiency as currently achieved on existing smaller systems.

Most simulations which are expected to deliver economic, societal and scientific impact from Exascale systems contain time-stepping in some form and present-day codes make little or no use of parallelism in the time domain; time stepping is currently treated as a serial process.

For time dependent problems, either pure differential systems or coupled with partial differential equations, the time direction leads to new families of algorithms that might allow to provide full efficiencies and speed ups. The parareal (parallel in time) algorithm and the waveform relaxation methods have been introduced to fill this gap and have the potential to extract very large additional parallelism from a wide range of time-stepping application codes. This is a disruptive technology which will deliver performance speed-ups of between 10 and 100. By comparison, optimisations of current algorithms typically yield benefits in the range of tens of percent, or at most a factor of 2-3 improvement.

In this talk, we shall introduce the basics of the approach, taking care of the only time direction. We shall present the efficiency that can be expected, the drawbacks of the original approach and the way these can be circumvented. We shall then present the way to combine this algorithm with other iterative procedures such as algebraic linear or nonlinear solvers, domain decomposition methods or as control problems. A current state of the art including the numerical analysis of these combined schemes will also be presented so as the challenges that need to be addressed now.

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Emergence of macroscopic behavior in complex systems
Sebastien Motsch, Arizona State University, USA

In a human crowd or in a shoal of fish, thousands of individuals interact and form large scale structures. Although the interaction among individuals might be simple, the resulting dynamics is quite complex. Modeling is an essential tool to understand such dynamics. For instance, Agent-based models, also referred to as microscopic models, are widely developed to analyze various dynamics such as swarming and opinion formations.

In this talk, we investigate the emergence of macroscopic behavior for such models. The challenge is to link "microscopic models" describing each agent with "macroscopic models" describing the evolution of a fluid. To achieve this transition, we present a novel approach based on kinetic theory and asymptotic analysis. Numerical simulations are presented to illustrate the results.

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A steady state preserving spectral method for Boltzmann equations
Lorenzo Pareschi, Università degli Studi di Ferrara, Italy

We present a general way to construct spectral methods for the collision operator of the Boltzmann equation which preserve exactly the Maxwellian steady-state of the system. We show that the resulting method is able to approximate with spectral accuracy the solution uniformly in time. Extension of the steady state technique here developed to other problems is also discussed.

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Discontinuous Galerkin finite element methods
Jianxian Qiu, Xiamen University, China

The discontinuous Galerkin (DG) method, which is a finite element method suitable for solving convection dominated partial di erential equations (PDEs). This method has gained a lot of popularity in recent years because of its nice mathematical properties in stability and convergence, its
exibilit to many different PDEs from applications, and its efficiency for adaptivity and parallel implementation.

In this 4 hours tutorial, we will give a general introduction and a detail implementation of the DG methods for solving time-dependent, convection-dominated PDEs, including the hyperbolic conservation laws, convection-diffusion equations, and Hamilton-Jacobi equations. We will discuss cell entropy inequalities, nonlinear stability of DG methods.

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Weighted essentially non-oscillatory limiters for Runge-Kutta discontinuous Galerkin methods
Jianxian Qiu, Xiamen University, China

In the presentation we will describe our recent work on a class of new limiters, called WENO (weighted essentially non-oscillatory) type limiters, for Runge-Kutta discontinuous Galerkin (RKDG) methods. The goal of designing such limiters is to obtain a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition for the RKDG method. We adopt the following framework: first we identify the "troubled cells", namely those cells which might need the limiting procedure; then we replace the solution polynomials in those troubled cells by reconstructed polynomials using WENO methodology which maintain the original cell averages (conservation), have the same orders of accuracy as before, but are less oscillatory. These methods work quite well in our numerical tests for both one and two dimensional cases, which will be shown in the presentation.

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Discrete dislocation dynamics model for deformation in polycrystals
Siu Sin Quek, Institute of High Performance Computing

Grain boundaries (GBs) play important roles during the deformation of polycrystalline materials. While it is well known that GBs obstruct dislocations, they in fact do much more during deformation. Depending on the characteristics of the GB and the applied load, they can also transmit dislocations; serve as dislocation sources and sinks; migrate; and serve as interfaces where grains slide against one another. These mechanisms often operate synergistically together to relieve elastic stress buildup, even though the degree of their effects varies according to factors like grain size, sample size, material system and so on. In this work, we extend the classical discrete dislocation dynamics (DDD) framework to include some of the above mechanisms at GBs. We demonstrate the implementation of these mechanisms in first, a bi-crystalline system, followed by studying grain size effects in nanocrystalline systems. We found that the framework enables us to reproduce a crossover from a "smaller is stronger" to a "smaller is weaker" behavior as the grain size is reduced. This lends further insight on the inverse Hall-Petch behavior observed in nanocrystalline materials.
Joint work with Siu Sin Quek, Zheng Hoe Chooi, Zhaoxuan Wu, Yong-Wei Zhang and David J. Srolovitz.

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Modelling and simulation of moving contact lines
Weiqing Ren, National University of Singapore and Institute of High Performance Computing, A*STAR

I will present a sharp-interface model for moving contact lines. The continuum model consists of the Navier-Stokes equation, the conventional interface conditions, a slip boundary condition and a condition for the dynamic contact angle. The boundary conditions are derived from thermodynamic principles and molecular dynamics simulations. The continuum model is solved using the level set method. Numerical examples for the detachment of droplets and moving contact lines on rough surfaces will be presented.

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Parallel multigrid solvers for geophysical flow
Ulrich Rüde, University Erlangen-Nuremberg, Germany

Hierarchical hybrid grids (HHG) are a parallel data stucture that supports the design of scalable and fast multigrid solvers for finite element problems. A semi-structured mesh is constructed by the successive refinement of an unstructured tetrahedral base mesh. Here the algorithms and data structures of HHG are extended to solve the incompressible Stokes equations. The saddle point system can be treated either by a Schur complement approach that uses multigrid for the repeated approximate inversion of the Laplace-operator, or directly by multigrid using special smoothers. In this presentation, we investigate and analyze experiments for Earth Mantle Convection and demonstrate that finite element systems with a trillion (10^12) unknowns can be solved on state of the art supercomputers.

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Parallel spectral-element direction splitting method for incompressible Navier-Stokes equations and phase-field models of multiphase flows
Jie Shen, Purdue University, USA

An efficient parallel algorithm for the time dependent incompressible Navier-Stokes equations is developed in this paper. The time discretization is based on a direction splitting method which only requires solving a sequence of one-dimensional Poisson type equations at each time step. Then, a spectral-element method is used to approximate these one-dimensional problems. A Schur-complement approach is used to decouple the computation of interface nodes from that of interior nodes, allowing an efficient parallel implementation. The unconditional stability of the full discretized scheme is rigorously proved for the two-dimensional case. Numerical results are presented to show that this algorithm retains the same order of accuracy as a usual spectral-element projection type schemes but it is much more efficient, particularly on massively parallel computers. We shall also discuss how to extend the above approach to deal with phase-field models of multiphase flows.

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Introduction to defects in crystals
David Srolovitz, University of Pennsylvania, USA and Penn Institute for Computational Science, USA

This tutorial is designed as an introduction to defects in crystals. I'll begin with a quick review of crystal structure and symmetry, then go through the basic ideas behind point defects, line defects (dislocations) and interfaces in materials (grain boundaries, twins, surfaces, stacking faults, anti-phase boundaries). The focus will be on the general ideas and the language of defects. No prior materials science or physics background will be assumed.

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Rethinking defect structure
David Srolovitz, University of Pennsylvania, USA and Penn Institute for Computational Science, USA

We often think of defects such as grain boundaries, dislocation cores, surfaces,... as having specific, well-defined structures. There is considerable evidence that in fact the structure of individual defects is not well-defined and is statistically distributed. I will discuss our recent atomistic simulation studies that enumerates the possible grain boundary structural states as a function of bicrystallography for a range of grain boundary types and materials. Next, I will discuss a systematic statistical mechanics framework for understanding these structures that puts bounds on their thermodynamic properties. Then I will show how the plethora of metastable grain boundary states give rise to glass-like grain boundary behavior and show how grain boundaries explore the energy landscape during kinetic processes.

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Fault tolerant multigrid
Linda Stals, The Australian National University, Australia

If time permits, I will also talk about new work with Ulrich Ruede on fault tolerant multigrid. In particular, the work looks at how to design a parallel multigrid algorithm to recover from a fault that may occur due to a loss of a processor. The talk will address this issue in terms of both how to reconstruct the grid layers, especially in the case of adaptive refinement, and how to resume the iterative process with minimal disruption.

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The parallel solution of discrete thin plate splines
Linda Stals, The Australian National University, Australia

Data fitting is an integral part of a number of applications including data mining, 3D reconstruction of geometric models, image warping and medical image analysis. A commonly used method for fitting functions to data is the thin-plate spline method. This method is popular because it is not sensitive to noise in the data.

We have developed a discrete thin-plate spline approximation technique that uses local basis functions. With this approach the system of equations is sparse and its size depends only on the number of points in the discrete grid, not the number of data points.

In this talk I will discuss the parallel implementation of the discrete thin-plate spline method. As the resulting system of equations is a saddle point problem, the talk also touches on some of the general issues related to the parallel solution of such types of systems.

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Numerical study of quantized vortex dynamics and interaction in superfluidity and superconductivity
Qinglin Tang, Université de Lorraine, France

The appearance of quantized vortices is regarded as the key signature of superfluidity and superconductivity, and their phenomenological properties have been well captured by the Ginzburg-Landau-Schrodinger (GLSE) equation. In this talk, we will propose accurate and efficient numerical methods for simulating GLSE. Then we apply them to study various issues about the quantized vortex phenomena, including vortex dynamics, sound-vortex interaction, radiation, pinning effect and the validity of the reduced dynamical law (RDL) which govern the motion of the vortex centers in GLSE.

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Quantum continuum mechanics for many-body systems
Giovanni Vignale, University of Missouri, USA

Classical continuum mechanics is a theory of the dynamics of classical liquids and solids in which the state of the body is described by a small set of collective fields, such as the displacement field in elasticity theory; density, velocity, and temperature in hydrodynamics. A similar description is possible for quantum many-body systems, at all length scales, and indeed its existence is guaranteed by the basic theorems of time-dependent current density functional theory. In this talk I show how the exact Heisenberg equation of motion for the current density of a many-body system can be closed by expressing the quantum stress tensor as a functional of the current density. I then introduce an "anti-adiabatic" approximation scheme for this functional. I show that this approximation schemes emerges naturally from a variational Ansatz for the time-dependent many-body wave function. The anti-adiabatic approximation scheme allows us to bypass the solution of the time-dependent Schrodinger equation, resulting in an equation of motion for the displacement field that requires only ground-state properties as input. This approach may have significant advantages over the conventional Kohn-Sham density- and current-density functional approaches for large systems, particularly for systems that exhibit strongly collective behavior. I illustrate the formalism by applying it to the calculation of excitation energies in a few model systems. I discuss strategies for improvement and generalizations, for example, to systems at high magnetic field.

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Phonon and electron transport in nanostructures and crystal solids
Jian-Sheng Wang, National University of Singapore

We start with a brief review of the formulism and computational methods for phonon and electron transport in nanostructures. Two major approaches will be discussed, the nonequilibrium Green's function method and quantum master equation. We also discuss a molecular dynamics method based on a generalized Langevin equation and quantum heat baths. We present some results of the applications. Finally we present some recent results on the electron transport in EuTiO3 perovskite.

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An efficient numerical method for large scale simulation of two-phase flows with moving contact lines
Xiaoping Wang, Hong Kong University of Science and Technology, Hong Kong

Moving contact line (MCL) problem, where the fluid-fluid interface intersects the solid wall, can be described by a phase field model consists of the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition. We designed an efficient numerical method for the model based on convex splitting and projection method. Accurate simulations for the MCL problem require very fine meshes, and the computation in 3d is challenging because of the complex computational domain and the boundary condition. Thus the use of high performance computers and scalable parallel algorithms becomes indispensable. In our algorithm, a geometrical additive Schwarz preconditioned GMRES iterative method is used to solve the disceretized systems in parallel. Numerical experiments show that the algorithm is efficient and is scalable on a large number of processors.

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Fast high-order methods for wave scattering and transformation electromagnetics
Li-Lian Wang, Nanyang Technological University

In this talk, we present fast and accurate spectral/spectral-element methods for acoustic and electromagnetic wave scattering problems in both frequency and time domains. We advocate the use of Dirichlet-to-Neumann (DtN) transparent boundary conditions to reduce the unbounded problem to an equivalent boundary value problem. We provide semi-analytic techniques to seamless integrate global DtN boundary conditions with local spectral elements in both 2D and 3D by using special elemental transformations of curved elements. We highlight the applications of the solvers in transformation electromagnetics, e.g., the invisibility cloaks. In particular, we introduce new cloaking boundary conditions along the interface of the inner cloaking layer, and show that the perfect cloaking effects can be achieved. We also demonstrate the robustness of the solver to large wave number

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Solving linear systems and eigenvalue problems on GPU clusters
Weichung Wang, National Taiwan University, Taiwan

Large sparse linear systems and eigenvalue problems are kernels of computational sciences and engineering. How can the latest many-core accelerators such as GPU benefit these matrix computations? We address this question by presenting some of our recent results. For the linear system solvers, we fit the multifrontal method to CPU-GPU heterogeneous system with performance model based workload distribution. We then describe our hierarchical Schur complement method that is designed for parallel systems with multiple-GPU. As an application example, we present an efficient algorithm and implementation of three dimensional finite-difference frequency-domain simulation of photonic devices. For eigenvalue solvers, we focus on a challenging eigenvalue problem arising in the simulation of three dimensional photonic crystals. We demonstrate how the eigenvalue problem can be solved in a ultrafast manner by the proposed FFT-based null space free algorithm on a multiple-GPU cluster.

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Optimization in electronic structure calculation
Zaiwen Wen, Peking University, China

Minimizing the Kohn-Sham total energy functional with respect to electron wave functions, or orthogonality constraints, is a fundamental nonlinear eigenvalue problem in electronic structure calculation. These problems are very challenging due to non-convex orthogonality constraints. This talk presents a few recent advance on analyzing the convergence of the widely used self-consistent field iteration methods. A few efficient optimization algorithms will also be introduced.

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Dual and hybrid hierarchical grids for fast stokes solvers
Barbara Wohlmuth, Technische Universität München, Germany

Fast iterative solution methods for Stokes-type systems are a crucial ingredient for coupled geophysical solvers, e.g., in mantle-convection- or ice-sheet modeling. For this reason, it is of importance to co-design discretization concepts and solvers for this type of problem, that both satisfy desirable physical, mathematical and computational properties, and potentially scale up to current peta-scale architectures. We emphasize the importance of the interplay between physical conservation properties and suitable discretization concepts. Low order standard finite element approaches are very attractive from the computational point of view due to the sparsity and as a consequence to the relative small communication cost. However they fail to guarantee strong mass conservation and thus may result in poor accuracy. We propose a computationally inexpensive local recovery step and illustrate the effect on coupled multi-physics problems. Additionally we adress new challenges in peta-scale computation such as fault tolerence and scheduling for uncertainty.

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A regularized newton method for computing ground states of Bose-Einstein condensates
Xinming Wu, Fudan University, China

The Gross-Pitaevskii equation (GPE) for describing the Bose-Einstein condensation (BEC) at zero or very low temperature can be formulated as an energy minimization problem with a spherical constraint. In this paper, we propose a regularized Newton method for solving models obtained from finite difference, sine or Fourier pseudospectral discretization. First, an initial solution is constructed by using a feasible gradient type method, which is an explicit scheme and maintains the spherical constraint automatically. To accelerate the convergence of the gradient type method, we approximate the energy functional by its second-order Taylor expansion with a regularized term at each Newton iteration. It leads to a standard trust-region subproblem and we solve it again by the feasible gradient type method. The convergence of the regularized Newton method is established by adjusting the regularization parameter as the standard trust-region strategy. Extensive numerical experiments on difficult 2D and 3D examples, including rotating BEC with large parameters, show that our method is efficient and robust.

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Dislocation climb in discrete dislocation dynamics
Yang Xiang, Hong Kong University of Science and Technology, Hong Kong

We derive a Green's function formulation for the climb of curved dislocations and multiple dislocations in three-dimensions. In this new dislocation climb formulation, the dislocation climb velocity is determined from the Peach-Koehler force on dislocations through vacancy diffusion in a non-local manner. The long-range contribution to the dislocation climb velocity is associated with vacancy diffusion rather than from the climb component of the well-known, long-range elastic effects captured in the Peach-Koehler. Both long-range effects are important in determining the climb velocity of dislocations. Analytical and numerical examples show that the widely used local climb formula, based on straight infinite dislocations, is not generally applicable, except for a small set of special cases. We also present a numerical discretization method of this Green's function formulation appropriate for implementation in discrete dislocation dynamics (DDD) simulations. In DDD implementations, the long-range Peach-Koehler force is calculated as is commonly done, then a linear system is solved for the climb velocity using these forces.

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An augmented singular transform and its partial Newton method for finding new solutions
Ziqing Xie, Hunan Normal University, China

Using the information provided by previously found solutions, a new augmented singular transform is developed to change the local basin-barrier structure of the original problem for finding new solutions in an infinite-dimensional space. Then an augmented partial Newton method is proposed to solve the new problem on the solution set. Mathematical justification of the new formulation and method is established. Details on algorithm implementation are presented. Numerical results for two very different variational PDE problems are obtained and displayed with their profile-contour plots and data on convergence and errors.

Keywords: Multiple solutions, Barriers, Augmented Singular Transform, Augmented Partial Newton Method.

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Numerical methods for abnormal diffusion equations
Chuanju Xu, Xiamen University, China

In this talk, we present some efficient numerical methods for fractional differential equations modelling abnormal diffusion. The main ingredients include:
1) High order methods for time space abnormal diffusion equations;
2) Spectral methods for space fractional partial differential equations.
We will discuss the existence and uniqueness of the weak solution of a kind of boundary value problems, and their spectral approximations based on the weak formulations. Particularly, We will talk about fast spectral methods for high dimensional fractional diffusion equations.

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On two numerical models of nonlinear water wave equations
Liwei Xu, Chongqing University, China

In this talk, we first introduce two numerical models, the Craig-Sulem model and the Green-Naghdi model, describing the propagation of nonlinear water waves. Then we discuss the design of corresponding numerical methods, spectral methods and discontinuous Galerkin methods, to solve these equations. Numerical solutions will be presented to show the efficiency and accuracy of both numerical models and methods. This is a joint work with Mr. Yongping Cheng at CQU, Prof. Philippe Guyenne at UD, Prof. Fengyan Li at RPI, Dr. Maojun Li at CQU.

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DNS of colloidal dispersions using the smoothed profile method: formulation and applications
Ryoichi Yamamoto, Kyoto University, Japan

We developed a unique numerical method for direct numerical simulations (DNS) of dense dispersions of spherical and arbitanary shaped rigid bodies, called the smoothed profile (SP) method [1,2]. Recentry, the SP method is extended to study the rheological behavior of colloidal dispersions so that it can be used with the Lees-Edwards boundary condition, under DC or AC shear flow. By this reformulation, all the resultant physical quantities including local and total shear stresses become available through direct calculation. Three rheological simulations are then performed for (A) a spherical particle, (B) a rod of beads under flow, and (C) collision of two spherical particles. Quantitative validity of these simulation is confirmed by comparing the viscosity with that obtained from theory and Stokesian Dynamics calculation.

[1] Y. Nakayama and RY, Phys. Rev. E, 71, 036707 (2005); Y. Nakayama, K. Kim, and RY, Eur. Phys. J. E, 26, 361-368 (2008); J. J. Molina and RY, J. Chem. Phys., 139, 234105 (2013).
[2] KAPSEL website,
http://www-tph.cheme.kyoto-u.ac.jp/kapsel/

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High performance multigrid software
Ulrike Meier Yang, Lawrence Livermore National Laboratory, USA

Computational science is facing several major challenges with future architectures: non-increasing clock speeds are being offset with added concurrency (more cores) and limited power resources are leading to reduced memory per core, complex heterogeneous architectures, and higher levels of hardware failures (faults). To meet these challenges and yield fast and efficient performance, solvers need to exhibit extreme levels of parallelism, minimize data movement, and demonstrate resilience to faults, properties exhibited by well-designed multigrid methods. At the Center for applied Scientific Computing at Lawrence Livermore National Laboratory, we are developing two high performance multigrid software packages: hypre, a well-established linear solvers library, and XBraid, an implementation of a parallel multigrid-in-time method.

In this talk, we will summarize recent efforts to improve hypre's performance for exascale computers, such as efforts to reduce communication, improve numerical and computational scalability and increase structure. We will then describe Xbraid, which uses an approach based on multigrid reduction methods and is designed to be as non-intrusive as possible, while maintaining a high degree of versatility. We will present numerical results, obtained by applying Xbraid to a compressible computational fluid application.

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Fast numerical methods for electronic structure calculations
Chao Yang, Lawrence Berkeley National Laboratory, USA

The Kohn-Sham density functional theory (KSDFT) is the most widely used theory for studying electronic properties of molecules and solids. It reduces the need to solve a many-body Schrodinger's equation to the task of solving a system of single-particle equations coupled through the electron density. These equations can be viewed as a nonlinear eigenvalue problem. Although they contain far fewer degrees of freedom, Kohn-Sham equations are more difficult in terms of their mathematical structures. A significant amount of computational resources are often required to perform a Kohn-Sham DFT calculation for complex materials including materials with defects. In this talk, I will give an overview on efficient algorithms for solving this type of problem. I will describe discretization techniques that can potentially reduce the number of degrees of freedom required to represent the Kohn-Sham Hamiltonian while keeping it sparse. I will also present recently developed algebraic techniques for reducing the computational complexity of the Kohn-Sham DFT calculation. A key concept that is important for understanding these algorithms is a nonlinear map known as the Kohn Sham map. The ground state electron density is a fixed point of this map. I will examine properties of this map and its Jacobian. These properties can be used to develop effective strategies for accelerating Broyden's method for finding the optimal solution. They can also be used to reduce the computational complexity associated with the evaluation of the Kohn Sham map, which is the most expensive step in a Broyden iteration.

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The challenges of multiscale flow simulations
Stéphane Zaleski, UPMC University of Paris 6, France

In recent years the improvements of numerical methodology have made ever larger simulations possible. However even with this tremendous increase in computational power some important problems remain difficult to investigate. High speed jet atomization, two phase flows in ducts, droplet splashing, cavitating flows involve large numbers of complex objects and widely varying length and time scales. The talk will descirbe the various strategies that have been used and could be used in the future to deal with this situation. These involve adaptive mesh refinement, subscale physics modelling, coupling reduced models for boundary layers and particles with the direct simulation of the flow.

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The toolbox PHG and its recent applications
Linbo Zhang, Chinese Academy of Sciences, China

PHG (Parallel Hierarchical Grid, http://lsec.cc.ac.cn/phg/index_en.htm) is an open source toolbox for writing parallel adaptive finite element programs. The key features of PHG include:
(1) Newest-vertex bisection based parallel mesh adaptation algorithm for conforming tetrahedral meshes;
(2) Two-level parallelsm, namely MPI based mesh partitioning and OpenMP based multi-threading, scalable to tens of thousands of submeshes and handreds of thousands of CPU cores;
In this talk I will first give an introduction to the infrastructure and core algorithms of PHG. Then I will present some recent applications of PHG, including some benchmark results obtained on the current World's No. 1 supercomputer, the Tianhe-2.

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Modeling and simulation of moving contact line problem for two-phase complex fluids flow
Zhen Zhang, National University of Singapore

We introduce the sharp interface models for moving contact lines with insoluble surfactants and polymeric fluids. A continuous model is derived for the dynamics of two immiscible fluids with moving contact lines and insoluble surfactants based on thermodynamic principles. The continuum model consists of the Navier-Stokes equations for the dynamics of the two fluids and a convection-diffusion equation for the evolution of the surfactant on the fluid interface. The interface condition, the boundary condition for the slip velocity, and the condition for the dynamic contact angle are derived from the consideration of energy dissipations. Different types of energy dissipations, including the viscous dissipation, the dissipations on the solid wall and at the contact line, as well as the dissipation due to the diffusion of surfactant, are identified from the analysis. The model is extended to moving contact lines for polymeric fluids. Front tracking based numerical methods are developed for the continuum models. Numerical experiments show that the addition of surfactant enhances the contact line movement while the addition of polymer retard the contact line movement.

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The role of substrate impurity on catalytic activity of supported gold catalysts: an ab initio computational study
Chun Zhang, National University of Singapore

Density functional theory (DFT) based computational modelling has proven to be an effective tool for discovering new phenomena and designing novel functional materials. In this talk, I will use an example to demonstrate how computational modelling can be used in investigating catalytic properties of complex materials, and how these studies lead us to a new class of catalysts. We will show that after impurity atoms are doped, a so-called metal-insulator transition (MIT) is caused in SrTiO3 (STO), turning the material from insulator to metal. The doping induced MIT then greatly enhances the reactivity of the STO surface, which in turn improve the catalytic activity of STO supported gold clusters by several orders. The interesting mechanism of CO oxidation catalysed by STO supported gold clusters are revealed by detailed DFT based computational studies. Implications of these studies for future design of catalysts are also discussed.

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Accurate and efficient computation of some nonlocal potentials based on Gaussian-sum approximation
Yong Zhang, Wolfgang Pauli Institute, Austria

We introduce an accurate and efficient method for the evaluation of a class of free space nonlocal potentials, such as the free space Coulomb and Poisson potential in 2D/3D and dipolar potential in 2D/3D. Starting from the convolution formulation, for smooth and fast decaying densities, we make a full use of the Fourier pesudospectral (plane wave) approximation of the density and a separable Gaussian sums approximation of the kernel in an interval where the singularity point (the origin) is excluded. Hence, the potential is split into a regular integral and a correction integral, and the latter is well resolved utilising a low-order Taylor expansion of the density. Both can be computed with fast Fourier transforms (FFT). The method is accurate (14-16 digits), efficient (O(N log N) complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelable.

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Design and implementation of fast explicit integration factor method for phase field simulations
Jian Zhang, Chinese Academy of Sciences, China

We devised fast Explicit Integration Factor(fEIF) method for phase-field simulations. It combines efficient decompositions of compact spatial difference operators with stable and accurate exponential time integrators and can deal with stiff nonlinearity in the phase field equations by use of multistep approximations and analytic evaluations of time integrals. In addition, stable high order splitting schemes are designed for typical phase field equations. The fEIF method is implemented on a heterogeneous computing node consists of two CPUs and two MICs. We are able to achieve over 1,200 Gflops performance (double precision, 45% peak) in a phase separation simulation.

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Weak Galerkin finite element scheme and its applications
Ran Zhang, Jilin University, China

The weak Galerkin finite element method (WG) is a newly developed and efficient numerical technique for solving partial differential equations (PDEs). It was first introduced and analyzed for second order elliptic equations. The central idea of WG is to interpret partial differential operators as generalized distributions, called weak differential operators, over the space of discontinuous functions including boundary information. The weak differential operators are further discretized and applied to the corresponding variational formulations of the underlying PDEs. This talk introduces the basic principle and the theoretical foundation for the WG method by using the second order elliptic equation. The WG method is further applied to several other model equations, such as the biharmonic, Stokes equations to demonstrate its power and efficiency as an emerging new numerical method.

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Theory based construction of atomistic/continuum coupling methods
Lei Zhang, Shanghai Jiao Tong University, China

We discuss the construction of quasi-optimal energy based atomistic/continuum (A/C) coupling methods for crystalline solids with defects, based on tools from numerical analysis. For general multi-body interactions on the 2D triangular lattice (and potentially for 3D lattices), we show that ghost force removal (patch test consistent) A/C methods can be constructed for arbitrary interface geometries. We prove that all methods within this class are first-order consistent at the atomistic/continuum interface. The rigorous error analysis of the coupling methods leads to their optimal implementation. For a range of benchmark problems, the resulting consistent coupling methods have the same convergence rates as optimal force-based coupling schemes. This is a joint work with Christoph Ortner (Warwick).

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Defects of liquid crystals
Pingwen Zhang, Peking University, China

Defects in liquid crystals are of great practical importance and theoretical interest. Despite tremendous efforts, predicting the location and transition of defects under various topological constraint and external field remains to be a challenge. We investigate defect patterns of nematic liquid crystals confined in three-dimensional spherical droplet and two-dimensional disk under different boundary conditions, within the Landau-de Gennes model. A spectral method that numerically solves the Landau-de Gennes model with high accuracy is implemented, which allows us to study the detailed static structure of defects. We observe five types of defect structures. Among them the 1/2 disclination lines are the most stable structure at low temperature. Inspired by numerical results, we obtain the profile of disclination lines analytically. Moreover, the connection and difference between defect patterns under the Landau-de Gennes model and the Oseen-Frank model is discussed. Finally, four conjectures are made to summarize the common characteristics of defects in the Landau-de Gennes theory, in the hope of providing a deeper understanding of the defect pattern in nematic liquid crystals.

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A tutorial on PHG
Linbo Zhang, Chinese Academy of Sciences, China

This is a tutorial on the parallel adaptive finite element toolbox Parallel Hierarchical Grid (PHG, http://lsec.cc.ac.cn/phg/index_en.htm). Its purpose is to get started with using PHG to write parallel adaptive finite element programs for solving 3-d partial differential equations.

The tutorial consists of four parts.
Part I is a brief introduction to the basic concepts and algorithms of adaptive finite element methods.
Part II contains instructions on how to get, compile and install PHG.
Part III introduces the basic structure of PHG programs with a sample program. And the last part,
Part IV, is a systematic presentation of PHG's application programming interface.

We hope that the attendees of the tutorial will get acquainted with the basic usage of PHG, and can write their own parallel adaptive finite element programs with PHG.

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A mathematical theory of super-resolution by using Helmholtz resonators
Hai Zhang, Ecole Normale Superieure, France

We frist briefly review existing super-resolution techniques in practise. We then introduce the time reversal experiment by using Helmholtz resonators [2] and propose a mathematical theory for the super-focusing observed. The super-focusing shall lead to super-resolution when imaging in the resonant media. The key ingredient of our analysis is a novel method for calculating the resonance frequencies for Helmholtz resonators and the asymtotics for the Green's function. This is a joint work with Habib Ammari.

References
[1] H. Ammari and H. Zhang. A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators, Comm. Math. Physics, accepted.

[2] F. Lemoult, M. Fink and G. Lerosey. Acoustic resonators for far-field control of sound on a subwavelength scale, Phys. Rev. Lett., 107 (2011), 064301.

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Solving highly-oscillatory NLS with SAM: numerical efficiency and geometric properties
Yong Zhang, Wolfgang Pauli Institute, Austria

In this paper, we present the Stroboscopic Averaging Method (SAM), which aims at numerically solving highly-oscillatory differential equations. More specifically, we first apply SAM to the Schrodinger equation on the $1$-dimensional torus and on the real line with harmonic potential, with the aim of assessing its efficiency: as compared to the well-established standard splitting schemes, the stiffer the problem is, the larger the speed-up grows (up to a factor $100$ in our tests). The geometric properties of SAM are also explored: on very long time intervals, symmetric implementations of the method show a very good preservation of the mass invariant and of the energy. In a second series of experiments on $2$-dimensional equations, we demonstrate the ability of SAM to capture qualitatively the long-time evolution of the solution (without spurring high oscillations).

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Multiscale methods and analysis for highly oscillatory differential equations
Xiaofei Zhao , National University of Singapore

The oscillatory phenomena happen almost everywhere in our life, ranging from macroscopic to microscopic level. They are usually described and governed by some highly oscillatory nonlinear differential equations from either classical mechanics or quantum mechanics. Effective and accurate approximations to the highly oscillatory equations become the key way of further studies of the nonlinear phenomena with oscillations in different scientific research fields. In this thesis, we propose and analyze some efficient numerical methods for approximating a class of highly oscillatory differential equations arising from quantum or plasma physics. The methods here include classical numerical discretizations and the multiscale methods with numerical implementations. Special attentions are paid to study the error bound of each numerical method in the highly oscillatory regime, which are geared to understand how the step size should be chosen in order to resolve the oscillations, and eventually to find out the uniformly accurate methods that could totally ignore the oscillations when approximating the equations.

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Adaptive finite element approximations for density functional models
Aihui Zhou, Chinese Academy of Sciences, China

In this presentation, we will talk about adaptive finite element approximations of orbital-free and Kohn-Sham density functional models. We will demonstrate several typical numerical experiments that show the robustness and efficiency of the adaptive finite element computations in electronic structure calculations for materials defects. This presentation is based on some joint works with H. Chen, X. Dai, X. Gong, L. He,and J. Zhu.

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