# Institute for Mathematical Sciences Programs & Activities

# ~ ABSTRACTS ~

**Control of Shock Positions with Applications to Sonic
Booms**

*Olivier Pironneau, University of Paris VI (Pierre et Marie
Curie)*

There are many fluid flow problems with discontinuities in the data or in the flow. Among them two are quite important for applications:

- transonic and supersonic flow with shocks and buffeting
- acoustics with sonic boom

Optimisation of these systems by standard gradient methods requires the application of the techniques of the Calculus of Variations and an implicit assumption that a Taylor expansion exists with respect to the degrees of freedom of the problem. Take for example the flow in a transonic nozzle and the variation of the flow with respect to the inflow conditions; when these vary the shock moves and the derivative of the flow variables with respect to inflow conditions is a Dirac measure and so the Taylor expansion does not exists.

By extending the calculus of variation via the theory of distribution it is possible to show however that the derivatives exist. But the result has serious numerical implications; in particular it favors the mixed finite element methods.

We shall give numerical illustrations using the finite element method for an inverse problem for Burger's equation, for the design of a transonic nozzle and for the design of a business supersonic airplane for sonic boom minimization.

**Post-Fisherian Experimentation**

*Jeff Wu, School of Industrial and Systems Engineering,
Georgia Institute of Technology, USA*

It has been more than 70 years since Fisherâ€™s pioneering work on experimental design. His impacts on the field of DOE continue to be felt to these days. In this talk I will review the major advances in DOE since Fisher and speculate on future development. A common theme underlying these advances was an emerging technological development that called for novel ideas and methods to address these challenges. Fisher's principles of replication, blocking and randomization arose in the context of agricultural experiments. Since the 50's attention had turned to industrial experiments with many factors. Basic principles that govern the relationships among factorial effects include: effect hierarchy, effect sparsity, and effect heredity. The minimum aberration criterion has lately become the primary criterion for selecting optimal fractional factorial designs. Another development was Box's response surface methodology, which came from chemical industries. It emphasized response model fitting and optimization instead of factorial effect estimation. The quality movement in the 80's has inspired the development of robust parameter design (Taguchi). Its central idea is to reduce variation by exploiting control-by-noise interactions. Factor asymmetry caused by the differential treatment of control and noise factors has led to new strategies in design and modeling. Looking into the future, the current trend in rapid prototyping through CAD/CAM provides a golden opportunity for research on computer modeling and experiments. Recent breakthroughs in biotechnology may call for the development of other novel techniques.

**Choice of Technique in the Robinson-Solow-Srinivasan Model**

M. Ali Khan, Johns Hopkins University

We report results on the optimal "choice of technique" in a model originally formulated by Robinson, Solow and Srinivasan (henceforth, the RSS model) and further discussed by Okishio and Stiglitz. By viewing this vintage-capital model without discounting as a specific instance of the general theory of intertemporal resource allocation associated with Brock, Gale and McKenzie, we resolve long-standing conjectures in the form of theorems on the existence and price support of optimal paths, and of conditions sufficient for the optimality of a policy first identified by Stiglitz. We dispose of the necessity of these conditions in surprisingly simple examples of economies in which (i) an optimal path is periodic, (ii) a path following Stiglitz' policy is bad, and (iii) there is optimal investment in different vintages at different times.

**Some Problems and Recent Developments in Preference and
Utility Theory**

*Ghanshyam Mehta, School of Economics, University of
Queensland, Australia*

In this talk, the speaker will give an overview from an economic point of view of some problems and recent developments in the branch of utility theory that deals with the problem of representing a preference relation by a utility function. This topic is very closely related to certain purely mathematical problems in the fields of topology and order. It also has many applications in other applied fields such as mathematical psychology, physics (thermodynamics), ordinal data analysis in statistics and so on. The talk has been designed to appeal to a broad group of economists with different interests, mathematicians, applied scientists in other related fields and also students at the advanced undergraduate and graduate levels.

**An Introduction to Parallel Computing
Message Passing Interface & Application**

*Bud Fox,
University of Western Australia*

An overview of parallel computing, the Message Passing Interface (MPI) and its applications, is provided in a three-part seminar to introduce the relative simplicity of obtaining the maximum benefit from parallel computing and in particular on networks of workstations. Part 1 explores the architectures, models and environments concerning parallel computation and may allow researchers to identify the possibility for parallelism in their own work. Part 2 provides an in-depth look at MPI centred on a practical perspective of installation, building and running code, and the usage of supporting tools for debugging and graphical display. Part 3 concerns applications for MPI and includes code excerpts, in a "show & tell" approach.