This equipment will be used to support research and graduate and undergraduate education in the mathematical sciences. The Department of Mathematics at Carnegie Mellon has a strong program in numerical analysis which attracts some of the most talented graduate students in our Ph.D. program. This equipment will be used by the principal investigators, visitors, postdoctoral fellows, and their graduate students to enhance ongoing research programs in computational material science and multilevel computational techniques. The CMU program in material science has application in computing the properties of materials that exhibit complex fine scale phenomena. These problems appear in the study of metallurgy, magnetism, optimal design, etc. where fine scale (microscopic) variations are frequently observed. Recent mathematical advances suggest that these oscillations are governed by energy considerations, and can often be characterized by a macroscopic (slowly varying) quantity, the Young measure. Such characterizations provide a basis for the efficient numerical modeling and approximation of these materials and their properties. The development, analysis, and testing of algorithms for the solution of such problems is considered in this proposal. The research program on efficient algorithms - especially multilevel algorithms- is aimed at making optimal design calculations in acoustics, electromagnetism, shape optimization, and nondestructive testing computationally feasible. Efficient solution of such problems requires multiple solutions for the underlying problem in order to isolate an optimal solution: the problems we consider would be impractical without powerful computation algorithms of the kind under investigation. Multilevel computational techniques are very effective solution methods in a variety of different fields of science and engineering. The authors have been involved in developing these methods for a variety of problems with great success. The proposed research deals with new developments of these techniques for boundary related optimization problems governed by non-elliptic systems, and for the efficient discretization and multilevel time marching techniques for hyperbolic evolution problems. Applications of these techniques include problems arising in electromag netism, design of microwave devices, acoustic noise reduction, non-destructive testing, and the control of fluid flows.
