Numerical Analysis Seminar

Wednesday, March 26, 2008

Abstract

The accurate representation everywhere of complex physical processes requires the high fidelity solution of systems of partial differential equations (PDEs). In many engineering applications, however, not the entire PDE solution is of interest, but one only needs an accurate description of input-to-output maps. Examples are flow control, where one is interested, e.g., in actuation of the flow on a small section of the wing to improve the lift, and reservoir optimization, where one is interested, e.g., in injection and production rates at wells to maximize net present value. In such cases model reduction is of interest.

Model reduction seeks to systematically extract the important features from a high fidelity model to accurately represent the input-to-output maps of interest. This talk begins with an overview of projection-based model reduction techniques, including proper orthogonal decomposition (POD) and balanced truncation model reduction. We discuss the ideas behind these methods, some theoretical results, computational aspects, and their application to some example problems derived from flow control.

We then discuss the integration of balanced truncation model reduction and domain decomposition for dynamical systems of PDEs with spatially localized nonlinearities. The problem is decomposed into subproblems such that the nonlinear part is concentrated in one subdomain. Balanced truncation model reduction is applied to the linear subdomain problems. Here the interface conditions lead to auxiliary inputs and outputs in addition to the original inputs and outputs. The reduced subdomain problems are combined with the nonlinear subdomain problem to form the reduced coupled problem. Estimates for the error of the input-to-output map of the original problem and that of the reduced problem are given and numerical results are presented that illustrate the applicability of the approach.