Multilevel methods in PDE constrained optimization

University Of Maryland Baltimore County, Baltimore MD

Investigators

Abstract

The objective of this project is to develop efficient multilevel algorithms for large-scale optimization problems constrained by partial differential equations (PDEs) with additional inequality constraints (ICs) on the controls and states. The computational revolution of the last two decades has fostered not only high-resolution numerical computations based on PDE models, but also a shift from model based simulation to model based design. The latter translates into the question of solving optimization problems in order to identify initial and/or boundary values, material properties, sources, and other parameters for which the PDE models behave in a desired way. However, in general, by increasing resolution not only do optimization problems get larger, but they also become more difficult to solve, thus rendering an ever widening gap between the resolution of PDEs and that of associated parameter identification problems that can be solved using state of the art resources; in order to take full advantage of these resources, highly efficient algorithms are critical. While such efficient algorithms have been developed over the past few years, they are mostly restricted to problems without ICs. The addition of ICs on the controls and/or states normally increases the difficulty of the problem due to the presence of Lagrange multipliers that have lower regularity than the solution. Recent years have witnessed a sensible progress in the optimization algorithms that target such problems, however, it is expected that significant efficiency can further be gained by improvements in the linear algebra technology needed during the optimization process. In this project the PI specifically aims to develop optimal order multilevel preconditioners for the linear systems arising in the interior point method and semismooth Newton method solution processes of optimization problems constrained by linear and semilinear elliptic or parabolic PDEs with ICs on the controls and/or states. For the more difficult problem of state ICs, both Lavrentiev and Moreau-Yosida regularizations will be considered. The long term goal is to develop efficient multilevel algorithms for large-scale control problems for fluid flows (Stokes, and Navier-Stokes systems). The results of this project are expected to enable end users of the software - engineers, applied scientists - to solve high-resolution, relevant optimization problems at a cost that is comparable (a small multiple of) to that of performing a single simulation. Long-term targeted applications include data assimilation for weather prediction and air contamination modeling. Fast data assimilation for high resolution models would enable, for example, gaining in a timely manner a better quantitative understanding of the current state of the atmosphere around a hurricane, thus potentially improving the current predictive capabilities. From an educational perspective, the successful project will help the PI's efforts in promoting this field of research at University of Maryland Baltimore County (UMBC), and it will allow graduate and undergraduate UMBC students to gain experience in a research area of strategic interest, which is likely to increase their opportunities of finding a good position in a research university or laboratory.