Past Event: Oden Institute Seminar

Low-rank tensor methods for PDE-constrained optimization under uncertainty

Peter Benner, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

Abstract

Work with Sergey Dolgov, University of Bath, UK, Akwum Onwunta and Martin Stoll, Faculty of Mathematics, TU Chemnitz, Germany We discuss optimization and control of unsteady partial differential equations (PDEs), where some coefficient of the PDE as well as the control may be uncertain. This may be due to the lack of knowledge about the exact physical parameters like material properties describing a real-world problem ("epistemic uncertainty") or the inability to apply a computed optimal control exactly in practice. Using a stochastic Galerkin space-time discretization of the optimality system resulting from such PDE-constrained optimization problems under uncertainty leads to large-scale linear or nonlinear systems of equations in saddle point form. Nonlinearity is treated witha Picard-type iteration in which linear saddle point systems have to be solved in each iteration step. Using data compression based on separation of variables and the tensor train (TT) format, we show how these large-scale indefinite and (non)symmetric systems that typically have $10^8$ to $10^{15}$ unknowns can be solved without the use of HPC technology. The key observation is that the unknown and the data can be well approximated in a new block TT format that reduces complexity by several orders of magnitude. As examples, we consider control and optimization problems for the linear heat equation, the unsteady Stokes and Stokes-Brinkman equations, as well as the incompressible unsteady Navier-Stokes equations. The talk reviews the results published in {BenOS16,BenDOS16} and provides new results for the Navier-Stokes case. Bibliography {BenOS16} P. Benner, A. Onwunta, M. Stoll, Block-diagonal preconditioning for optimal control problems constrained by {PDE}s with uncertain inputs, SIAM Journal on Matrix Analysis and Application, 37(2):491--518, 2016. {BenDOS16} P. Benner, S. Dolgov, A. Onwunta, M. Stoll, Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data, Computer Methods in Applied Mechanics and Engineering, 304:26--54, 2016. Bio Peter Benner received the Diplom in mathematics from RWTH Aachen, Germany, in 1993. From 1993 to 1997, he worked on his Ph.D. at the University of Kansas, Lawrence, KS USA, and the TU Chemnitz-Zwickau, Germany, where he received the Ph.D. degree in February 1997. In 2001, he received the Habilitation in Mathematics from the University of Bremen, Germany, where he held an Assistant Professor position from 1997 to 2001. After spending a term as a Visiting Associate Professor at TU Hamburg-Harburg, Germany, he was a Lecturer in mathematics at TU Berlin 2001–2003. Since 2003, he has been a Professor for "Mathematics in Industry and Technology" at TU Chemnitz. In 2010, he was appointed as one of the four directors of the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany. Since 2011, he has also been an Honorary Professor at the Otto-von-Guericke University of Magdeburg. Benner's research interests are in the areas of scientific computing, numerical mathematics, systems theory, and optimal control. A particular emphasis has been on applying methods from numerical linear algebra and matrix theory in systems and control theory. Recent research focuses on numerical methods for optimal control of systems modeled by evolution equations (PDEs, DAEs, SPDEs), model order reduction, preconditioning in optimal control and UQ problems, and Krylov subspace methods for structured or quadratic eigenproblems. Research in all these areas is accompanied by the development of algorithms and mathematical software suitable for modern and high-performance computer architectures.