Constrained Optimization with Low-Rank Tensors and Applications to Parametric Problems with PDEs
Abstract: Low-rank tensor methods provide efficient representations and computations for high-dimensional problems and are able to break the curse of dimensionality when dealing with systems involving multiple parameters. We present algorithms for constrained nonlinear optimization problems that use low-rank tensors and apply them to optimal control of PDEs with uncertain parameters and to parametrized variational inequalities. These methods are tailored to the usage of low-rank tensor arithmetics and allow to solve huge scale optimization problems. In particular, we consider a semismooth Newton method for an optimal control problem with pointwise control constraints and an interior point algorithm for an obstacle problem, both with uncertainties in the coefficients.
Keywords: nonlinear optimization, low-rank tensors, PDEs with uncertainties, optimal control under uncertainty, parametric variational inequalities, uncertainty quantification, semismooth Newton methods, interior point methods
Category 1: Nonlinear Optimization (Systems governed by Differential Equations Optimization )
Category 2: Applications -- Science and Engineering (Optimization of Systems modeled by PDEs )
Category 3: Stochastic Programming
Citation: S. Garreis and M. Ulbrich: Constrained Optimization with Low-Rank Tensors and Applications to Parametric Problems with PDEs, Preprint (submitted), Department of Mathematics, Technical University of Munich, 2016.
Entry Submitted: 01/25/2016
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