Constrained Optimization with Low-Rank Tensors and Applications to Parametric Problems with PDEs

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.