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. Motivated by optimal control problems with PDEs under uncertainty, we present algorithms for constrained nonlinear optimization problems that use low-rank tensors. They are applied to optimal control of PDEs with uncertain parameters and to parametrized variational inequalities of obstacle type. These methods are tailored to the usage of low-rank tensor arithmetics and allow us 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.