Multilevel preconditioning and low-rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of instationary parabolic evolution equations simultaneously in space and time. As a model problem we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov-Galerkin variational formulation of the heat equation in space-time results in a generalized least squares problem. This formulation admits a unique, quasi-optimal solution in the natural space-time Hilbert space and serves as a basis for the development of space-time compressive algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from the well-known BPX preconditioner, multilevel space-time preconditioners for the normal equations are derived. The resulting “parabolic BPX preconditioners” render the normal equations well-conditioned uniformly in the discretization level. In order to reduce the complexity of the full space-time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format supposing that the input data is available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for Matlab are presented.