UC Santa Cruz

The numerical simulation of high-dimensional partial differential equations (PDEs) is a challenging and important problem in science and engineering. Classical methods based on tensor product representations are not viable in high-dimensions, as the number of degrees of freedom grows exponentially fast with the problem dimension. In this dissertation we present low-rank tensor methods for approximating high-dimensional PDEs, which have a number of degrees of freedom and computational cost that grow linearly with the problem dimension. These methods are based on projecting a given PDE onto a low-rank tensor manifold and then constructing an approximate PDE solution as a path on the manifold. In order to control the accuracy of the low-rank tensor approximation we present a rank-adaptive algorithm that can add or remove tensor modes adaptively from the PDE solution during time integration. We also present a tensor rank reduction method based on coordinate transformations that can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is to determine a coordinate transformation of a given functions domain so that the function in the new coordinate system has smaller tensor rank. We demonstrate each of the presented low-rank tensor methods by providing several numerical applications to multivariate functions and PDEs.