Multilevel Sparse Tensor Approximation for High-Dimensional Parametric PDEs
This addresses efficiency issues in computational science for high-dimensional parametric PDEs, offering a method that mitigates exponential cost growth, though it is incremental as it builds on existing tensor and multilevel techniques.
The paper tackles the computational cost of solving high-dimensional parametric PDEs by combining multilevel Galerkin discretization with sparse tensor train approximations, achieving significant cost reductions with work overhead independent of discretization level, as validated in numerical experiments.
In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the recently presented Sparse Alternating Least Squares (SALS) algorithm is employed to construct adaptive tensor train (TT) approximations of quantities of interest (QoI). By combining this tensor-based approach with a multilevel Galerkin discretization strategy, the solution's regularity can be exploited to significantly reduce computational costs by level-adapted sample sizes. A rigorous theoretical analysis is derived, demonstrating that the work overhead for the proposed multilevel method remains independent of the discretization level, which stands in stark contrast to the exponential growth observed in single-level approaches. The presented analysis is quite general and not constrained to the sparse TT format but uses a generic framework that can be extended to other model classes. Numerical experiments validate the predicted efficiency gains in high-dimensional settings.