High-order Adaptive Rank Integrators for Multi-scale Linear Kinetic Transport Equations in the Hierarchical Tucker Format

TL;DR

This paper introduces an adaptive low-rank tensor method using hierarchical Tucker format for efficiently solving high-dimensional linear kinetic transport equations, improving computational efficiency and solution accuracy.

Contribution

It develops a novel adaptive rank approximation technique with high-order discretizations, addressing the curse of dimensionality in kinetic transport equations.

Findings

Significant reduction in computational cost compared to full-grid methods

High-order discretizations improve solution accuracy within the low-rank framework

Effective low-rank truncation maintains solution quality in benchmark problems

Abstract

In this paper, we present a new adaptive rank approximation technique for computing solutions to the high-dimensional linear kinetic transport equation. The approach we propose is based on a macro-micro decomposition of the kinetic model in which the angular domain is discretized with a tensor product quadrature rule under the discrete ordinates method. To address the challenges associated with the curse of dimensionality, the proposed low-rank method is cast in the framework of the hierarchical Tucker decomposition. The adaptive rank integrators we propose are built upon high-order discretizations for both time and space. In particular, this work considers implicit-explicit discretizations for time and finite-difference weighted-essentially non-oscillatory discretizations for space. The high-order singular value decomposition is used to perform low-rank truncation of the high-dimensional time-dependent distribution function. The methods are applied to several benchmark problems, where we compare the solution quality and measure compression achieved by the adaptive rank methods against their corresponding full-grid methods. We also demonstrate the benefits of high-order discretizations in the proposed low-rank framework.