A Tensor-Train accelerated solver for integral equations in complex geometries

TL;DR

This paper introduces a QTT-based hierarchical solver for 3D integral equations that significantly reduces computational and storage costs, enabling efficient solutions in complex geometries.

Contribution

It develops a novel QTT decomposition framework for solving volume and boundary integral equations in 3D, with analysis of rank bounds and practical performance demonstrations.

Findings

QTT ranks are bounded for translation-invariant systems.

The QTT solver requires less memory than existing methods.

QTT preconditioners improve iterative solution efficiency.

Abstract

We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest $O(\log N)$ and once the inverse is computed, it can be applied in $O(N \log N)$. We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.