Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains

TL;DR

This paper introduces a novel QTT-based solver for high-resolution full-field simulations of multiscale elliptic equations, achieving unprecedented data compression and stability in 2D and 3D.

Contribution

The paper develops an original, stable solver using Quantized Tensor Trains that handles extremely fine meshes and large degrees of freedom beyond classical methods.

Findings

Successfully solves a 3D elliptic problem with up to 10^37 virtual DoFs.

Achieves accurate solutions and gradients with exponential data compression.

Provides an a posteriori error estimator for solution reliability.

Abstract

In this article, we design an original solver based on Quantized Tensor Trains (QTT) for linear elliptic equations with heterogeneous coefficient field, that allows for extremely fine meshes. It can achieve full-field simulations in dimensions $d=2$ and $d=3$ with a number of Degrees of Freedom (DoFs) up to $20$ orders of magnitude beyond the classical solvers, recovering accurately the solution as well as its gradient in the $\LL^2$ norm. For treating such an enormous amount of data, the solver crucially relies on the exponential compression properties of QTTs. This significantly improves upon the existing literature. The main ingredient of the proposed solver consists in the introduction of a penalization term involving the Helmholtz--Leray projector in the equation governing the gradient unknown. For practical reasons related to the expression of the Helmholtz--Leray projector, the penalized equation is solved in Fourier space. The primal solution is then obtained from the gradient via the Green operator. A core property of the solver is that it is unconditionally stable with respect to the mesh size. Based on numerical evidence supported by mathematical analysis, we show that reliable gradients and solutions can be obtained, and guaranteed by the proposed a posteriori error estimator. As an illustration, we successfully solve an elliptic equation in a microstructured material with up to $10^{37}$ virtual degrees of freedom in dimension $d=3$.