A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids

TL;DR

This paper introduces a tensor-train reduced basis method for efficiently solving parameterized PDEs on Cartesian grids, reducing offline costs and computational complexity while maintaining accuracy.

Contribution

The paper presents a novel tensor-train based reduced basis approach that improves efficiency and reduces computational costs for parameterized PDEs compared to existing methods.

Findings

Effective in solving benchmark PDE problems like Poisson and heat equations.

Reduces computational complexity and offline costs significantly.

Validated accuracy with a posteriori error estimates.

Abstract

In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely used for their computational efficiency compared to full-order models, they often involve significant offline computational costs. Our proposed approach mitigates this limitation by leveraging the tensor train format to efficiently represent high-dimensional finite element quantities. This method offers several advantages, including a reduced number of operations for constructing the reduced subspaces, a cost-effective hyper-reduction strategy for assembling the PDE residual and Jacobian, and a lower dimensionality of the projection subspaces for a given accuracy. We provide a posteriori error estimates to validate the accuracy of the method and evaluate its computational performance on benchmark problems, including the Poisson equation, heat equation, and transient linear elasticity in two- and three-dimensional domains. Although the current framework is restricted to problems defined on Cartesian grids, we anticipate that it can be extended to arbitrary shapes by integrating the tensor-train reduced basis method with unfitted finite element techniques.