[Paper Review] Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in $L_2$

Markus Bachmayr, Wolfgang Dahmen|arXiv (Cornell University)|Dec 12, 2014

Tensor decomposition and applications1 references3 citations

TL;DR

This paper introduces an adaptive low-rank tensor method for high-dimensional elliptic PDEs that ensures rigorous error control in the $L_2$-norm rather than the energy (H¹) norm. By employing an asymmetric preconditioning scheme, the method achieves significant computational savings and scalability—enabling problems up to dimension $d=256$—while maintaining reliable $L_2$ error bounds, albeit at the cost of losing $H^1$-norm control.

ABSTRACT

Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such approximations, not with respect to a fixed finite dimensional discrete background problem, but with respect to the exact solution of the continuous problem. While the energy norm offers a natural error measure corresponding to the underlying operator considered as an isomorphism from the energy space onto its dual, this norm requires a careful treatment in its interplay with the tensor structure of the problem. In this paper we build on our previous work on energy norm-convergent subspace-based tensor schemes contriving, however, a modified formulation which now enforces convergence only in $L_2$. In order to still be able to exploit the mapping properties of elliptic operators, a crucial ingredient of our approach is the development and analysis of a suitable asymmetric preconditioning scheme. We provide estimates for the computational complexity of the resulting method in terms of the solution error and study the practical performance of the scheme in numerical experiments. In both regards, we find that controlling solution errors in this weaker norm leads to substantial simplifications and to a reduction of the actual numerical work required for a certain error tolerance.

Motivation & Objective

• To develop an adaptive low-rank tensor method for high-dimensional elliptic PDEs with rigorous error control in the $L_2$-norm, rather than the standard energy (H¹) norm.
• To address the challenge of balancing subspace approximation and adaptive refinement of tensor factor representations in a computationally efficient manner.
• To simplify the computational framework by relaxing the need for symmetric preconditioning while maintaining convergence and error control.

Proposed method

• The method reformulates the elliptic problem using an asymmetric preconditioning scheme to enable convergence in the $L_2$-norm, leveraging the mapping properties of the elliptic operator.
• It employs a residual-based a posteriori error estimator that reflects the accuracy of the approximate solution in the $L_2$-norm.
• Adaptive refinement is driven by error estimates derived from the residual in the dual space, ensuring convergence to the exact solution in $L_2$.
• The approach uses subspace-based tensor formats with solution-dependent basis functions, avoiding fixed discretizations.
• A computational complexity analysis is provided, relating the number of operations to the desired $L_2$-error tolerance.
• Numerical experiments validate the method on Poisson problems and tridiagonal diffusion matrices, demonstrating scalability up to $d=256$.

Experimental results

Research questions

• RQ1 Can adaptive low-rank tensor methods achieve reliable error control in the $L_2$-norm for high-dimensional elliptic PDEs without relying on the energy norm?
• RQ2 How does asymmetric preconditioning affect the convergence and computational efficiency of low-rank tensor schemes?
• RQ3 What is the trade-off between $L_2$-norm error control and loss of $H^1$-norm control in terms of computational complexity and scalability?
• RQ4 How does the method perform on problems with varying diffusion matrix structures, such as diagonal versus tridiagonal matrices?
• RQ5 To what extent can the computational cost be reduced by shifting from $H^1$-norm to $L_2$-norm error control?

Key findings

• The method achieves reliable $L_2$-error bounds for high-dimensional problems, with numerical experiments confirming convergence up to dimension $d=256$, a significant improvement over prior $H^1$-norm methods.
• For the Poisson problem on $(0,1)^d$, the new scheme allows $d=256$ with similar operation counts and error bounds as previous $d=64$ results in $H^1$-norm control.
• The use of asymmetric preconditioning simplifies the computational framework and reduces the need for symmetric structure, leading to substantial practical gains.
• For tridiagonal diffusion matrices with $a=1/2$, the method maintains good convergence and scalability up to $d=64$, while performance degrades for $a=1$ due to increasing condition numbers.
• The $L_2$-error control is verified numerically: the computed error bounds match the actual $L_2$-errors to reference solutions, while $H^1$-errors are no longer bounded by the theory.