Tensor train solution to uncertain optimization problems with shared sparsity penalty

TL;DR

This paper introduces tensor train-based numerical methods for solving high-dimensional uncertain optimization problems with shared sparsity penalties, demonstrating efficient convergence and reduced tensor ranks in complex engineering applications.

Contribution

It develops first and second order optimization algorithms using tensor approximations for non-smooth, uncertain PDE-constrained problems with shared sparsity, improving computational efficiency.

Findings

Error converges linearly with iterations and smoothing parameter

Tensor ranks decrease in topology optimization with shared sparsity penalty

Achieves sparse high-resolution designs under high-dimensional uncertainty

Abstract

We develop both first and second order numerical optimization methods to solve non-smooth optimization problems featuring a shared sparsity penalty, constrained by differential equations with uncertainty. To alleviate the curse of dimensionality we use tensor product approximations. To handle the non-smoothness of the objective function we employ a smoothed version of the shared sparsity objective. We consider both a benchmark elliptic PDE constraint, and a more realistic topology optimization problem in engineering. We demonstrate that the error converges linearly in iterations and the smoothing parameter, and faster than algebraically in the number of degrees of freedom, consisting of the number of quadrature points in one variable and tensor ranks. Moreover, in the topology optimization problem, the smoothed shared sparsity penalty actually reduces the tensor ranks compared to the unpenalised solution. This enables us to find a sparse high-resolution design under a high-dimensional uncertainty.