Convergence of a greedy algorithm for high-dimensional convex nonlinear problems

TL;DR

This paper introduces a greedy tensor product decomposition algorithm for high-dimensional convex nonlinear problems, proving its convergence and demonstrating its application to uncertainty propagation in obstacle problems.

Contribution

The paper presents a novel greedy algorithm based on tensor decompositions for solving high-dimensional convex nonlinear problems, with proven convergence.

Findings

Algorithm converges under Lipschitz gradient condition.

Effective for high-dimensional nonlinear convex problems.

Demonstrated on uncertainty propagation in obstacle problems.

Abstract

In this article, we present a greedy algorithm based on a tensor product decomposition, whose aim is to compute the global minimum of a strongly convex energy functional. We prove the convergence of our method provided that the gradient of the energy is Lipschitz on bounded sets. The main interest of this method is that it can be used for high-dimensional nonlinear convex problems. We illustrate this method on a prototypical example for uncertainty propagation on the obstacle problem.