A low-rank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis

TL;DR

This paper introduces a novel low-rank tensor method combined with hierarchical Bayesian modeling to efficiently reconstruct sparse initial states in PDEs, specifically for identifying heat sources in 3D domains.

Contribution

It develops a new approach integrating low-rank tensor computations with hyperpriors to improve sparse PDE state reconstruction accuracy and efficiency.

Findings

Effective reconstruction of sparse heat sources in 3D PDEs

Enhanced computational efficiency through tensor train formats

Successful incorporation of hyperpriors for sparsity promotion

Abstract

When working with PDEs the reconstruction of a previous state often proves difficult. Good prior knowledge and fast computational methods are crucial to build a working reconstruction. We want to identify the heat sources on a three dimensional domain from later measurements under the assumption of small, distinct sources, such as hot chippings from a milling tool. This leads us to the need for a Prior reflecting this a priori information. Sparsity-inducing hyperpriors have proven useful for similar problems with sparse signal or image reconstruction. We combine the method of using a hierarchical Bayesian model with gamma hyperpriors to promote sparsity with low-rank computations for PDE systems in tensor train format.