IETI-based Low-Rank method for PDE-constrained optimization

TL;DR

This paper extends low-rank tensor-train methods to multi-patch isogeometric analysis (IgA) for PDE-constrained optimization, ensuring efficient handling of complex geometries and interface continuity.

Contribution

It introduces a novel generalization of low-rank tensor-train methods within the IETI framework for multi-patch IgA, improving computational efficiency.

Findings

Effective tensor-train based low-rank methods for multi-patch IgA.

Enhanced handling of complex geometries with local refinements.

Maintained interface continuity across patches.

Abstract

Isogeometric Analysis (IgA) is a versatile method for the discretization of partial differential equations on complex domains, which arise in various applications of science and engineering. Some complex geometries can be better described as a computational domain by a multi-patch approach, where each patch is determined by a tensor product Non-Uniform Rational Basis Splines (NURBS) parameterization. This allows on the one hand to consider the problem of the complex assembly of mass or stiffness matrices (or tensors) over the whole geometry locally on the individual smaller patches, and on the other hand it is possible to perform local mesh refinements independently on each patch, allowing efficient local refinement in regions of high activity where higher accuracy is required, while coarser meshes can be used elsewhere. Furthermore, the information about differing material models or properties that are to apply in a subdomain of the geometry can be included in the patch in which this subdomain is located. For this it must be ensured that the approximate solution is continuous over the entire computational domain and therefore at the interfaces of two (or more) patches. The most promising approach for this problem, which transfers the idea of Finite Element Tearing and Interconnecting (FETI) methods into the isogeometric setup, was the IsogEometric Tearing and Interconnecting (IETI) method, where by introducing a constraints matrix and associated Lagrange multipliers and formulating it into a dual problem, depending only on the Lagrange multipliers, continuity at the interfaces was ensured in solving the resulting system. In this paper we illustrate that low-rank methods based on the tensor-train format can be generalised for a multi-patch IgA setup, which follows the IETI idea.