Tensor train based isogeometric analysis for PDE approximation on parameter dependent geometries

TL;DR

This paper introduces a novel tensor train-based isogeometric analysis method for efficiently solving PDEs on geometries that depend on parameters, significantly reducing computational costs and storage requirements.

Contribution

The work combines tensor train decomposition with isogeometric analysis to handle parameter-dependent geometries in PDE approximation, enabling efficient, low-rank solutions.

Findings

High computational efficiency demonstrated on test cases

Achieves significant compression ratios in operator and solution representation

Effectively incorporates parameters into the TT-IGA framework

Abstract

This work develops a numerical solver based on the combination of isogeometric analysis (IGA) and the tensor train (TT) decomposition for the approximation of partial differential equations (PDEs) on parameter-dependent geometries. First, the discrete Galerkin operator as well as the solution for a fixed geometry configuration are represented as tensors and the TT format is employed to reduce their computational complexity. Parametric dependencies are included by considering the parameters that control the geometry configuration as additional dimensions next to the physical space coordinates. The parameters are easily incorporated within the TT-IGA solution framework by introducing a tensor product basis expansion in the parameter space. The discrete Galerkin operators are accordingly extended to accommodate the parameter dependence, thus obtaining a single system that includes the parameter dependency. The system is solved directly in the TT format and a low-rank representation of the parameter-dependent solution is obtained. The proposed TT-IGA solver is applied to several test cases which showcase its high computational efficiency and tremendous compression ratios achieved for representing the parameter-dependent IGA operators and solutions.