DeepAI AI Chat
Log In Sign Up

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

04/06/2022
by   Ion Gabriel Ion, et al.
0

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.

READ FULL TEXT
06/29/2021

Tensor-train approximation of the chemical master equation and its application for parameter inference

In this work, we perform Bayesian inference tasks for the chemical maste...
08/03/2020

A parameter-dependent smoother for the multigrid method

The solution of parameter-dependent linear systems, by classical methods...
02/23/2021

Solving high-dimensional parabolic PDEs using the tensor train format

High-dimensional partial differential equations (PDEs) are ubiquitous in...
10/08/2021

Randomized algorithms for rounding in the Tensor-Train format

The Tensor-Train (TT) format is a highly compact low-rank representation...
12/24/2022

Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

Petrov-Galerkin formulations with optimal test functions allow for the s...
10/26/2022

A robust GMRES algorithm in Tensor Train format

We consider the solution of linear systems with tensor product structure...
04/09/2018

Scalar and Tensor Parameters for Importing the Notation in Differential Geometry into Programming

This paper proposes a method for importing tensor index notation, includ...