# An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains

## TL;DR

This paper introduces an adaptive stochastic Galerkin method using tensor train formats to efficiently solve PDEs on randomly perturbed domains, with error estimation and refinement capabilities.

## Contribution

It develops a novel tensor train-based adaptive Galerkin framework for high-dimensional PDEs on random domains, including an a posteriori error estimator for refinement.

## Key findings

- Efficient handling of high-dimensional randomness via tensor train compression.
- Successful numerical benchmarks demonstrating accuracy and adaptivity.
- Effective error estimation enabling iterative refinement.

## Abstract

A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding Karhunen-Lo\`eve expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined high-dimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problem-dependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems.