An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains
For researchers in uncertainty quantification, this provides a computationally feasible adaptive framework for PDEs with random domain perturbations, though it is an incremental extension of existing tensor train and Galerkin methods.
This paper develops an adaptive stochastic Galerkin method using tensor train format to solve linear PDEs on randomly perturbed domains, enabling efficient high-dimensional computation. Numerical benchmarks demonstrate the method's effectiveness in error reduction and adaptive refinement.
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-Loève 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.