High-dimensional stochastic finite volumes using the tensor train format

TL;DR

This paper introduces a novel hybrid tensor train approach for uncertainty quantification in nonlinear hyperbolic conservation laws, efficiently handling many stochastic parameters while maintaining full physical space and time tensors.

Contribution

The paper presents a new hybrid tensor train method combining stochastic finite volume schemes with tensor trains, enabling efficient high-dimensional uncertainty quantification in hyperbolic PDEs.

Findings

Demonstrates convergence for scalar Burgers' equation with multiple stochastic parameters.

Shows feasibility and efficiency in solving Euler equations with high-dimensional uncertainty.

Provides comparative analysis with full tensor train format for validation.

Abstract

A method for the uncertainty quantification of nonlinear hyperbolic conservation laws with many uncertain parameters is presented. The method combines stochastic finite volume methods and tensor trains in a novel way: the dimensions of physical space and time are kept as full tensors, while all stochastic dimensions are compressed together into a tensor train. The resulting hybrid format has one tensor train for each spatial cell and each time step. The MUSCL scheme is adapted to the proposed hybrid format, and its feasibility is demonstrated through several test cases. For the scalar Burgers' equation, we conduct a convergence study and compare the results with those obtained using the full tensor train format with three stochastic parameters. The equation is then solved for an increasing number of stochastic dimensions.For systems of conservation laws, we focus on the Euler equations. A parameter study and a comparison with the full tensor train format are carried out for the Sod shock tube problem. As a more complex application, we investigate the Shu-Osher problem, which involves intricate wave interactions. The presented method opens new avenues for integrating uncertainty quantification with established numerical schemes for hyperbolic conservation laws.