High-dimensional stochastic finite volumes using the tensor train format

TL;DR

The paper tackles uncertainty quantification for nonlinear hyperbolic PDEs with many uncertain parameters by introducing a hybrid stochastic finite volume method that keeps the physical space and time in full and compresses the stochastic space into a tensor train (TT). The stochastic finite volume formulation reformulates the problem in $(t,x,\boldsymbol{\xi})$, coupling cell-averaged fluxes with TT-based representations to control the curse of dimensionality, while the MUSCL scheme is adapted to operate within this hybrid TT framework. Numerical results on the stochastic Burgers' equation, the Sod problem, and the Shu-Osher problem demonstrate the approach's feasibility, revealing how TT ranks concentrate near shocks and how the hybrid method compares favorably with the full-TT variant in accuracy and scalability. The work provides a practical pathway to integrate high-dimensional uncertainty quantification with classical finite-volume solvers for conservation laws, and it suggests directions for adaptive rank strategies and mesh refinement to further enhance efficiency and robustness.

Abstract

A method for the uncertainty quantification of nonlinear hyperbolic equations with many uncertain parameters is presented. The method combines the stochastic finite volume method and tensor trains in a novel way: the physical space and time dimensions 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 this hybrid format and the feasibility of the approach using several classical test cases is shown. For the Burgers' equation a convergence study and a comparison with the full tensor train format are done with three stochastic parameters. The equation is then solved for an increasing number of stochastic dimensions. The Euler equations are then considered. A parameter study and a comparison with the full tensor train format are performed with the Sod problem. For a complex application we consider the Shu-Osher problem. The presented method opens new avenues for combining uncertainty quantification with well-known numerical schemes for conservation laws.

Figures (19)

• Figure 1: Relative $L^1$-errors of the expectation and variance for the shock wave, computed with $N_\xi=N_x$ using the hybrid method.
• Figure 2: Relative $L^1$-errors of the expectation and variance for the shock wave, computed with $N_\xi=20$ using the hybrid method.
• Figure 3: Relative $L^1$-error of the expectation for the shock wave, using the full-TT (plain line) and hybrid (dashed line) formats.
• Figure 4: Number of coefficients of the solution at the final time: hybrid format, full-TT format, and theoretical value for the full tensor.
• Figure 5: Number of coefficients of the solution at the final time for increasing stochastic dimensions: hybrid format and theoretical value for the full tensor

Theorems & Definitions (1)

• Remark 1