High-dimensional stochastic finite volumes using the tensor train format

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.