High-Dimensional Stochastic Finite Volumes Using the Tensor Train Format

SIAM/ASA Journal on Uncertainty Quantification, vol. 14, pp. 482–507

Abstract

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 (full-TT) 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-TT 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.

Authors 2

1. Michaël Herty Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University · University of Pretoria

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany, and Extraordinary Professor, Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield 0028, South Africa

2. Siegfried Müller Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany