The Tensor-Train Stochastic Finite Volume Method for Uncertainty Quantification

TL;DR

This paper introduces a tensor-train based stochastic finite volume method with WENO reconstruction to efficiently quantify uncertainty in hyperbolic conservation laws, especially those with shocks, addressing high-dimensional challenges.

Contribution

It presents the first tensor-train framework for hyperbolic systems with shocks, integrating WENO reconstruction into stochastic finite volume methods to improve efficiency and accuracy.

Findings

Effective handling of high-dimensional stochastic variables.

Successful integration of tensor-train with WENO for hyperbolic systems.

Potential for improved uncertainty quantification in complex systems.

Abstract

The stochastic finite volume method offers an efficient one-pass approach for assessing uncertainty in hyperbolic conservation laws. Still, it struggles with the curse of dimensionality when dealing with multiple stochastic variables. We introduce the stochastic finite volume method within the tensor-train framework to counteract this limitation. This integration, however, comes with its own set of difficulties, mainly due to the propensity for shock formation in hyperbolic systems. To overcome these issues, we have developed a tensor-train-adapted stochastic finite volume method that employs a global WENO reconstruction, making it suitable for such complex systems. This approach represents the first step in designing tensor-train techniques for hyperbolic systems and conservation laws involving shocks.