The Tensor-Train Stochastic Finite Volume Method for Uncertainty Quantification

TL;DR

This work tackles uncertainty quantification for hyperbolic conservation laws by marrying the stochastic finite volume (SFV) method with tensor-train (TT) representations. The authors develop TT-SFV, a low-rank framework that employs a global WENO reconstruction to accurately capture discontinuities while mitigating the curse of dimensionality through TT cores. They establish a global matrix formulation for WENO3, introduce the TT-SFV algorithm with cell-wise TT initialization, physical and stochastic reconstructions, and TT-based flux evaluations, and validate the method on linear advection, Burgers, Sod, and shock-bubble problems up to 12 stochastic dimensions. Results show accurate shock capturing, controlled rank growth, and competitive scalability compared with traditional SFV implementations, indicating TT-SFV as a promising pathway for high-dimensional uncertainty quantification in hyperbolic PDEs.

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.

Figures (11)

• Figure 1: Illustration of reconstruction points for the SFV method; "L" and "R" label the "left" and "right" sides of the cell interface located at $x_{i+\frac{1}{2}}$, and ${{\bf U}}^{m,L}_{i+\frac{1}{2},j}$ and ${{\bf U}}^{m,R}_{i+\frac{1}{2},j}$ are the reconstructed values at the quadrature node with coordinates $(x_{i+\frac{1}{2}},y_m)$.
• Figure 1: Graphical depiction of a three dimensional tensor $U_\ell$, representing a conserved quantity, and its tensor-train approximation via the tensor-train cores ${\mathcal{U}}_{\ell,k}$ for $k=1,2,3$, cf. \ref{['eqn:tt-format:def']}. Here, the rank set is $r=\{r_1,r_2\}$ and all cores have an equal number of modes $N=M$.
• Figure 1: Results from a h-refinement convergence study for the problem \ref{['eqn::smooth_linear_advection']} demonstrating that the SFV in TT format obtains the expected 3rd order convergence.
• Figure 2: Reconstruction stencil of a scalar quantity for 1D WENO3.
• Figure 2: Graphical representation of applying reconstruction in physical domain for the case $n=1$ and $m=2$. The left and right reconstruction matrices are applied to the core of $\bar{\mathcal{U}}_{\ell}$ representing the physical dimension, $\bar{\mathcal{U}}_{\ell,1}$, resulting in reconstruction along the modes. The result is $\mathbf{L}\bar{\mathcal{U}}_{\ell,1}$ and $\mathbf{R}\bar{\mathcal{U}}_{\ell,1}$ which are then recombined with the stochastic cores to produce $\bar{\mathcal{U}}^1_{\ell,L}$ and $\bar{\mathcal{U}}^1_{\ell,R}$, as desired.