Dynamical low-rank approximation for Burgers' equation with uncertainty

Abstract

Quantifying uncertainties in hyperbolic equations is a source of several challenges. First, the solution forms shocks leading to oscillatory behaviour in the numerical approximation of the solution. Second, the number of unknowns required for an effective discretization of the solution grows exponentially with the dimension of the uncertainties, yielding high computational costs and large memory requirements. An efficient representation of the solution via adequate basis functions permits to tackle these difficulties. The generalized polynomial chaos (gPC) polynomials allow such an efficient representation when the distribution of the uncertainties is known. These distributions are usually only available for input uncertainties such as initial conditions, therefore the efficiency of this ansatz can get lost during runtime. In this paper, we make use of the dynamical low-rank approximation (DLRA) to obtain a memory-wise efficient solution approximation on a lower dimensional manifold. We investigate the use of the matrix projector-splitting integrator and the unconventional integrator for dynamical low-rank approximation, deriving separate time evolution equations for the spatial and uncertain basis functions, respectively. This guarantees an efficient approximation of the solution even if the underlying probability distributions change over time. Furthermore, filters to mitigate the appearance of spurious oscillations are implemented, and a strategy to enforce boundary conditions is introduced. The proposed methodology is analyzed for Burgers' equation equipped with uncertain initial values represented by a two-dimensional random vector. The numerical experiments validate that the results of a standard filtered Stochastic-Galerkin (SG) method are consistent with the numerical results obtained via the use of numerical integrators for dynamical low-rank approximation. Significant reduction of the memory requirements is obtained, and the important characteristics of the original system are well captured.

On the Stability of Robust Dynamical Low-Rank Approximations for Hyperbolic Problems

“…Note that such a discretization has been discussed in [33,23]. Let us investigate L 2 -stability for the above system.…”

“…This stems mainly from its ability to mitigate the curse of dimensionality in terms of computational costs and memory requirements. Problems in which dynamical low-rank approximation has proven its efficiency include, e.g., kinetic theory [11,12,33,32,13,9,10,19] as well as uncertainty quantification [14,29,30,34,23]. In both fields the high-dimensional phase space implies that obtaining numerical solutions is extremely expensive both in terms of memory and computational cost.…”

“…It has been shown in numerical experiments that the unconventional integrator yields smoother solution profiles for first order moments such as the scalar flux in radiation transport or the expected value in uncertainty quantification [23]. Higher-order moments, however, are dampened heavily by the unconventional integrator and the projector-splitting integrator allows for a more adequate representation [23].…”

A low-rank power iteration scheme for neutron transport criticality problems

“…This approach does not only provide the ability to derive stable discretizations [19], but in the radiation transfer context allows an efficient implementation of scattering [19,20]. Problems in which dynamical low-rank is successfully applied to reduce memory and computational costs are, e.g., kinetic theory [7,8,31,30,9,5,6,13,20] as well as uncertainty quantification [10,28,29,33,18]. Furthermore, DLRA allows for adaptive model refinement [4,2,32], where the main idea is to pick the rank of the solution representation adaptively.…”

Efficient 6D Vlasov simulation using the dynamical low-rank framework Ensign

“…Recently, a dynamical algorithm that is conservative from first principle has been constructed [13]. Because of these advances, dynamical low-rank approximations have received significant interest lately and methods for problems from plasma physics [14,17,20,23], radiation transport [11,26,32,33], and uncertainty quantification for hyperbolic problems [25] have been proposed. These schemes have the potential to enable the 6D simulation of such systems on small clusters or even desktop computers.…”