Oblique projection for scalable rank-adaptive reduced-order modelling of nonlinear stochastic partial differential equations with time-dependent bases

Abstract

Time-dependent basis reduced-order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive and highly parallelizable. These favourable properties are achieved via oblique projections that require evaluating the MDE at a small number of rows and columns. The columns and rows are selected using the discrete empirical interpolation method (DEIM), which yields near-optimal matrix low-rank approximations. We show that the proposed algorithm is equivalent to a CUR matrix decomposition. Numerical results demonstrate the accuracy, efficiency and robustness of the new method for a diverse set of problems.

A rank-adaptive robust integrator for dynamical low-rank approximation

“…Dynamical low-rank approximation of time-dependent matrices [37] has proven to be an efficient model order reduction technique for applications from widely varying fields including plasma physics [24,26,20,8,22,14,27,15,23,56], radiation transport [49,16,47,41,48,57,21,4], radiation therapy [40], chemical kinetics [32,50,25], wave propagation [31,58], kinetic shallow water models [38], uncertainty quantification [51,3,28,43,44,46,39,33,17,2], and machine learning [54,59,52,53]. These problems can be written as a prohibitively large matrix differential equation for A(t) ∈ R m×n , .…”

“…Additional integrators, expected to be robust to small singular values based on numerical evidence but without proof, are discussed in [5,17,30] and [45].…”

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

“…Though dynamical low-rank approximation (DLRA) [33] offers a significant reduction of computational costs and memory consumption when solving tensor differential equations [26,10,54], the use of DLRA to solve matrix differential equations has sparked immense interest in several communities. Research fields in which DLRA for matrix differential equations has a considerable impact include plasma physics [21,23,16,5,18,12,24,13,20,55], radiation transport [47,14,45,39,46,56,17,3,38], chemical kinetics [29,48,22], wave propagation [28,57], uncertainty quantification [49,2,25,41,42,43,36,30,15,1], and machine learning [52,58,50,51]. These application fields commonly require memory-intensive and computational costly numerical simulations due to the solution's prohibitively large phase space.…”