Existence of dynamical low rank approximations for random semi-linear evolutionary equations on the maximal interval

Yoshihito Kazashi; Fabio Nobile

doi:10.1007/s40072-020-00177-4

Existence of dynamical low rank approximations for random semi-linear evolutionary equations on the maximal interval

Stoch Partial Differ Equ. 2021;9(3):603-629. doi: 10.1007/s40072-020-00177-4. Epub 2020 Aug 5.

Authors

Yoshihito Kazashi¹, Fabio Nobile¹

Affiliation

¹ Institute of Mathematics, École Polytechnique Fédérale de Lausanne, CSQI, Station 8, CH-1015 Lausanne, Switzerland.

Abstract

An existence result is presented for the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations. The DLR solution approximates the true solution at each time instant by a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time. A key to our proof is to find a suitable equivalent formulation of the original problem. The so-called Dual Dynamically Orthogonal formulation turns out to be convenient. Based on this formulation, the DLR approximation is recast to an abstract Cauchy problem in a suitable linear space, for which existence and uniqueness of the solution in the maximal interval are established.

Keywords: Dynamical low rank approximation; Non-linear evolution equation; Singular value decomposition; Well-posedness.