Reduced order modeling with time-dependent bases for PDEs with stochastic boundary conditions

TL;DR

This paper introduces a novel methodology for determining boundary conditions for time-dependent bases in reduced-order models of stochastic PDEs with non-homogeneous boundary conditions, preserving orthonormality and orthogonality constraints.

Contribution

The work develops a boundary condition determination method for TDBs that requires no extra computational cost and maintains the constraints of the bases, applicable to various boundary types.

Findings

Method effectively handles stochastic boundary conditions in reduced-order models.

Preserves orthonormality and orthogonality constraints of TDBs.

Validated on multiple PDEs including advection-diffusion and Burgers' equation.

Abstract

Low-rank approximation using time-dependent bases (TDBs) has proven effective for reduced-order modeling of stochastic partial differential equations (SPDEs). In these techniques, the random field is decomposed to a set of deterministic TDBs and time-dependent stochastic coefficients. When applied to SPDEs with non-homogeneous stochastic boundary conditions (BCs), appropriate BC must be specified for each of the TDBs. However, determining BCs for TDB is not trivial because: (i) the dimension of the random BCs is different than the rank of the TDB subspace; (ii) TDB in most formulations must preserve orthonormality or orthogonality constraints and specifying BCs for TDB should not violate these constraints in the space-discretized form. In this work, we present a methodology for determining the boundary conditions for TDBs at no additional computational cost beyond that of solving the same SPDE with homogeneous BCs. Our methodology is informed by the fact the TDB evolution equations are the optimality conditions of a variational principle. We leverage the same variational principle to derive an evolution equation for the value of TDB at the boundaries. The presented methodology preserves the orthonormality or orthogonality constraints of TDBs. We present the formulation for both the dynamically bi-orthonormal (DBO) decomposition as well as the dynamically orthogonal (DO) decomposition. We show that the presented methodology can be applied to stochastic Dirichlet, Neumann, and Robin boundary conditions. We assess the performance of the presented method for linear advection-diffusion equation, Burgers' equation, and two-dimensional advection-diffusion equation with constant and temperature-dependent conduction coefficient.