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

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 [1] as well as the dynamically orthogonal (DO) decomposition [2]. 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.

Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases

“…The optimality conditions of the variational principle result in closed-form evolution equations for U, Σ, and Y. As indicated in [47], the closed-form evolution equations of the DBO decomposition are defined by:…”

“…They all extract identical low-rank subspaces and they only differ in an in-subspace scaling and rotation. In this study, we utilize the DBO decomposition since the efficiency of this approach for quantifying the uncertainty of highly ill-conditioned physical systems has been established by multiple research studies [18,[20][21][22]. Extracting correlated structures using TDB has been established in the chemical physics literature for solving high-dimensional deterministic problems before the application of TDB for solving SPDE.…”