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

12/28/2021
by   Prerna Patil, et al.
0

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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 21

08/07/2019

Absorbing boundary conditions for the time-dependent Schrödinger-type equations in R^3

Absorbing boundary conditions are presented for three-dimensional time-d...
11/03/2021

Estimación y Análisis de Sensibilidad para el Coeficiente de Difusividad en un Problema de Conducción de Calor

The aim of this article is to discuss the estimation of the diffusivity ...
01/11/2021

On-the-fly Reduced Order Modeling of Passive and Reactive Species via Time-Dependent Manifolds

One of the principal barriers in developing accurate and tractable predi...
09/02/2020

A numerical study of third-order equation with time-dependent coefficients: KdVB equation

In this article we present a numerical analysis for a third-order differ...
01/10/2022

Scalable In Situ Compression of Transient Simulation Data Using Time-Dependent Bases

Large-scale simulations of time-dependent problems generate a massive am...
03/11/2018

Optimal Estimation of Dynamically Evolving Diffusivities

The augmented, iterated Kalman smoother is applied to system identificat...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.