A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient

Abstract

In this work, a Reduced Basis (RB) approach is used to solve a large number of boundary value problems parametrized by a stochastic input -expressed as a Karhunen-Loève expansion -in order to compute outputs that are smooth functionals of the random solution fields. The RB method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the RB approach developed previously for deterministic systems. However, the stochastic framework requires the development of new a posteriori estimates for "statistical" outputs -such as the first two moments of integrals of the random solution fields; these error bounds, in turn, permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many-query context.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

“…Another active field relates to the development and application of the reduced basis methodology in the context of the quantification of uncertainty, offing another example of application where many-query problems arise naturally [13,111,62]. Such problems are often characterized by having a high-dimensional parameter space and recent work has focused on the development of efficient ways to explore the parameters space, e.g., modified greedy algorithms and combined adaptive techniques [56,21,22], and hp-reduced basis method [40,39].…”

“…The efficient implementation of the reduced basis approximation then becomes: find u rb (µ) = n=1 (u µ rb ) n ξ n ∈ V rb such that 12) which translates into the N -dimensional system of nonlinear algebraic equations 13) for the unknown u µ rb , where b q rb ∈ R N with (b q rb ) n = b q (ξ n ). This can then be solved with Newton's method for example.…”

A Two-Step Certified Reduced Basis Method

“…We consider in Sect. 4 and Sect. 5 the new approach in the context of focus calculations and in the context of hp-RB approximations, respectively.…”

“…In particular, the RB method provides rapid and certifiable computation of linear functional outputs-such as average field values or average fluxes-associated with the solution to the PDE for any set of input parameter values that configure the PDE in terms of (say) applied forces, material properties, geometry, or boundary conditions. The RB method is of interest in two particular contexts: real-time-such as parameter estimation [23] and optimal control [13]-and many-query-such as multiscale [3,20] or stochastic simulation [4]. In these contexts, a computational preprocessing (offline) stage is typically justified.…”

Stochastic Galerkin Methods and Model Order Reduction for Linear Dynamical Systems

“…In case of large numbers of random parameters, a reduction of the random space based on a sensitivity analysis is discussed in [16]. Furthermore, reduced basis methods have been investigated for partial differential equations with stochastic influences in [17,18], for example.…”