A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

Abstract

Abstract. We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced-basis approximations -Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation -relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures -methods which decouple the generation and projection stages of the approximation process. This component is ideally suited -considering the operation count of the online stage -for the repeated and rapid evaluation required in the context of parameter estimation, design, optimization, and real-time control. The second component is a framework for distributed simulations. This framework comprises a library providing the necessary abstractions/concepts for distributed simulations and a small set of tools -namely SimT E X and SimLaB-allowing an easy manipulation of those simulations. While the library is the backbone of the framework and is therefore general, the various interfaces answer specific needs. We shall describe both components and present how they interact.Mathematics Subject Classification. 65N15, 65N30, 68U01, 68U20, 68M14, 68M15.

Reduced basis method for finite volume approximations of parametrized linear evolution equations

“…Studies on eigenvalue problems are treated with RB-methods in [22]. The general methodology for linear elliptic equations is the subject of [33], details on algorithmic aspects can be found in [32]. The stationary advection-diffusion problem has been treated in the context of optimal control in [34] and with view on geometry optimization in [39].…”

The Reduced Basis Element Method for Fluid Flows

“…Remark It should be noticed here that the reduced basis element method, applied to the fin problem, has a lot a similarity with the plain reduced basis method that has been extensively used on this example for illustrating the power of the method (see [17], [25]). However, note that there is an additional dimension to the reduced basis element method due to the possibility of varying the number of stages.…”

The localized reduced basis multiscale method for two‐phase flows in porous media

“…In numerous publications, these methods have proven to be versatile tools for reducing the computational effort for stationary problems [17,18,19,20,21] as well as for linear [22,23] and nonlinear [24] parabolic problems, and linear [25,26] and nonlinear [27] hyperbolic problems. Applications to two-phase flow in porous media exist [28].…”