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

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 )( prov- ably) 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 inex- pensive 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 ap- proximation 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 sim- ulations and a small set of tools — namely SimTEX 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.