A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations

Abstract

In this paper, we extend the earlier work [M. Barrault, Y. Maday, N. C. Nguyen, A.T. Patera, An ?empirical interpolation? method application to efficient reduced-basis discretization of partial differential equations, C.R. Acad. Sci. Paris, Serie I 339 (2004) 667?672; M.A. Grepl, Y. Maday, N.C. Nguyen, A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, M2AN Math. Model. Numer. Anal. 41 (3) (2007) 575?605.] to provide a posteriori error estimation and basis adaptivity for reduced-basis approximation of linear elliptic partial differential equations with nonaffine parameter dependence. The essential components are (i) rapidly convergent reduced-basis approximations ? (Galerkin) projection onto a space spanned by N global hierarchical basis functions which are constructed from solutions of the governing partial differential equation at judiciously selected points in parameter space; (ii) stable and inexpensive interpolation procedures ? methods which allow us to replace nonaffine parameter functions with a coefficient-function expansion as a sum of M products of parameter-dependent coefficients and parameter-independent functions; (iii) a posteriori error estimation ? relaxations of the error-residual equation that provide inexpensive yet sharp error bounds for the error in the outputs of interest; (iv) optimal basis construction ? processes which make use of the error bounds as an inexpensive surrogate for the expensive true error to explore the parameter space in the quest for an optimal sampling set; and (v) offline/online computational procedures ? methods which decouple the generation and projection stages of the approximation process. The operation count for the online stage - in which, given a new parameter value, we calculate the output of interest and associated error bounds - depends only on N, M, and the affine parametric complexity of the problem; the method is thus ideally suited for repeated and reliable evaluation of input?output relationships in the many-query or real-time contexts.