We propose certified reduced basis methods for the efficient and reliable evaluation of a general output that is implicitly connected to a given parameterized input through the harmonic Maxwell's equations. The truth approximation and the development of the reduced basis through a greedy approach is based on a discontinuous Galerkin approximation of the linear partial differential equation. The formulation allows the use of different approximation spaces for solving the primal and the dual

more »

... approximation problems to respect the characteristics of both problem types, leading to an overall reduction in the off-line computational effort. The main features of the method are the following: (i) rapid convergence on the entire representative set of parameters, (ii) rigorous a posteriori error estimators for the output, and (iii) a parameter independent off-line phase and a computationally very efficient on-line phase to enable the rapid solution of many-query problems arising in control, optimization, and design. The versatility and performance of this approach is shown through a numerical experiment, illustrating the modeling of material variations and problems with resonant behavior. 971 plicit relationship between the input and the output through the partial differential equation. Our primary goal is to develop a systematic approach to obtain an accurate and reliable approximation of the output of interest at very low computational cost for applications where many queries, i.e., solutions, are needed. We will explore the use of a reduced basis method by recognizing, and implicitly assuming, that the parameter dependent solution u e (ν) is not simply an arbitrary member of the infinite-dimensional space associated with the partial differential equation, but rather that it evolves on a lower-dimensional manifold induced by the parametric dependence. Under this assumption we can expect that as ν (∈ D ⊂ R q ) varies, the set of all solutions u e (ν) can be well approximated by a finite-and low-dimensional vector space. Hence, for a well chosen set of N parameters ν i , there exist coefficients c i = c N i (ν) for any ν ∈ D such that N i=1 c i u e (ν i ) is very close to u e (ν) when measured in an appropriate norm. The reduced basis method was first introduced in the 1970s for nonlinear structural analysis [1, 27] , and it was subsequently abstracted, analyzed [5, 32] , and generalized to other types of parameterized partial differential equations [12, 28] . Most of these earlier works focus on arguments that are local in the parameter space. Expansions to a low-dimensional manifold are typically defined around a particular point of interest, and the associated a priori analysis relies on asymptotic arguments on sufficiently small neighborhoods [8, 30] . In such cases, the computational improvements are quite modest. In [3, 17] a global approximation space was built by using solutions of the governing PDE at globally sampled points in the parameter space, resulting in a vastly improved method. However, no a priori theory or a posteriori error estimators were developed in this early work. In recent years, a number of novel ideas and essential new features have been presented [22, 21, 39, 31, 38, 4, 10, 37, 34] . In particular, global approximation spaces have been introduced and uniform exponential convergence of the reduced basis approximation has been numerically observed and confirmed in [23] , where the first theoretical a priori convergence result for a one-dimensional parametric space problem is presented. The development of rigorous a posteriori error estimators have also been presented, thereby transforming the reduced basis methods from an experimental technique to a computational approach with a true predictive value. Furthermore, in cases where the problem satisfies an affine assumption, that is, the operators and the data can be written as a linear combination of functions with separable dependence of the parameter and the spatial variation of the data, an offline/on-line computational strategy can be formulated. The off-line part of the algorithm, consisting of the generation of the reduced basis space, is ν-independent and can be done in preprocessing. The computational cost of the on-line part depends solely on the dimension of the reduced basis space and the parametric complexity of the problem, while the dependence on the complexity of the truth approximation has been removed, resulting in a highly efficient approach. When the data of the PDE are not affine, this computational strategy cannot be directly applied anymore and the on-line computational cost of the algorithm may be rather high. Recently, in [4], a procedure allowing the treatment of some of these nonaffine operators has been presented and shown to recover the off-line/on-line efficiency of the original algorithm. This technique, which also provides asymptotic a posteriori error estimators, has been successfully used in several applications [10, 9, 26, 36] .