Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are as follows: (i) efficient empirical port approximation spaces—the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system; and (ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditioner—the error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization. Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context, which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.Research Council of NorwayUnited States. Office of Naval Research (Grant N00014-11-0713

Approximation of Parametric Derivatives by the Empirical Interpolation Method

We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory

Parameter multi-domain ‘hp’ empirical interpolation

International audienc

An “hp” Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations

We present a new “hp” parameter multi-domain certified reduced basis method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations. We propose, and provide theoretical justification for, a new procedure for adaptive partition (“h”-refinement) of the parameter domain into smaller parameter subdomains: we pursue a hierarchical splitting of the parameter (sub)domains based on proximity to judiciously chosen parameter anchor points within each subdomain. Subsequently, we construct individual standard RB approximation spaces (“p”-refinement) over each subdomain. Greedy parameter sampling procedures and a posteriori error estimation play important roles in both the “h”-type and “p”-type stages of the new algorithm. We present illustrative numerical results for a convection-diffusion problem: the new “hp”-approach is considerably faster (respectively, more costly) than the standard “p”-type reduced basis method in the online (respectively, offline) stage.Norges teknisk-naturvitenskapelige universitetUnited States. Air Force Office of Scientific Research (Grant number FA 9550-07-1-0425 and OSD/AFOSR Grant number FA 9550-09-1-0613

A posteriori error bounds for the empirical interpolation method Un estimateur a posteriori d'erreur pour la méthode d'interpolation empirique

We present rigorous a posteriori error bounds for the Empirical Interpolation Method (EIM). The essential ingredients are (i) analytical upper bounds for the parametric derivatives of the function to be approximated, (ii) the EIM “Lebesgue constant,” and (iii) information concerning the EIM approximation error at a finite set of points in parameter space. The bound is computed “off-line” and is valid over the entire parameter domain; it is thus readily employed in (say) the “on-line” reduced basis context. We present numerical results that confirm the validity of our approach.United States. Air Force Office of Scientific Research (Grant No. FA9550-07-1-0425)United States. Air Force Office of Scientific Research (OSD/AFOSR Grant No. FA9550-09-1-0613)Norges teknisk-naturvitenskapelige universitetGermany. Federal Ministry of Education and Research (Excellence Initiative