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In this paper we present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameters vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, for the cost functional, its gradient and the optimal parameters. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.

The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques – such as the reduced basis method – traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k-means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized leftventricle geometry shows the capability of the proposed ROM to face complex problems.

The ensemble Kalman filter is a computationally efficient technique to solve state and/or parameter estimation problems in the framework of statistical inversion when relying on a Bayesian paradigm. Unfortunately its cost may become moderately large for systems de- scribed by nonlinear time-dependent PDEs, because of the cost entailed by each PDE query. In this paper we propose a reduced basis ensemble Kalman filter technique to address the above problems. The reduction stage yields intrinsic approximation errors, whose propagation through the filtering process might affect the accuracy of state/parameter estimates. For an efficient evaluation of these errors, we equip our reduced basis ensemble Kalman filter with a reduction error model (or error surrogate). The latter is based on ordinary kriging for functional-valued data, to gauge the effect of state reduction on the whole filtering process. The accuracy and efficiency of the resulting method is then verified on the estimation of uncer- tain parameters for a FitzHugh-Nagumo model and uncertain fields for a Fisher-Kolmogorov model.