Reduction of complex models for simulation and estimation Application to cardiac modelling

Abstract : This report analyzes and validates possible applications of some model reduction methods for direct simulations, and the solving of inverse problems of parameter estimation on complex models. It focuses on the reduction by proper orthogonal decomposition (POD), and its extensions. We start by proving new a priori estimates for the reduction error on typical abstract problems (parabolic and hyperboic, linear or with Lipschitzcontinuous nonlinearities), also validated in various nonlinear cases. In particular, we avoid the issue of controlling the high-order terms by using a specific sequence of projector norms. Then, in order to tackle parameter-dependent systems, and using some interpolation results, we adapt the previous method in a multi-POD reduction strategy. We also extend the previous estimates for the maximum reduction error over a given parameter range, at the cost of an additive term. We illustrate the power of the method on the electrophysiology FitzHugh-Nagumo system, known to be highly parameter-sensitive. Finally, we numerically validate the reduced versions, still with the multi-POD reduction, of some parameter estimation problems : of variational kind with the FitzHugh-Nagumo system, and of sequential kind (Kalmanian filtering) with a mechanical model of a heart (multi-scale, 3D, large displacements). In particular, we exhibit similar efficiency and robustness of the method as with direct problems.
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Asven Gariah. Reduction of complex models for simulation and estimation Application to cardiac modelling. Complex Variables [math.CV]. Université Pierre et Marie Curie - Paris VI, 2011. English. ⟨NNT : 2011PA066497⟩. ⟨tel-00824615⟩

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