A moment-matching method to study the variability of phenomena described by partial differential equations
| Autor: | Jean-Frédéric Gerbeau, Damiano Lombardi, Eliott Tixier |
|---|---|
| Přispěvatelé: | Numerical simulation of biological flows (REO), Sorbonne Université (SU)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical simulation of biological flows ( REO ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Tixier, Eliott |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Matching (statistics)
Partial differential equation Differential equation Applied Mathematics Principle of maximum entropy Backward uncertainty quantification 010103 numerical & computational mathematics [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA] [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA] 01 natural sciences 010101 applied mathematics Moment (mathematics) Computational Mathematics Moment matching Maximum entropy Stochastic inverse problem Calculus Mathematics::Metric Geometry Applied mathematics 0101 mathematics [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] Mathematics |
| Zdroj: | SIAM Journal on Scientific Computing SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2018, 40 (3) SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2018, 40 (3), 〈https://epubs.siam.org/doi/10.1137/16M1103476〉 SIAM Journal on Scientific Computing, 2018, 40 (3) |
| ISSN: | 1064-8275 |
| Popis: | International audience; Many phenomena are modeled by deterministic differential equations , whereas the observation of these phenomena, in particular in life sciences, exhibits an important variability. This paper addresses the following question: how can the model be adapted to reflect the observed variability? Given an adequate model, it is possible to account for this variability by allowing some parameters to adopt a stochastic behavior. Finding the parameters probability density function that explains the observed variability is a difficult stochastic inverse problem, especially when the computational cost of the forward problem is high. In this paper, a non-parametric and non-intrusive procedure based on offline computations of the forward model is proposed. It infers the probability density function of the uncertain parameters from the matching of the statistical moments of observable degrees of freedom (DOFs) of the model. This inverse procedure is improved by incorporating an algorithm that selects a subset of the model DOFs that both reduces its computational cost and increases its robustness. This algorithm uses the pre-computed model outputs to build an approximation of the local sensitivities. The DOFs are selected so that the maximum information on the sensitivities is conserved. The proposed approach is illustrated with elliptic and parabolic PDEs. In the Appendix, an nonlinear ODE is considered and the strategy is compared with two existing ones. |
| Databáze: | OpenAIRE |
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