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Journal of Convex Analysis 30 (2023), No. 3, 851--885
Copyright Heldermann Verlag 2023

Stochastic Elliptic Inverse Problems. Solvability, Convergence Rates, Discretization, and Applications

Marc Dambrine
Université de Pau et des Pays de l'Adour, Lab. de Mathématiques et de leurs Applications, Pau, France

Akhtar A. Khan
School of Math. Sciences, Rochester Institute of Technology, Rochester, New York, U.S.A.

Miguel Sama
Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Madrid, Spain

Hans-Jörg Starkloff
Institute of Stochastics, Technische Universität Bergakademie Freiberg, Germany

Motivated by the necessity to identify stochastic parameters in a wide range of stochastic partial differential equations, an abstract inversion framework is designed. The stochastic inverse problem is studied in a stochastic optimization framework. The essential properties of the solution map are derived and used to prove the solvability of the stochastic optimization problems. Novel convergence rates for the stochastic inverse problem are presented in the abstract formulation without requiring the so-called smallness condition. Under the assumption of finite-dimensional noise, the stochastic inverse problem is parametrized and solved by using the Stochastic Galerkin discretization scheme. The developed framework is applied to estimate stochastic Lam\'e parameters in the system of linear elasticity. We present numerical results that are quite encouraging and show the feasibility and efficacy of the developed framework.

Keywords: Stochastic inverse problems, partial differential equations with random data, stochastic Galerkin method, regularization, finite-dimensional noise, convergence rates.

MSC: 35R30, 49N45, 65J20, 65J22, 65M30.

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