Prof. Fabio Nobile (École Polytechnique Fédérale de Lausanne)
Abstract
In this talk we present few instances of multilevel approximation methods involving PDEs with random parameters and associated scalar output quantities of interest (QoI). Multilevel methods aim at optimally combining multiple discretizations in space and probability of the underlying PDE at different resolution levels.
In the first instance [1], we propose a multilevel polynomial approximation in the random parameters of the output QoI by the weighted discrete least squares method based on evaluations in random points.
In the second instance [2], we focus on dimension reduction of the input parameter space and propose a multilevel active subspace approximation of the output QoI based on gradient information and SVD compression of the input-output map.
In the third instance [3] we consider an optimal control problem constrained by a random PDE, where the optimal control minimizes the expected value of the QoI, and propose a multilevel method that combines optimal controls computed with different approximation levels in space and probability.
References
[1] A-L. Haji-Ali, F. Nobile, S. Wolfers, R. Tempone, "Multilevel weighted least squares polynomial approximation", M2AN, 2020(54) p. 649-677. DOI : 10.1051/m2an/2019045.
[2] F. Nobile, M. Raviola,R. Tempone, "Multilevel active subspaces method", in preparation
[3] F. Nobile, T. Vanzan, "Multilevel quadrature formulae for the optimal control of random PDEs", 2024, arXiv:2407.06678