Ph.D. student Yang Liu attended the WCCM-APCOM Yokohama 2022 and presented the work Goal-oriented adaptive MLMC method for elliptic random PDEs by Joakim Beck, Yang Liu, Erik von Schwerin, and Prof. Raul Tempone.
Abstract: Multilevel Monte Carlo methods (MLMC) can dramatically reduce the computational cost of Monte Carlo simulations where each sample is computed using a discretization based numerical method, for example, when computing the expected value of a quantity of interest (QoI) depending on the solution to a partial differential equation with stochastic data.
Goal-oriented adaptive finite element refines the mesh based on the error contribution to the QoI. The method is effective, for instance, when the geometry presents a singularity, such as a non-convex domain.
The purpose of this work is to combine MLMC and adaptive finite element solvers, to efficiently solve a boundary-value problem of an elliptic partial differential equation with random coefficients on a non- convex domain. The QoI is a linear functional of the PDE solution, and the coefficient field is efficiently sampled from a regular coefficient random field. The adaptive refinement algorithm is based on [1]. This work can also be seen as an extension of [2].
REFERENCES
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[1] Moon, K-S., Erik von Schwerin, Anders Szepessy, and Rau ́l Tempone. ”Convergence rates for an adaptive dual weighted residual finite element algorithm.” BIT Numerical Mathematics 46, no. 2 (2006): 367-407.
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[2] Hoel, Ha ̊kon, Erik von Schwerin, Anders Szepessy, and Rau ́l Tempone. ”Implementation and anal- ysis of an adaptive multilevel Monte Carlo algorithm.” Monte Carlo Methods and Applications 20, no. 1 (2014): 1-41.
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