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Multilevel adaptive sparse Leja approximations for Bayesian inverse problems. (English) Zbl 1432.35250
An adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parametrized functions with high-dimensional parameters were proposed in [A. Narayan and J. D. Jakeman, SIAM J. Sci. Comput. 36, No. 6, A2952–A2983 (2014; Zbl 1316.65018)].
For Bayesian inverse problems the authors formulate and test a multilevel adaptive sparse Leja algorithm. A very detailed description is given how the Leja approximation is used. As the model discretization is coarse, the construction of the sparse grid is computationally effecient.
In numerical examples such as elliptic inverse problems it seems superior to Markov Chain Monte Carlo Sampling and a standard multilevel approximation.
##### MSC:
 35R30 Inverse problems for PDEs 65D30 Numerical integration 35J25 Boundary value problems for second-order elliptic equations 62F15 Bayesian inference 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 68T05 Learning and adaptive systems in artificial intelligence 62-08 Computational methods for problems pertaining to statistics
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