A stochastic algorithm for fault inverse problems in elastic half space with proof of convergence
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A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [12]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic half-space and a slip on that fault have to be reconstructed from noisy surface displacement measurements. With the parameter giving the plane containing the fault denoted by m, we prove that the reconstructed posterior marginal of m is convergent as the number of measurement points and the dimension of the space for discretizing slips increase. Our proof relies on a recent result on the stability of the associated deterministic inverse problem [10], on trace operator theory, and on the existence of exact quadrature rules for a discrete scheme involving the underlying integral operator. The existence of such a discrete scheme was proved by Yarvin and Rokhlin, [16]. Our algorithm models the regularization constant C for the linear part of the inverse problem as a random variable allowing us to sweep through a wide range of possible values. We show in simulations that this is crucial when the noise level is not known. We also show numerical simulations that illustrate the numerical convergence of our algorithm.
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