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For solving Bayesian inverse problems governed by largescale forward pr...
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Enabling and interpreting hyperdifferential sensitivity analysis for Bayesian inverse problems
Inverse problems constrained by partial differential equations (PDEs) pl...
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Bayesian inverse problems with unknown operators
We consider the Bayesian approach to linear inverse problems when the un...
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The SPDE Approach to Matérn Fields: Graph Representations
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Optimal design of largescale Bayesian linear inverse problems under reducible model uncertainty: good to know what you don't know
We consider optimal design of infinitedimensional Bayesian linear inver...
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On Bayesian posterior mean estimators in imaging sciences and HamiltonJacobi Partial Differential Equations
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Connecting Hamilton–Jacobi partial differential equations with maximum a posteriori and posterior mean estimators for some nonconvex priors
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Variational Bayes' method for functions with applications to some inverse problems
Bayesian approach as a useful tool for quantifying uncertainties has been widely used for solving inverse problems of partial differential equations (PDEs). One of the key difficulties for employing Bayesian approach is how to extract information from the posterior probability measure. Variational Bayes' method (VBM) is firstly and broadly studied in the field of machine learning, which has the ability to extract posterior information approximately by using much lower computational resources compared with the sampling type method. In this paper, we generalize the usual finitedimensional VBM to infinitedimensional space, which makes the usage of VBM for inverse problems of PDEs rigorously. General infinitedimensional meanfield approximate theory has been established, and has been applied to abstract linear inverse problems with Gaussian and Laplace noise assumptions. Finally, some numerical examples are given which illustrate the effectiveness of the proposed approach.
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