
Bayesian inverse problems with partial observations
We study a nonparametric Bayesian approach to linear inverse problems un...
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Efficient Marginalizationbased MCMC Methods for Hierarchical Bayesian Inverse Problems
Hierarchical models in Bayesian inverse problems are characterized by an...
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Continuum Limit of Posteriors in Graph Bayesian Inverse Problems
We consider the problem of recovering a function input of a differential...
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Probabilistic solution of chaotic dynamical system inverse problems using Bayesian Artificial Neural Networks
This paper demonstrates the application of Bayesian Artificial Neural Ne...
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Stabilities of Shape Identification Inverse Problems in a Bayesian Framework
A general shape identification inverse problem is studied in a Bayesian ...
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Inverse heat source problem and experimental design for determining iron loss distribution
Iron loss determination in the magnetic core of an electrical machine, s...
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Posterior contraction rates for nonparametric state and drift estimation
We consider a combined state and drift estimation problem for the linear...
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Allatonce formulation meets the Bayesian approach: A study of two prototypical linear inverse problems
In this work, the Bayesian approach to inverse problems is formulated in an allatonce setting. The advantages of the allatonce formulation are known to include the avoidance of a parametertostate map as well as numerical improvements, especially when considering nonlinear problems. In the Bayesian approach, prior knowledge is taken into account with the help of a prior distribution. In addition, the error in the observation equation is formulated by means of a distribution. This method naturally results in a whole posterior distribution for the unknown target, not just point estimates. This allows for further statistical analysis including the computation of credible intervals. We combine the Bayesian setting with the allatonce formulation, resulting in a novel approach for investigating inverse problems. With this combination we are able to chose a prior not only for the parameter, but also for the state variable, which directly influences the parameter. Furthermore, errors not only in the observation equation, but additionally, in the model can be taken into account. reconstructions of the unknown parameter but also to maximize the information gained from measurements through combining it with prior knowledge, obtained either from certain expertise or former investigation in the model. We analyze this approach with the help of two linear standard examples, namely the inverse source problem for the Poisson equation and the backwards heat equation, i.e. a stationary and a time dependent problem. Appropriate function spaces and derivation of adjoint operators are investigated. To assess the degree of illposedness, we analyze the singular values of the corresponding allatonce forward operators. priors are designed and numerically tested.
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