
Connecting Hamilton–Jacobi partial differential equations with maximum a posteriori and posterior mean estimators for some nonconvex priors
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Optimally weighted loss functions for solving PDEs with Neural Networks
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D3M: A deep domain decomposition method for partial differential equations
A stateoftheart deep domain decomposition method (D3M) based on the v...
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Variational Bayes' method for functions with applications to some inverse problems
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PDE Evolutions for MSmoothers in One, Two, and Three Dimensions
Local Msmoothers are interesting and important signal and image process...
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Robust estimation in controlled branching processes: Bayesian estimators via disparities
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On the cost of Bayesian posterior mean strategy for logconcave models
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On Bayesian posterior mean estimators in imaging sciences and HamiltonJacobi Partial Differential Equations
Variational and Bayesian methods are two approaches that have been widely used to solve image reconstruction problems. In this paper, we propose original connections between Hamilton–Jacobi (HJ) partial differential equations and a broad class of Bayesian methods and posterior mean estimators with Gaussian data fidelity term and logconcave prior. Whereas solutions to certain firstorder HJ PDEs with initial data describe maximum a posteriori estimators in a Bayesian setting, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimators. These connections allow us to establish several representation formulas and optimal bounds involving the posterior mean estimate. In particular, we use these connections to HJ PDEs to show that some Bayesian posterior mean estimators can be expressed as proximal mappings of twice continuously differentiable functions, and furthermore we derive a representation formula for these functions.
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