scholarly journals Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

2018 ◽  
Vol 34 (4) ◽  
pp. 045002 ◽  
Author(s):  
Sergios Agapiou ◽  
Martin Burger ◽  
Masoumeh Dashti ◽  
Tapio Helin
Keyword(s):  
Inverse Problems ◽  
Maximum A Posteriori ◽  
A Posteriori ◽  
Edge Preserving ◽  
Bayesian Inverse Problems ◽  
Non Parametric

scholarly journals Γ -convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems

2021 ◽  
Vol 38 (2) ◽  
pp. 025005
Author(s):  
Birzhan Ayanbayev ◽  
Ilja Klebanov ◽  
Han Cheng Lie ◽  
T J Sullivan
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Convergence Theory ◽  
A Posteriori ◽  
Edge Preserving ◽  
Bayesian Inverse Problems ◽  
Map Estimator ◽  
Γ Convergence ◽  
The Stability ◽  
Statistical Inverse Problem

Abstract The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

scholarly journals Technical Note: Improving the computational efficiency of sparse matrix multiplication in linear atmospheric inverse problems

2016 ◽  
Author(s):  
Vineet Yadav ◽  
Anna M. Michalak
Keyword(s):  
Inverse Problems ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
Matrix Multiplication ◽  
Technical Note ◽  
Covariance Matrices ◽  
A Posteriori ◽  
Bayesian Inverse Problems ◽  
Dense Matrices ◽  
The Cost

Abstract. Matrix multiplication of two sparse matrices is a fundamental operation in linear Bayesian inverse problems for computing covariance matrices of observations and a posteriori uncertainties. Applications of sparse-sparse matrix multiplication algorithms for specific use-cases in such inverse problems remain unexplored. Here we present a hybrid-parallel sparse-sparse matrix multiplication approach that is more efficient by a third in terms of execution time and operation count relative to standard sparse matrix multiplication algorithms available in most libraries. Two modifications of this hybrid-parallel algorithm are also proposed for the types of operations typical of atmospheric inverse problems, which further reduce the cost of sparse matrix multiplication by yielding only upper triangular and/or dense matrices.

scholarly journals A BAYESIAN APPROACH FOR THE SOLUTION OF INVERSE PROBLEMS TO ESTIMATE RADIATIVE PROPERTIES

2020 ◽  
Vol 5 (6) ◽  
Author(s):  
Cairo Martins Da Silva ◽  
Gustavo Antunes Guedes ◽  
Luiz Alberto da Silva Abreu ◽  
Diego Campos Knupp ◽  
Antônio José Da Silva Neto
Keyword(s):  
Heat Transfer ◽  
Monte Carlo ◽  
Inverse Problems ◽  
Optical Thickness ◽  
Radiative Heat ◽  
Single Scattering ◽  
Radiative Properties ◽  
Maximum A Posteriori ◽  
A Posteriori ◽  
Numerical Application

The main objective of the present work is related to the formulation and solutionof inverse problems in radiative heat transfer phenomena. The analysis consists in estimating parameters and functions of a participanting medium, such as optical thickness, single scattering albedo, diffusive reflectivities and phase function coefficients. It is performed with the numerical application of a Bayesian framework, which includes “Maximum a Posteriori” (MAP) and "Markov Chains Monte Carlo"(MCMC), within the Metropolis-Hastings procedure. These methodologiesproved to be effective for solving such problems.

scholarly journals Regularization, Bayesian Inference and Machine Learning methods for Inverse Problems†

2021 ◽  
Author(s):  
Ali Mohammad-Djafari
Keyword(s):  
Inverse Problems ◽  
Data Model ◽  
Prior Probability ◽  
Maximum A Posteriori ◽  
A Posteriori ◽  
Model Matching ◽  
Regularization Theory ◽  
Probability Models ◽  
Machine Learning Methods ◽  
Map Interpretation

Classical methods for inverse problems are mainly based on regularization theory. In particular those which are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two terms and great number of optimization algorithms have been proposed. When these two terms are distance or divergence measures, they can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond, respectively, to the likelihood and prior probability models.

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