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.

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

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1673
Author(s):  
Ali Mohammad-Djafari
Keyword(s):  
Machine Learning ◽  
Neural Networks ◽  
Inverse Problems ◽  
Data Model ◽  
Hidden Variables ◽  
Model Matching ◽  
Regularization Theory ◽  
Probability Models ◽  
Map Interpretation ◽  
And Inversion

Classical methods for inverse problems are mainly based on regularization theory, in particular those, that 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 a 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 to the likelihood and prior-probability models, respectively. The Bayesian approach gives more flexibility in choosing these terms and, in particular, the prior term via hierarchical models and hidden variables. However, the Bayesian computations can become very heavy computationally. The machine learning (ML) methods such as classification, clustering, segmentation, and regression, based on neural networks (NN) and particularly convolutional NN, deep NN, physics-informed neural networks, etc. can become helpful to obtain approximate practical solutions to inverse problems. In this tutorial article, particular examples of image denoising, image restoration, and computed-tomography (CT) image reconstruction will illustrate this cooperation between ML and inversion.

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 Threshold Computation for Spatially Coupled Turbo-Like Codes on the AWGN Channel

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 240
Author(s):  
Muhammad Umar Farooq ◽  
Alexandre Graell i Amat ◽  
Michael Lentmaier
Keyword(s):  
Gaussian Noise ◽  
Belief Propagation ◽  
Additive White Gaussian Noise ◽  
Maximum A Posteriori ◽  
Prediction Methods ◽  
Threshold Analysis ◽  
A Posteriori ◽  
Awgn Channel ◽  
Efficient Prediction ◽  
Binary Erasure Channel

In this paper, we perform a belief propagation (BP) decoding threshold analysis of spatially coupled (SC) turbo-like codes (TCs) (SC-TCs) on the additive white Gaussian noise (AWGN) channel. We review Monte-Carlo density evolution (MC-DE) and efficient prediction methods, which determine the BP thresholds of SC-TCs over the AWGN channel. We demonstrate that instead of performing time-consuming MC-DE computations, the BP threshold of SC-TCs over the AWGN channel can be predicted very efficiently from their binary erasure channel (BEC) thresholds. From threshold results, we conjecture that the similarity of MC-DE and predicted thresholds is related to the threshold saturation capability as well as capacity-approaching maximum a posteriori (MAP) performance of an SC-TC ensemble.

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