Variational Bayesian approximation of inverse problems using sparse precision matrices

Povala, Jan; Kazlauskaite, Ieva; Febrianto, Eky; Cirak, Fehmi; Girolami, Mark

doi:10.1016/j.cma.2022.114712

Cited by 14 publications

(4 citation statements)

References 54 publications

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“…Existing iterative inference algorithms are either to find a point-estimate or to approximate the posterior pdf itself. The former mainly includes gradientbased, heuristic and data assimilation methods for computing the maximum a posterior (MAP) or maximum likelihood estimate (MLE), while the latter includes MCMC and variational Bayes approaches (Posselt, 2013;Emerick and Reynolds, 2013;Povala et al, 2022). All these methods need repetitive evaluations of the forward model in the inference process, which can be computationally expensive for high-dimensional problems.…”

Section: The Inverse Problem and Bayesian Uncertainty Quantificationmentioning

confidence: 99%

“…On the other hand, methods to approximate the true posterior pdf mainly include Markov Chain Monte Carlo (MCMC), variational Bayes and ensemble-based methods (e.g. ensemble Kalman filter and ensemble smoother) (Olalotiti-Lawal and Datta-Gupta, 2018;Emerick and Reynolds, 2013;Song et al, 2021;Povala et al, 2022). These methods approximate the posterior pdf using sample distributions, in the case of MCMC and ensemble-based methods, or trial functions in the case of variational Bayes.…”

Section: Introductionmentioning

confidence: 99%

“…For linear Gaussian inverse problems, analytical solutions exist for Kalman filter and variational Bayes (Emerick and Reynolds, 2013;Povala et al, 2022). For nonlinear and/or non-Gaussian inverse problems, almost all existing methods approximate the posterior (MAP, MLE or pdf) via updating the parameters iteratively, where forward simulation runs are needed in each iteration to evaluate the objective function.…”

Section: Introductionmentioning

confidence: 99%

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We discuss three developments in the study of the tangential Casimir force. The first one generalizes the force from dielectrics to magnetodielectrics. This generalization is readily realized with the help of working out the total zero-point energy of multilayered magnetodielectrics. The second one revisits the tangential force for real conductors by taking into account the temperature dependence of their dielectric constants, and provides needed theoretical results for experimental investigations that are expected to be conducted at room temperature. The third investigates the Casimir torque from plates made of isotropic media, which offers a simple way to realize torques for uncharged surfaces.

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Section: The Inverse Problem and Bayesian Uncertainty Quantificationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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“…e Bayesian perspective to inverse problems is developed rapidly in the past two decades. ere are two main points about the algorithms in the Bayesian inversion: MCMC-based sampling methods [19][20][21][22][23][24][25][26][27] and variational methods [28][29][30]. For the MCMCbased sampling algorithms, some fast numerical techniques are studied by many authors.…”

Section: Introductionmentioning

confidence: 99%

A Homotopy Bayesian Approach for Inverse Problems

Yang

Deng

2022

Mathematical Problems in Engineering

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In solving Bayesian inverse problems, it is often desirable to use a common density parameterization to characterize the prior and posterior. Typically, we seek an optimal approximation of the true posterior from the same distribution family as the prior. As one of the most important classes of distributions in statistics, the exponential family is considered as the parameterization. The optimal parameter values for representing the approximated posterior are achieved by minimizing the deviation between the parameterized density and a homotopy that deforms the prior density into the posterior density. Instead of the usual moment parameters, we introduce a new parameter system in the process, by which we reduce the dimension of the parameter and therefore the computational cost. The parameters are ruled by a system of explicit first-order differential equations. Solving this system over finite “time” interval yields the desired optimal density parameters. This method is proven to be effective using some numerical examples.

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VI-DGP: A Variational Inference Method with Deep Generative Prior for Solving High-Dimensional Inverse Problems

Xia,

Liao,

2023

J Sci Comput

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Variational Bayesian approximation of inverse problems using sparse precision matrices

Cited by 14 publications

References 54 publications

Research on the Current Situation of Korean Literature Teaching in Qingdao National University, Shandong Province, China

Research on the Current Situation of Korean Literature Teaching in Qingdao National University, Shandong Province, China

A Homotopy Bayesian Approach for Inverse Problems

VI-DGP: A Variational Inference Method with Deep Generative Prior for Solving High-Dimensional Inverse Problems

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