Bayesian inversion of log-normal eikonal equations

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the...

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Main Authors: Yeo, Zhan Fei, Hoang, Viet Ha
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2023
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Online Access:https://hdl.handle.net/10356/170119
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-1701192023-08-29T01:08:22Z Bayesian inversion of log-normal eikonal equations Yeo, Zhan Fei Hoang, Viet Ha School of Physical and Mathematical Sciences Science::Mathematics Eikonal Equation Bayesian Inverse Problems We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen-Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results. Ministry of Education (MOE) Nanyang Technological University Zhan Fei Yeo’s research is supported by the Nanyang President’s Graduate Scholarship. Viet Ha Hoang’s research is supported by the Singapore Ministry of Education Academic Research Fund Tier 2 Grant MOE2017-T2-2-144. 2023-08-29T01:08:22Z 2023-08-29T01:08:22Z 2023 Journal Article Yeo, Z. F. & Hoang, V. H. (2023). Bayesian inversion of log-normal eikonal equations. Inverse Problems, 39(6), 065007-. https://dx.doi.org/10.1088/1361-6420/acc888 0266-5611 https://hdl.handle.net/10356/170119 10.1088/1361-6420/acc888 2-s2.0-85153871892 6 39 065007 en MOE2017-T2-2-144 Inverse Problems © 2023 IOP Publishing Ltd. All rights reserved.
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics
Eikonal Equation
Bayesian Inverse Problems
spellingShingle Science::Mathematics
Eikonal Equation
Bayesian Inverse Problems
Yeo, Zhan Fei
Hoang, Viet Ha
Bayesian inversion of log-normal eikonal equations
description We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen-Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Yeo, Zhan Fei
Hoang, Viet Ha
format Article
author Yeo, Zhan Fei
Hoang, Viet Ha
author_sort Yeo, Zhan Fei
title Bayesian inversion of log-normal eikonal equations
title_short Bayesian inversion of log-normal eikonal equations
title_full Bayesian inversion of log-normal eikonal equations
title_fullStr Bayesian inversion of log-normal eikonal equations
title_full_unstemmed Bayesian inversion of log-normal eikonal equations
title_sort bayesian inversion of log-normal eikonal equations
publishDate 2023
url https://hdl.handle.net/10356/170119
_version_ 1779156254433738752