An optimization problem based on a Bayesian approach for the 2D Helmholtz equation
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ORIGINAL ARTICLE
An optimization problem based on a Bayesian approach for the 2D Helmholtz equation Lili Guadarrama1
•
Carlos Prieto1 • Elijah Van Houten2
Received: 25 July 2019 / Accepted: 4 July 2020 Ó Sociedad Matemática Mexicana 2020
Abstract Elastography is a ill-posed inverse problem that aims at recovering the Lame´ and density of the domain of interest from finite number of observations, we consider as model the Helmoltz equation. We present an implementation for solving the Helmholtz inverse problem in two dimensions via an optimization problem based on Bayesian approach. In addition, the accuracy of the method is also investigated with respect to the amount of information taken from the generalized Hermitian Eigenvalue problem and by comparing the maximum a posterior estimate to the true parameter distribution in simulated experiments. Keywords Helmholtz equation Inverse problem Bayesian inference Elastography
Mathematics Subject Classification 35R30 65N21
1 Introduction We consider the inverse problem of recovering coefficients in the Helmholtz equation from noisy observations of the solution in a 2D region. The main motivations of this work are due to the above equation is one of the most common mathematical models for elasticity imaging or elastography which is an emerging technique for non-invasive imaging that has a broad application arising in & Lili Guadarrama [email protected] Carlos Prieto [email protected] Elijah Van Houten [email protected] 1
Centro de Investigacio´n en Matema´ticas, Guanajuato, Mexico
2
Universite´ de Sherbrooke, Sherbrooke, Canada
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biomedical imaging [3], non-destructive testing [11], exploration geophysics [2] among others. Elastography is a very active filed that has evolved during the past 2 decades, and many approaches have been developed and studied. As it is well known, the inverse problems are ill-posed problems and regularization is required to ameliorate this behavior; in the presence of noisy data, the Bayesian approach has been increasingly used for regularization [8, 15, 19], because it also allows for quantification of uncertainty. An inverse problem can be expressed in a statistical inference framework by defining a set of observations (dobs 2 Rn ), the forward problem that governs the phenomena and a priori information of parameters (l 2 Rk ). The solution of the inverse problem is the posterior probability distribution, ppost : ppost ðljdobs Þ / plike ðdobs jlÞpprior ðlÞ;
ð1Þ
where plike and pprior are the likelihood and the a priori probability distributions, respectively. To explore the posterior probability density, we need sampling methods. In this work, an additive Gaussian noise model for the observations and a Gaussian random field with constant mean was chosen as prior model, so the posterior probability distribution, ppost , is a Gaussian multivariate probability distribution; to compute the parameters of the Gaussian posterior distribution, an optimization problem is proposed. Gi
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