Iglesias, Marco A.; Lu, Yulong; Stuart, Andrew M. A Bayesian level set method for geometric inverse problems. (English) Zbl 1353.65050 Interfaces Free Bound. 18, No. 2, 181-217 (2016). Summary: We introduce a level set based approach to Bayesian geometric inverse problems. In these problems the interface between different domains is the key unknown, and is realized as the level set of a function. This function itself becomes the object of the inference. Whilst the level set methodology has been widely used for the solution of geometric inverse problems, the Bayesian formulation that we develop here contains two significant advances: firstly it leads to a well-posed inverse problem in which the posterior distribution is Lipschitz with respect to the observed data, and may be used to not only estimate interface locations, but quantify uncertainty in them; and secondly it leads to computationally expedient algorithms in which the level set itself is updated implicitly via the Markov chain Monte Carlo methodology applied to the level set function – no explicit velocity field is required for the level set interface. Applications are numerous and include medical imaging, modelling of subsurface formations and the inverse source problem; our theory is illustrated with computational results involving the last two applications. Cited in 1 ReviewCited in 31 Documents MSC: 65J15 Numerical solutions to equations with nonlinear operators 65J22 Numerical solution to inverse problems in abstract spaces 65C05 Monte Carlo methods 65C40 Numerical analysis or methods applied to Markov chains 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 92C55 Biomedical imaging and signal processing Keywords:inverse problems; Bayesian level set method; Markov chain Monte Carlo method; numerical example; Poisson equation; medical imaging PDFBibTeX XMLCite \textit{M. A. Iglesias} et al., Interfaces Free Bound. 18, No. 2, 181--217 (2016; Zbl 1353.65050) Full Text: DOI arXiv References: [1] Adler, R. J. & Taylor, J. E.,Random Fields and Geometry, vol. 115 of Springer Monographs in Mathematics. Springer, 2007.Zbl1149.60003 MR2319516 · Zbl 1149.60003 [2] Alessandrini, G., On the identification of the leading coefficient of an elliptic equation. Boll. Un. Mat. Ital. C (6) 4 (1985), 87–111.Zbl0598.35129 MR0805207 · Zbl 0598.35129 [3] Ameur, H. B., Burger, M. & Hackl, B., Level set methods for geometric inverse problems in linear elasticity. Inverse Problems 20 (2004), 673.Zbl1086.35117 MR2067495 · Zbl 1086.35117 [4] Angenent, S., Ilmanen, T. & Chopp, D. L., A computed example of nonuniqueness of mean curvature flow in R3. Comm. Partial Differential Equations 20 (1995), 1937–1958.MR1361726 [5] Arbogast, T., Wheeler, M. F. & Yotov, I.,Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997), 828–852.Zbl0880. 65084 MR1442940 · Zbl 0880.65084 [6] Armstrong, M., Galli, A., Beucher, H., Le Loc’h, G., Renard, D., Doligez, B., Eschard, R. & Geffroy, F., Plurigaussian Simulations in Geosciences, 2nd revised edition. 2011.Zbl1047.86009 [7] Astrakova, A. & Oliver, D., Conditioning truncated pluri-Gaussian models to facies observations in ensemble-Kalman-based data assimilation. Mathematical Geosciences (2014), Published online April 2014.Zbl1323.86039 · Zbl 1323.86039 [8] Bogachev, V. I., Gaussian Measures, vol. 62 of Mathematical Surveys and Monographs. American Mathematical Society, 1998.Zbl0913.60035 MR1642391 [9] Bonito, A., DeVore, R. A. & Nochetto, R. H., Adaptive finite element methods for elliptic problems with 216M.IGLESIAS,Y.LU AND A.M.STUART · Zbl 1285.65078 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.