A Bayesian Level Set Method for Geometric Inverse Problems

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 secondly it leads to computationally expedient algorithms in which the level set itself is updated implicitly via the MCMC 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.

[1]  S. B. Childs,et al.  INVERSE PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS. , 1968 .

[2]  Eric T. Chung,et al.  Electrical impedance tomography using level set representation and total variational regularization , 2005 .

[3]  Yalchin Efendiev,et al.  Uncertainty Quantification in History Matching of Channelized Reservoirs using Markov Chain Level Set Approaches , 2011, ANSS 2011.

[4]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[5]  Faming Liang,et al.  Statistical and Computational Inverse Problems , 2006, Technometrics.

[6]  F. Santosa.  A Level-set Approach Inverse Problems Involving Obstacles , 1995 .

[7]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[8]  W. Rundell,et al.  Iterative methods for the reconstruction of an inverse potential problem , 1996 .

[9]  Margaret Armstrong,et al.  Plurigaussian Simulations in Geosciences , 2014 .

[10]  Marco A. Iglesias,et al.  Level-set techniques for facies identification in reservoir modeling , 2011 .

[11]  Omar Ghattas,et al.  An Analysis of Infinite Dimensional Bayesian Inverse Shape Acoustic Scattering and Its Numerical Approximation , 2014, SIAM/ASA J. Uncertain. Quantification.

[12]  Jing Ping,et al.  History Matching of Channelized Reservoirs With Vector-Based Level-Set Parameterization , 2014 .

[13]  T. Chan,et al.  Multiple level set methods with applications for identifying piecewise constant functions , 2004 .

[14]  Marco A. Iglesias,et al.  Well-posed Bayesian geometric inverse problems arising in subsurface flow , 2014, 1401.5571.

[15]  Tony F. Chan,et al.  A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model , 2002, International Journal of Computer Vision.

[16]  Stanley Osher,et al.  A survey on level set methods for inverse problems and optimal design , 2005, European Journal of Applied Mathematics.

[17]  Charles M. Elliott,et al.  Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient , 2015, 1504.01935.

[18]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[19]  G. Roberts,et al.  MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster , 2012, 1202.0709.

[20]  Rolf Johan Lorentzen,et al.  Estimating Facies Fields by Use of the Ensemble Kalman Filter and Distance Functions--Applied to Shallow-Marine Environments , 2013 .

[21]  Marco Antonio Iglesias-Hernandez.  An iterative representer-based scheme for data inversion in reservoir modeling , 2008 .

[22]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 3. Application to Synthetic and Field Data , 1986 .

[23]  O. Dorn,et al.  Level set methods for inverse scattering , 2006 .

[24]  Dean S. Oliver,et al.  Ensemble Kalman filter for automatic history matching of geologic facies , 2005 .

[25]  V. Bogachev.  Gaussian Measures on a , 2022 .

[26]  D. Oliver,et al.  Conditioning Truncated Pluri-Gaussian Models to Facies Observations in Ensemble-Kalman-Based Data Assimilation , 2015, Mathematical Geosciences.

[27]  Stanley Osher,et al.  Level Set Methods, with an Application to Modeling the Growth of Thin Films , 2019, Free boundary problems:.

[28]  Rolf Johan Lorentzen,et al.  History Matching Channelized Reservoirs Using the Ensemble Kalman Filter , 2012 .

[29]  M. Burger,et al.  Level set methods for geometric inverse problems in linear elasticity , 2004 .

[30]  A. Stuart,et al.  The Bayesian Approach to Inverse Problems , 2013, 1302.6989.

[31]  Alper Yilmaz,et al.  Level Set Methods , 2007, Wiley Encyclopedia of Computer Science and Engineering.

[32]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[33]  D. Chopp,et al.  A Computed example of nonuniqueness of mean curvature flow in R3 , 1995 .

[34]  F. FRÜHAUF,et al.  Analysis of Regularization Methods for the Solution of Ill-Posed Problems Involving Discontinuous Operators , 2005, SIAM J. Numer. Anal..

[35]  E. Stein,et al.  Real Analysis: Measure Theory, Integration, and Hilbert Spaces , 2005 .

[36]  Andrew Gelman,et al.  Handbook of Markov Chain Monte Carlo , 2011 .

[37]  M. Wheeler,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences , 1997 .

[38]  M. Burger.  A framework for the construction of level set methods for shape optimization and reconstruction , 2003 .

[39]  M. Hanke.  A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems , 1997 .

[40]  Ricardo H. Nochetto,et al.  Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients , 2013, SIAM J. Numer. Anal..

[41]  Ning Liu,et al.  Inverse Theory for Petroleum Reservoir Characterization and History Matching , 2008 .

[42]  M. Burger.  A level set method for inverse problems , 2001 .

[43]  Lin Wang,et al.  Binary Tomography Reconstructions With Stochastic Level-Set Methods , 2015, IEEE Signal Processing Letters.

[44]  Marco A. Iglesias,et al.  Evaluation of Gaussian approximations for data assimilation in reservoir models , 2012, Computational Geosciences.

[45]  Victor Isakov,et al.  Inverse Source Problems , 1990 .

[46]  Lea Fleischer,et al.  Regularization of Inverse Problems , 1996 .

[47]  G. Richter.  An Inverse Problem for the Steady State Diffusion Equation , 1981 .

[48]  A. Stuart,et al.  Ensemble Kalman methods for inverse problems , 2012, 1209.2736.

[49]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[50]  Xue-Cheng Tai,et al.  A variant of the level set method and applications to image segmentation , 2006, Math. Comput..

[51]  David Isaacson,et al.  Electrical Impedance Tomography , 1999, SIAM Rev..

[52]  Andrew Gelman,et al.  General methods for monitoring convergence of iterative simulations , 1998 .

[53]  Dominique Lesselier,et al.  Level set methods for inverse scattering—some recent developments , 2009 .