A nonparametric belief propagation method for uncertainty quantification with applications to flow in random porous media

A probabilistic graphical model approach to uncertainty quantification for flows in random porous media is introduced. Model reduction techniques are used locally in the graph to represent the random permeability. Then the conditional distribution of the multi-output responses on the low dimensional representation of the permeability field is factorized into a product of local potential functions. An expectation-maximization algorithm is used to learn the nonparametric representation of these potentials using the given input/output data. We develop a nonparametric belief propagation method for uncertainty quantification by employing the loopy belief propagation algorithm. The nonparametric nature of our model is able to capture non-Gaussian features of the response. The proposed framework can be used as a surrogate model to predict the responses for new input realizations as well as our confidence on these predictions. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed framework for solving uncertainty quantification problems in flows through porous media using stationary and non-stationary permeability fields.

[1]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[2]  Jiang Wan,et al.  A probabilistic graphical model approach to stochastic multiscale partial differential equations , 2013, J. Comput. Phys..

[3]  N. L. Johnson,et al.  Multivariate Analysis , 1958, Nature.

[4]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[5]  Brendan J. Frey,et al.  Graphical Models for Machine Learning and Digital Communication , 1998 .

[6]  Marco F. Huber,et al.  Gaussian mixture reduction via clustering , 2009, 2009 12th International Conference on Information Fusion.

[7]  Yuri Boykov,et al.  A Scalable graph-cut algorithm for N-D grids , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[8]  Edoardo M Airoldi.  Probabilistic graphical models in computational biology , 2007 .

[9]  Paul Geladi,et al.  Principal Component Analysis , 1987, Comprehensive Chemometrics.

[10]  Ilias Bilionis,et al.  Multi-output local Gaussian process regression: Applications to uncertainty quantification , 2012, J. Comput. Phys..

[11]  Thomas J. R. Hughes,et al.  Mixed Discontinuous Galerkin Methods for Darcy Flow , 2005, J. Sci. Comput..

[12]  Andy Podgurski,et al.  The Probabilistic Program Dependence Graph and Its Application to Fault Diagnosis , 2010, IEEE Trans. Software Eng..

[13]  Michael I. Jordan.  Graphical Models , 2003 .

[14]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[15]  Yalchin Efendiev,et al.  Multiscale Finite Element Methods: Theory and Applications , 2009 .

[16]  Xiang Ma,et al.  An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations , 2009, J. Comput. Phys..

[17]  William T. Freeman,et al.  Efficient Multiscale Sampling from Products of Gaussian Mixtures , 2003, NIPS.

[18]  Lior Rokach,et al.  Data Mining And Knowledge Discovery Handbook , 2005 .

[19]  Hilbert J. Kappen,et al.  Sufficient Conditions for Convergence of Loopy Belief Propagation , 2005, UAI.

[20]  Michael Isard,et al.  PAMPAS: real-valued graphical models for computer vision , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[21]  Uwe D. Hanebeck,et al.  Progressive Gaussian mixture reduction , 2008, 2008 11th International Conference on Information Fusion.

[22]  Eric O. Postma,et al.  Dimensionality Reduction: A Comparative Review , 2008 .

[23]  Tamara G. Kolda,et al.  An overview of the Trilinos project , 2005, TOMS.

[24]  Knut-Andreas Lie,et al.  Modelling of Multiscale Structures in Flow Simulations for Petroleum Reservoirs , 2007, Geometric Modelling, Numerical Simulation, and Optimization.

[25]  Michael I. Jordan,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008, Found. Trends Mach. Learn..

[26]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[27]  William T. Freeman,et al.  Learning low-level vision , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[28]  Anders Logg,et al.  DOLFIN: a C++/Python Finite Element Library , 2012 .

[29]  Michael Isard,et al.  Nonparametric belief propagation , 2010, Commun. ACM.

[30]  H. Künsch.  Gaussian Markov random fields , 1979 .

[31]  Leonhard Held,et al.  Gaussian Markov Random Fields: Theory and Applications , 2005 .

[32]  Klaus Hackl,et al.  IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials , 2010 .

[33]  Peng Chen,et al.  Adaptive Locally Weighted Projection Regression Method for Uncertainty Quantification , 2013 .

[34]  Richard B. Lehoucq,et al.  On finite element discretizations of the pure Neumann problem , 2001 .

[35]  Nanning Zheng,et al.  Stereo Matching Using Belief Propagation , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[36]  J.A. Bilmes,et al.  Graphical model architectures for speech recognition , 2005, IEEE Signal Processing Magazine.

[37]  Martin J. Wainwright,et al.  Tree-based reparameterization framework for analysis of sum-product and related algorithms , 2003, IEEE Trans. Inf. Theory.

[38]  William T. Freeman,et al.  Correctness of Belief Propagation in Gaussian Graphical Models of Arbitrary Topology , 1999, Neural Computation.

[39]  Nir Friedman,et al.  Probabilistic Graphical Models - Principles and Techniques , 2009 .

[40]  Kenneth Y. Goldberg,et al.  Nonparametric belief propagation for distributed tracking of robot networks with noisy inter-distance measurements , 2009, 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[41]  David J. Salmond.  Mixture reduction algorithms for target tracking in clutter , 1990 .

[42]  Hongyuan Zha,et al.  Adaptive Manifold Learning , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[43]  Jason L. Williams,et al.  Convergence of loopy belief propagation for data association , 2010, 2010 Sixth International Conference on Intelligent Sensors, Sensor Networks and Information Processing.

[44]  Dragomir Anguelov,et al.  A General Algorithm for Approximate Inference and Its Application to Hybrid Bayes Nets , 1999, UAI.

[45]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[46]  D. N. Geary.  Mixture Models: Inference and Applications to Clustering , 1989 .

[47]  Fabio Tozeto Ramos,et al.  An integrated probabilistic model for scan-matching, moving object detection and motion estimation , 2010, 2010 IEEE International Conference on Robotics and Automation.