Numerical Methods for Bayesian Inference in Hilbert Spaces

[1]  Albert Cohen,et al.  Approximation of high-dimensional parametric PDEs * , 2015, Acta Numerica.

[2]  G. Evensen.  Data Assimilation: The Ensemble Kalman Filter , 2006 .

[3]  D. Krige.  A statistical approach to some basic mine valuation problems on the Witwatersrand, by D.G. Krige, published in the Journal, December 1951 : introduction by the author , 1951 .

[4]  R. DeVore,et al.  Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs , 2010 .

[5]  John F. Pickens,et al.  Ground-water flow modeling of the Culebra dolomite , 1990 .

[6]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[7]  Robert Schaback,et al.  Interpolation of spatial data – A stochastic or a deterministic problem? , 2013, European Journal of Applied Mathematics.

[8]  Will Light,et al.  Approximation Theory in Tensor Product Spaces , 1985 .

[9]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[10]  Omar M. Knio,et al.  Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function , 2015, 1501.03323.

[11]  B. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[12]  Christoph Schwab,et al.  N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs , 2014 .

[13]  A. Gelman,et al.  Adaptively Scaling the Metropolis Algorithm Using Expected Squared Jumped Distance , 2007 .

[14]  Geir Evensen,et al.  The Ensemble Kalman Filter: theoretical formulation and practical implementation , 2003 .

[15]  N. Wiener.  The Homogeneous Chaos , 1938 .

[16]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[17]  R. Baierlein.  Probability Theory: The Logic of Science , 2004 .

[18]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[19]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[20]  Alexandros Beskos,et al.  Asymptotic analysis of the random walk Metropolis algorithm on ridged densities , 2015, The Annals of Applied Probability.

[21]  A. Stuart,et al.  ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE NONLINEAR CASE , 2006, math/0601092.

[22]  Non-asymptotic mixing of the MALA algorithm August 20 , 2010 , 2010 .

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

[24]  Andrew M. Stuart,et al.  Evaluating Data Assimilation Algorithms , 2011, ArXiv.

[25]  M. Rao.  Conditional measures and applications , 1993 .

[26]  Jens L. Eftang,et al.  Reduced basis methods for parametrized partial differential equations , 2011 .

[27]  W. H. Chung,et al.  5. Stochastic Processes and Stochastic Calculus , 2008 .

[28]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[29]  Claude Jeffrey Gittelson,et al.  Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* , 2011, Acta Numerica.

[30]  Ilse C. F. Ipsen,et al.  Randomized matrix-free trace and log-determinant estimators , 2016, Numerische Mathematik.

[31]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[32]  J. Rosenthal,et al.  Optimal scaling of discrete approximations to Langevin diffusions , 1998 .

[33]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[34]  Tiangang Cui,et al.  Dimension-independent likelihood-informed MCMC , 2014, J. Comput. Phys..

[35]  H. Bandemer,et al.  Mathematics of Uncertainty: Ideas, Methods, Application Problems (Studies in Fuzziness and Soft Computing) , 2006 .

[36]  H. Haario,et al.  An adaptive Metropolis algorithm , 2001 .

[37]  Hermann G. Matthies,et al.  Parameter Identification in a Probabilistic Setting , 2012, ArXiv.

[38]  Mario Ullrich,et al.  Positivity of hit-and-run and related algorithms , 2012, 1212.4512.

[39]  C. J. Gittelson.  STOCHASTIC GALERKIN DISCRETIZATION OF THE LOG-NORMAL ISOTROPIC DIFFUSION PROBLEM , 2010 .

[40]  L. Pratelli,et al.  Almost sure weak convergence of random probability measures , 2006 .

[41]  Andrew M. Stuart,et al.  Geometric MCMC for infinite-dimensional inverse problems , 2016, J. Comput. Phys..

[42]  E. Hille,et al.  Contributions to the theory of Hermitian series , 1939 .

[43]  Matjaž Omladič,et al.  Spectrum of the product of operators , 1988 .

[44]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[45]  K. A. Cliffe,et al.  Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients , 2011, Comput. Vis. Sci..

[46]  Charles J. Geyer,et al.  Introduction to Markov Chain Monte Carlo , 2011 .

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

[48]  R. Adler.  The Geometry of Random Fields , 2009 .

[49]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[50]  L. Tierney.  A note on Metropolis-Hastings kernels for general state spaces , 1998 .

[51]  Oliver G. Ernst,et al.  Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables - with Application to Lognormal Elliptic Diffusion Problems , 2016 .

[52]  Gideon Simpson,et al.  Algorithms for Kullback-Leibler Approximation of Probability Measures in Infinite Dimensions , 2014, SIAM J. Sci. Comput..

[53]  J. Potthoff.  Sample properties of random fields III : Differentiability , 2010 .

[54]  Yevgeniy Kovchegov,et al.  A class of Markov chains with no spectral gap , 2011, 1109.6050.

[55]  Gareth Roberts,et al.  Optimal scalings for local Metropolis--Hastings chains on nonproduct targets in high dimensions , 2009, 0908.0865.

[56]  A. Stuart,et al.  MAP estimators and their consistency in Bayesian nonparametric inverse problems , 2013, 1303.4795.

[57]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[58]  Giuseppe Da Prato,et al.  Second Order Partial Differential Equations in Hilbert Spaces: Bibliography , 2002 .

[59]  Dan Simon,et al.  Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches , 2006 .

[60]  A. M. Stuart,et al.  Sparse deterministic approximation of Bayesian inverse problems , 2011, 1103.4522.

[61]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[62]  A. Mandelbaum,et al.  Linear estimators and measurable linear transformations on a Hilbert space , 1984 .

[63]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[64]  Hamidou Tembine,et al.  Deterministic Mean-Field Ensemble Kalman Filtering , 2014, SIAM J. Sci. Comput..

[65]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

[66]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[67]  Hermann G. Matthies,et al.  A deterministic filter for non-Gaussian Bayesian estimation— Applications to dynamical system estimation with noisy measurements , 2012 .

[68]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[69]  S. Cambanis,et al.  Gaussian Processes and Gaussian Measures , 1972 .

[70]  G. Prato.  An Introduction to Infinite-Dimensional Analysis , 2006 .

[71]  P. Peskun,et al.  Optimum Monte-Carlo sampling using Markov chains , 1973 .

[72]  R. Tweedie,et al.  Exponential convergence of Langevin distributions and their discrete approximations , 1996 .

[73]  R. Freeze.  A stochastic‐conceptual analysis of one‐dimensional groundwater flow in nonuniform homogeneous media , 1975 .

[74]  D. Rudolf,et al.  Explicit error bounds for Markov chain Monte Carlo , 2011, 1108.3201.

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

[76]  Eileen Simkiss.  Weighing the Odds , 1950, Postgraduate medical journal.

[77]  A. Stuart,et al.  Sampling the posterior: An approach to non-Gaussian data assimilation , 2007 .

[78]  A. M. Stuart,et al.  Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems , 2016, SIAM/ASA J. Uncertain. Quantification.

[79]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[80]  O. L. Maître,et al.  Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics , 2010 .

[81]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[82]  Julia Charrier,et al.  Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients , 2012, SIAM J. Numer. Anal..

[83]  J. Rosenthal,et al.  Geometric Ergodicity and Hybrid Markov Chains , 1997 .

[84]  Oliver G. Ernst,et al.  Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case , 2014 .

[85]  F. Smithies.  Linear Operators , 2019, Nature.

[86]  Elisabeth Ullmann,et al.  Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients , 2012, Numerische Mathematik.

[87]  A. Stordal,et al.  Bridging the ensemble Kalman filter and particle filters: the adaptive Gaussian mixture filter , 2011 .

[88]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

[89]  Kody J. H. Law.  Proposals which speed up function-space MCMC , 2014, J. Comput. Appl. Math..

[90]  René L. Schilling.  Measures, Integrals and Martingales: Martingales , 2005 .

[91]  Ralph C. Smith,et al.  Uncertainty Quantification: Theory, Implementation, and Applications , 2013 .

[92]  D. Xiu,et al.  Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos , 2002 .

[93]  H. Widom.  Asymptotic behavior of the eigenvalues of certain integral equations , 1963 .

[94]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

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

[96]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

[97]  Michael Oberguggenberger,et al.  The mathematics of uncertainty: models, methods and interpretations , 2005 .

[98]  J. Mandel,et al.  On the convergence of the ensemble Kalman filter , 2009, Applications of mathematics.

[99]  E. Ullmann.  Solution strategies for stochastic finite element discretizations , 2008 .

[100]  F. Gland,et al.  Large sample asymptotics for the ensemble Kalman filter , 2009 .

[101]  Albert Cohen,et al.  Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients , 2015, 1509.07050.

[102]  Peter D. Hoff,et al.  A First Course in Bayesian Statistical Methods , 2009 .

[103]  Daniel Rudolf,et al.  On a Generalization of the Preconditioned Crank–Nicolson Metropolis Algorithm , 2015, Found. Comput. Math..

[104]  J. Boyd.  The rate of convergence of Hermite function series , 1980 .

[105]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[106]  S. Varadhan,et al.  Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions , 1986 .

[107]  James Martin,et al.  A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion , 2012, SIAM J. Sci. Comput..

[108]  G. Evensen,et al.  An ensemble Kalman smoother for nonlinear dynamics , 2000 .

[109]  Achim Klenke,et al.  Probability theory - a comprehensive course , 2008, Universitext.

[110]  Hans-Jörg Starkloff,et al.  ON THE CONVERGENCE OF THE STOCHASTIC GALERKIN METHOD FOR RANDOM ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 2013 .

[111]  Oliver G. Ernst,et al.  Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems , 2015, SIAM/ASA J. Uncertain. Quantification.

[112]  John P. Boyd,et al.  Asymptotic coefficients of hermite function series , 1984 .

[113]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[114]  W. Hackbusch,et al.  Hierarchical Matrices: Algorithms and Analysis , 2015 .

[115]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[116]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[117]  R. Douc,et al.  Practical drift conditions for subgeometric rates of convergence , 2004, math/0407122.

[118]  G. Szegő.  Zeros of orthogonal polynomials , 1939 .

[119]  Anthony Lee,et al.  Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation , 2012, 1210.6703.

[120]  José M. Bernardo,et al.  Bayesian Statistics , 2011, International Encyclopedia of Statistical Science.

[121]  A. Sokal,et al.  Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality , 1988 .

[122]  A. P. Dawid,et al.  Regression and Classification Using Gaussian Process Priors , 2009 .

[123]  R. C. Gilbert.  Introduction to Hilbert Space (S. K. Berberian) , 1963 .

[124]  Hermann G. Matthies,et al.  Sampling-free linear Bayesian update of polynomial chaos representations , 2012, J. Comput. Phys..

[125]  W. J. Padgett,et al.  Strong laws of large numbers for normed linear spaces , 1973 .

[126]  Kung-Sik Chan,et al.  Asymptotic Efficiency of the Sample Mean in Markov Chain Monte Carlo Schemes , 1996 .

[127]  J. Rosenthal,et al.  Optimal scaling for various Metropolis-Hastings algorithms , 2001 .

[128]  Christoph Schwab,et al.  Karhunen-Loève approximation of random fields by generalized fast multipole methods , 2006, J. Comput. Phys..

[129]  Juan Galvis,et al.  Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity , 2009, SIAM J. Numer. Anal..

[130]  Mark A. Girolami,et al.  Emulation of higher-order tensors in manifold Monte Carlo methods for Bayesian Inverse Problems , 2015, J. Comput. Phys..

[131]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[132]  A. Debussche,et al.  Weak truncation error estimates for elliptic PDEs with lognormal coefficients , 2013 .

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

[134]  T. J. Dodwell,et al.  A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow , 2013, SIAM/ASA J. Uncertain. Quantification.

[135]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[136]  Henning Omre,et al.  The Ensemble Kalman Filter and Related Filters , 2010 .

[137]  Colin Fox,et al.  Efficiency and computability of MCMC with Langevin, Hamiltonian, and other matrix-splitting proposals , 2015, 1501.03150.

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

[139]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[140]  Tiangang Cui,et al.  Optimal Low-rank Approximations of Bayesian Linear Inverse Problems , 2014, SIAM J. Sci. Comput..

[141]  Fabio Nobile,et al.  An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[142]  Oliver G. Ernst,et al.  Bayesian Inverse Problems and Kalman Filters , 2014 .

[143]  A. Stuart,et al.  Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions , 2011, 1112.1392.

[144]  Andrew M. Stuart,et al.  Analysis of the Ensemble Kalman Filter for Inverse Problems , 2016, SIAM J. Numer. Anal..

[145]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[146]  Nicola Stone,et al.  Gaussian Process Emulators for Uncertainty Analysis in Groundwater Flow , 2011 .

[147]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[148]  M. Burger,et al.  Maximum a posteriori estimates in linear inverse problems with log-concave priors are proper Bayes estimators , 2014, 1402.5297.

[149]  Roger Ghanem,et al.  Characterization of reservoir simulation models using a polynomial chaos‐based ensemble Kalman filter , 2009 .

[150]  Adrian Sandu,et al.  A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems , 2010 .

[151]  Youssef M. Marzouk,et al.  Bayesian inference with optimal maps , 2011, J. Comput. Phys..

[152]  Roger Ghanem,et al.  Robust System Identification of Strongly Non-linear Dynamics Using a Polynomial Chaos-Based Sequential Data Assimilation Technique , 2007 .

[153]  Martin Hairer,et al.  An Introduction to Stochastic PDEs , 2009, 0907.4178.

[154]  Elisabeth Ullmann,et al.  Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods , 2013, 1312.6047.

[155]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[156]  Jerzy Zabczyk,et al.  Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach , 2007 .

[157]  C. R. Dietrich,et al.  Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix , 1997, SIAM J. Sci. Comput..

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

[159]  G. Roberts,et al.  MCMC methods for diffusion bridges , 2008 .

[160]  O. Ernst,et al.  ON THE CONVERGENCE OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS , 2011 .

[161]  Jinglai Li,et al.  On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inference , 2017, J. Comput. Phys..

[162]  M. Girolami,et al.  Riemann manifold Langevin and Hamiltonian Monte Carlo methods , 2011, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[163]  S. Lakshmivarahan,et al.  Dynamic Data Assimilation: Index , 2006 .

[164]  G. Evensen.  The ensemble Kalman filter for combined state and parameter estimation , 2009, IEEE Control Systems.

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

[166]  Andrew M. Stuart,et al.  Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem , 2011, SIAM J. Numer. Anal..

[167]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[168]  Samuel Livingstone,et al.  Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance , 2015 .

[169]  Catherine E. Powell,et al.  An Introduction to Computational Stochastic PDEs , 2014 .