Multilevel approximation of Gaussian random fields: Fast simulation

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain 𝒟⊂ ℝn o...

[1]  Alan Demlow,et al.  Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces , 2009, SIAM J. Numer. Anal..

[2]  Christoph Schwab,et al.  Quasi-Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights , 2018, SIAM J. Numer. Anal..

[3]  L. Herrmann,et al.  Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients , 2019, ESAIM: Mathematical Modelling and Numerical Analysis.

[4]  Frances Y. Kuo,et al.  Fast random field generation with H-matrices , 2017, Numerische Mathematik.

[5]  E. Davies,et al.  Spectral Theory and Differential Operators: Index , 1995 .

[6]  Pierre Vandergheynst,et al.  Wavelets on Graphs via Spectral Graph Theory , 2009, ArXiv.

[7]  Joseph E. Pasciak,et al.  Numerical approximation of fractional powers of elliptic operators , 2013, Math. Comput..

[8]  Panayot S. Vassilevski,et al.  Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes , 2017, Numer. Linear Algebra Appl..

[9]  Rob Stevenson,et al.  Finite element wavelets with improved quantitative properties , 2009 .

[10]  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.

[11]  Christoph Schwab,et al.  QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights , 2018, Numerische Mathematik.

[12]  Panayot S. Vassilevski,et al.  A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields , 2017, SIAM J. Sci. Comput..

[13]  Annika Lang,et al.  Fast generation of isotropic Gaussian random fields on the sphere , 2017, Monte Carlo Methods Appl..

[14]  Frances Y. Kuo,et al.  Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients , 2018, Numerische Mathematik.

[15]  Joseph E. Pasciak,et al.  On the stability of the L2 projection in H1(Omega) , 2002, Math. Comput..

[16]  Elisabeth Ullmann,et al.  Fast sampling of parameterised Gaussian random fields , 2018, Computer Methods in Applied Mechanics and Engineering.

[17]  Nicholas J. Higham,et al.  Computing AAlpha, log(A), and Related Matrix Functions by Contour Integrals , 2008, SIAM J. Numer. Anal..

[18]  A. V. Balakrishnan,et al.  Fractional powers of closed operators and the semigroups generated by them. , 1960 .

[19]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[20]  Rob Stevenson,et al.  A quadratic finite element wavelet Riesz basis , 2018, Int. J. Wavelets Multiresolution Inf. Process..

[21]  Karl J. Friston,et al.  Bayesian fMRI time series analysis with spatial priors , 2005, NeuroImage.

[22]  James Demmel,et al.  A Supernodal Approach to Sparse Partial Pivoting , 1999, SIAM J. Matrix Anal. Appl..

[23]  Joseph E. Pasciak,et al.  New estimates for multilevel algorithms including the V-cycle , 1993 .

[24]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[25]  H. Rue,et al.  An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach , 2011 .

[26]  Robert J. Scherrer,et al.  Topology of large-scale structure in seeded hot dark matter models , 1992 .

[27]  Andrew M. Stuart,et al.  Uncertainty Quantification in Graph-Based Classification of High Dimensional Data , 2017, SIAM/ASA J. Uncertain. Quantification.

[28]  F. Lindgren,et al.  Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping , 2011, 1104.3436.

[29]  H. Rue,et al.  Spatio-temporal modeling of particulate matter concentration through the SPDE approach , 2012, AStA Advances in Statistical Analysis.

[30]  C. Schwab,et al.  Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations , 2013, 1305.1170.

[31]  Andrew T. A. Wood,et al.  Algorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields , 1997 .

[32]  W. Dahmen.  Multiscale and Wavelet Methods for Operator Equations , 2003 .

[33]  Ronald R. Coifman,et al.  Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems , 2008, Multiscale Model. Simul..

[34]  Yunrong Zhu,et al.  Multilevel Preconditioners for Reaction-Diffusion Problems with Discontinuous Coefficients , 2014, J. Sci. Comput..

[35]  Jinchao Xu,et al.  The method of alternating projections and the method of subspace corrections in Hilbert space , 2002 .

[36]  S. Torquato.  Random Heterogeneous Materials , 2002 .

[37]  David Bolin,et al.  The Rational SPDE Approach for Gaussian Random Fields With General Smoothness , 2017, Journal of Computational and Graphical Statistics.

[38]  Marie E. Rognes,et al.  Efficient White Noise Sampling and Coupling for Multilevel Monte Carlo with Nonnested Meshes , 2018, SIAM/ASA J. Uncertain. Quantification.

[39]  David Bolin,et al.  Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise , 2017, BIT Numerical Mathematics.

[40]  Stephan R. Sain,et al.  A spatial analysis of multivariate output from regional climate models , 2011, 1104.2703.

[41]  David Bolin,et al.  Numerical solution of fractional elliptic stochastic PDEs with spatial white noise , 2017, IMA Journal of Numerical Analysis.

[42]  Helmut Harbrecht,et al.  Covariance regularity and $$\mathcal {H}$$H-matrix approximation for rough random fields , 2017, Numerische Mathematik.

[43]  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.

[44]  Rob Stevenson,et al.  Adaptive wavelet methods for solving operator equations: An overview , 2009 .

[45]  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..