Approximation of probability density functions by the Multilevel Monte Carlo Maximum Entropy method

We develop a complete convergence theory for the Maximum Entropy method based on moment matching for a sequence of approximate statistical moments estimated by the Multilevel Monte Carlo method. Under appropriate regularity assumptions on the target probability density function, the proposed method is superior to the Maximum Entropy method with moments estimated by the Monte Carlo method. New theoretical results are illustrated in numerical examples.

[1]  P. Davis.  Interpolation and approximation , 1965 .

[2]  Michael B. Giles.  Multilevel Monte Carlo methods , 2015, Acta Numerica.

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

[4]  Andrea Barth,et al.  Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients , 2011, Numerische Mathematik.

[5]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[6]  S. Kullback,et al.  A lower bound for discrimination information in terms of variation (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[7]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[8]  Klaus Ritter,et al.  Multilevel Monte Carlo Approximation of Distribution Functions and Densities , 2015, SIAM/ASA J. Uncertain. Quantification.

[9]  L. Shapley,et al.  Geometry of Moment Spaces , 1953 .

[10]  Kody J. H. Law,et al.  Multilevel ensemble Kalman filtering , 2015, SIAM J. Numer. Anal..

[11]  R. Mnatsakanov.  Hausdorff moment problem: Reconstruction of distributions , 2008 .

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Jonathan M. Borwein,et al.  Convergence of Best Entropy Estimates , 1991, SIAM J. Optim..

[14]  Alexey Chernov,et al.  Estimation of arbitrary order central statistical moments by the multilevel Monte Carlo method , 2016 .

[15]  Aldo Tagliani,et al.  Numerical aspects of finite Hausdorff moment problem by maximum entropy approach , 2001, Appl. Math. Comput..

[16]  J. Borwein,et al.  Duality relationships for entropy-like minimization problems , 1991 .

[17]  Dirk P. Kroese,et al.  Kernel density estimation via diffusion , 2010, 1011.2602.

[18]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[19]  Alexey Chernov,et al.  Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems , 2015, Numerische Mathematik.

[20]  Ralf Kornhuber,et al.  Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities , 2014, SIAM J. Numer. Anal..

[21]  Jonas Sukys,et al.  Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions , 2012, J. Comput. Phys..

[22]  Gerassimos A. Athanassoulis,et al.  The truncated Hausdorff moment problem solved by using kernel density functions , 2002 .

[23]  A. Barron,et al.  APPROXIMATION OF DENSITY FUNCTIONS BY SEQUENCES OF EXPONENTIAL FAMILIES , 1991 .

[24]  Assyr Abdulle,et al.  Multilevel Monte Carlo methods for stochastic elliptic multiscale PDEs , 2012 .

[25]  A. Volokitin,et al.  On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. , 2005, Journal of physics. Condensed matter : an Institute of Physics journal.

[26]  A. Hoorfar,et al.  INEQUALITIES ON THE LAMBERTW FUNCTION AND HYPERPOWER FUNCTION , 2008 .

[27]  E. E. Tyrtyshnikov.  How bad are Hankel matrices? , 1994 .

[28]  C. Reisinger,et al.  Stochastic Finite Differences and Multilevel Monte Carlo for a Class of SPDEs in Finance , 2012, SIAM J. Financial Math..

[29]  D. Fasino.  Spectral properties of Hankel matrices and numerical solutions of finite moment problems , 1995 .