Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a co...

[1]  DickJosef,et al.  Infinite-Dimensional Integration in Weighted Hilbert Spaces , 2014 .

[2]  Robert Scheichl,et al.  Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods , 2013, SIAM J. Numer. Anal..

[3]  Frances Y. Kuo,et al.  Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces , 2003, J. Complex..

[4]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[5]  Henryk Wozniakowski,et al.  Finite-order weights imply tractability of multivariate integration , 2004, J. Complex..

[6]  I. H. SLOAN,et al.  Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces , 2002, SIAM J. Numer. Anal..

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

[8]  I. Sloan,et al.  QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND , 2011, The ANZIAM Journal.

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

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

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

[12]  I. Sloan.  Lattice Methods for Multiple Integration , 1994 .

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

[14]  Michael Gnewuch,et al.  Infinite-dimensional integration on weighted Hilbert spaces , 2012, Math. Comput..

[15]  G. Fix.  Review: Philippe G. Ciarlet, The finite element method for elliptic problems , 1979 .

[16]  Grzegorz W. Wasilkowski,et al.  Tractability of infinite-dimensional integration in the worst case and randomized settings , 2011, J. Complex..

[17]  Josef Dick.  On the convergence rate of the component-by-component construction of good lattice rules , 2004, J. Complex..

[18]  Dirk Nuyens,et al.  Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces , 2006, Math. Comput..

[19]  Christoph Schwab,et al.  Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients , 2007 .

[20]  Frances Y. Kuo,et al.  Constructing Embedded Lattice Rules for Multivariate Integration , 2006, SIAM J. Sci. Comput..

[21]  Albert Cohen,et al.  Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , 2010, Found. Comput. Math..

[22]  S. Joe.  Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy , 2006 .

[23]  M. Giles.  Improved Multilevel Monte Carlo Convergence using the Milstein Scheme , 2008 .

[24]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[25]  Xiaoqun Wang,et al.  Strong tractability of multivariate integration using quasi-Monte Carlo algorithms , 2003, Math. Comput..

[26]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

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

[28]  Henryk Wozniakowski,et al.  Liberating the weights , 2004, J. Complex..

[29]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[30]  Frances Y. Kuo,et al.  Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications , 2011, J. Comput. Phys..

[31]  Josef Dick,et al.  The construction of good extensible rank-1 lattices , 2008, Math. Comput..

[32]  I. Sloan,et al.  Brownian bridge and principal component analysis: towards removing the curse of dimensionality , 2007 .

[33]  Fred J. Hickernell,et al.  Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN , 2010, J. Complex..

[34]  Dirk Nuyens,et al.  Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points , 2006, J. Complex..

[35]  Hoang-Ngan Nguyen,et al.  Finite element wavelets for solving partial differential equations , 2005 .

[36]  Stephen Joe,et al.  Good lattice rules based on the general weighted star discrepancy , 2007, Math. Comput..

[37]  Henryk Wozniakowski,et al.  Liberating the dimension , 2010, J. Complex..

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

[39]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[40]  Fred J. Hickernell,et al.  Deterministic multi-level algorithms for infinite-dimensional integration on RN , 2011, J. Complex..

[41]  Gerhard Larcher,et al.  On the tractability of the Brownian Bridge algorithm , 2003, J. Complex..