Sparse Collocation Methods for Stochastic Interpolation and Quadrature

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

[2]  C. W. Clenshaw,et al.  A method for numerical integration on an automatic computer , 1960 .

[3]  W. Morven Gentleman,et al.  Implementing Clenshaw-Curtis quadrature, II computing the cosine transformation , 1972, Commun. ACM.

[4]  L. Brutman,et al.  ON THE LEBESGUE FUNCTION FOR POLYNOMIAL INTERPOLATION , 1978 .

[5]  V. K. Dzjadyk,et al.  On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points , 1983 .

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

[7]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[8]  W. Sweldens.  The Lifting Scheme: A Custom - Design Construction of Biorthogonal Wavelets "Industrial Mathematics , 1996 .

[9]  T. Sauer,et al.  On multivariate Lagrange interpolation , 1995 .

[10]  Wim Sweldens,et al.  The lifting scheme: a construction of second generation wavelets , 1998 .

[11]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

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

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

[14]  Åke Björck,et al.  The calculation of linear least squares problems , 2004, Acta Numerica.

[15]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

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

[17]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[18]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[19]  I. Babuska,et al.  Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation , 2005 .

[20]  Simon J. Smith,et al.  Lebesgue constants in polynomial interpolation , 2006 .

[21]  Baskar Ganapathysubramanian,et al.  Sparse grid collocation schemes for stochastic natural convection problems , 2007, J. Comput. Phys..

[22]  Clayton G. Webster.  Sparse grid stochastic collocation techniques for the numerical solution of partial differential equations with random input data , 2007 .

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

[24]  S. Robins,et al.  Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra , 2007 .

[25]  Michael Griebel,et al.  Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences , 1998, Computing.

[26]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

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

[28]  Thomas A. Zang,et al.  Stochastic approaches to uncertainty quantification in CFD simulations , 2005, Numerical Algorithms.

[29]  David R. Owen,et al.  A Fourier–Karhunen–Loève discretization scheme for stationary random material properties in SFEM , 2008 .

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

[31]  Gianluigi Rozza,et al.  Reduced basis method for linear elasticity problems with many parameters , 2008 .

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

[33]  N. Nguyen,et al.  A general multipurpose interpolation procedure: the magic points , 2008 .

[34]  Christoph Schwab,et al.  Sparse Tensor Discretization of Elliptic sPDEs , 2009, SIAM J. Sci. Comput..

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

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

[37]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2010, SIAM Rev..

[38]  Xiang Ma,et al.  An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations , 2010, J. Comput. Phys..

[39]  R. DeVore,et al.  ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S , 2011 .

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

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

[42]  Dongbin Xiu,et al.  Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids , 2011, J. Comput. Phys..

[43]  Howard C. Elman,et al.  ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA , 2011 .

[44]  R. Tempone,et al.  Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison , 2011 .

[45]  Guannan Zhang,et al.  Error Analysis of a Stochastic Collocation Method for Parabolic Partial Differential Equations with Random Input Data , 2012, SIAM J. Numer. Anal..

[46]  A. Patera,et al.  A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD , 2012 .

[47]  R. Tempone,et al.  ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS , 2012 .

[48]  Howard C. Elman,et al.  Stochastic Collocation With Kernel Density Estimation , 2012 .

[49]  Jean-Paul Calvi,et al.  Pseudo Leja sequences , 2012 .

[50]  Frances Y. Kuo,et al.  Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients , 2012, 1208.6349.

[51]  Moulay Abdellah Chkifa.  On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection , 2013, J. Approx. Theory.

[52]  Christoph Schwab,et al.  Sparse Tensor Galerkin Discretization of Parametric and Random Parabolic PDEs - Analytic Regularity and Generalized Polynomial Chaos Approximation , 2013, SIAM J. Math. Anal..

[53]  Christoph Schwab,et al.  Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs , 2013 .

[54]  Andrea Barth,et al.  Multilevel Monte Carlo method for parabolic stochastic partial differential equations , 2013 .

[55]  Albert Cohen,et al.  Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs , 2011 .

[56]  C. Schwab,et al.  Sparse Adaptive Approximation of High Dimensional Parametric Initial Value Problems , 2013 .

[57]  Guannan Zhang,et al.  An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Partial Differential Equations with Random Input Data , 2014 .

[58]  Fabio Nobile,et al.  Computers and Mathematics with Applications Convergence of Quasi-optimal Stochastic Galerkin Methods for a Class of Pdes with Random Coefficients , 2022 .

[59]  Guannan Zhang,et al.  Stochastic finite element methods for partial differential equations with random input data* , 2014, Acta Numerica.

[60]  Max Gunzburger,et al.  A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2014, SIAM/ASA J. Uncertain. Quantification.

[61]  Guannan Zhang,et al.  A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection , 2015, SIAM J. Numer. Anal..

[62]  Clayton G. Webster,et al.  A GRADIENT-BASED SAMPLING APPROACH FOR DIMENSION REDUCTION OF PARTIAL DIFFERENTIAL EQUATIONS WITH STOCHASTIC COEFFICIENTS , 2015 .

[63]  M. Gunzburger,et al.  A multilevel stochastic collocation method for SPDEs , 2015 .

[64]  Albert Cohen,et al.  Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs , 2015 .

[65]  Guannan Zhang,et al.  Accelerating Stochastic Collocation Methods for Partial Differential Equations with Random Input Data , 2015, SIAM/ASA J. Uncertain. Quantification.

[66]  Guannan Zhang,et al.  Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients , 2015, Comput. Math. Appl..

[67]  Guannan Zhang,et al.  Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients , 2015, Numerische Mathematik.