On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

In this work, we concern with the numerical comparison between different kinds of design points in least square (LS) approach on polynomial spaces. Such a topic is motivated by uncertainty quantification (UQ). Three kinds of design points are considered, which are the Sparse Grid (SG) points, the Monte Carlo (MC) points and the Quasi Monte Carlo (QMC) points. We focus on three aspects during the comparison: (i) the convergence properties; (ii) the stability, i.e. the properties of the resulting condition number of the design matrix; (iii) the robustness when numerical noises are present in function values. Several classical high dimensional functions together with a random ODE model are tested. It is shown numerically that (i) neither the MC sampling nor the QMC sampling introduce the low convergence rate, namely, the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used; (ii)The use of SG points admits better convergence properties only for very low dimensional problems (say d ≤ 2); (iii)The QMC points, being deterministic, seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view. AMS subject classifications: 65C20, 65M70

[1]  J. Collins,et al.  Construction of a genetic toggle switch in Escherichia coli , 2000, Nature.

[2]  J. Hammersley.  MONTE CARLO METHODS FOR SOLVING MULTIVARIABLE PROBLEMS , 1960 .

[3]  Fabio Nobile,et al.  Analysis of Discrete $$L^2$$L2 Projection on Polynomial Spaces with Random Evaluations , 2014, Found. Comput. Math..

[4]  R. Walters,et al.  Point-Collocation Nonintrusive Polynomial Chaos Method for Stochastic Computational Fluid Dynamics , 2010 .

[5]  Tao Zhou,et al.  On Sparse Interpolation and the Design of Deterministic Interpolation Points , 2013, SIAM J. Sci. Comput..

[6]  Tao Zhou,et al.  Multivariate Discrete Least-Squares Approximations with a New Type of Collocation Grid , 2014, SIAM J. Sci. Comput..

[7]  D. Xiu.  Efficient collocational approach for parametric uncertainty analysis , 2007 .

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

[9]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[11]  Adrian Sandu,et al.  Collocation least-squares polynomial chaos method , 2010, SpringSim.

[12]  Russel E. Caflisch,et al.  Quasi-Random Sequences and Their Discrepancies , 1994, SIAM J. Sci. Comput..

[13]  Jan S. Hesthaven,et al.  Efficient Solution of Ordinary Differential Equations with High-Dimensional Parametrized Uncertainty , 2011 .

[14]  Jan S. Hesthaven,et al.  Uncertainty analysis for the steady-state flows in a dual throat nozzle , 2005 .

[15]  Gianluca Iaccarino,et al.  A least-squares approximation of partial differential equations with high-dimensional random inputs , 2009, J. Comput. Phys..

[16]  Albert Cohen,et al.  On the Stability and Accuracy of Least Squares Approximations , 2011, Foundations of Computational Mathematics.

[17]  W. J. Whiten,et al.  Computational investigations of low-discrepancy sequences , 1997, TOMS.

[18]  M. Eldred.  Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design , 2009 .

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

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

[21]  Bruno Sudret,et al.  Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach , 2008 .

[22]  K. Ritter,et al.  High dimensional integration of smooth functions over cubes , 1996 .

[23]  Tao Tang,et al.  Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials , 2012, J. Sci. Comput..

[24]  K. Ritter,et al.  Simple Cubature Formulas with High Polynomial Exactness , 1999 .

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

[26]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .