Generation and application of multivariate polynomial quadrature rules

Jakeman, John D.; Narayan, Akil

doi:10.1016/j.cma.2018.04.009

Generation and application of multivariate polynomial quadrature rules

Journal Article · Wed Aug 01 00:00:00 EDT 2018 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2018.04.009· OSTI ID:1485818

^[1]; ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Univ. of Utah, Salt Lake City, UT (United States). Dept. of Mathematics, and Scientific Computing and Imaging (SCI) Inst.

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1485818

Alternate ID(s):: OSTI ID: 1682381

Report Number(s):: SAND--2018-13400J; 670536

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Journal Issue: C Vol. 338; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (30)

Spatially adaptive sparse grids for high-dimensional data-driven problems Pflüger, Dirk; Peherstorfer, Benjamin; Bungartz, Hans-Joachim Journal of Complexity, Vol. 26, Issue 5 https://doi.org/10.1016/j.jco.2010.04.001	journal	October 2010
An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations Ma, Xiang; Zabaras, Nicholas Journal of Computational Physics, Vol. 228, Issue 8 https://doi.org/10.1016/j.jcp.2009.01.006	journal	May 2009
Multi-element probabilistic collocation method in high dimensions Foo, Jasmine; Karniadakis, George Em Journal of Computational Physics, Vol. 229, Issue 5 https://doi.org/10.1016/j.jcp.2009.10.043	journal	March 2010
Sparse grids Bungartz, Hans-Joachim; Griebel, Michael Acta Numerica, Vol. 13 https://doi.org/10.1017/s0962492904000182	journal	May 2004
Transport Map Accelerated Markov Chain Monte Carlo Parno, Matthew D.; Marzouk, Youssef M. SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, Issue 2 https://doi.org/10.1137/17m1134640	journal	January 2018
Measure transformation and efficient quadrature in reduced-dimensional stochastic modeling of coupled problems: STOCHASTIC MODELING OF COUPLED PROBLEMS Arnst, M.; Ghanem, R.; Phipps, E. International Journal for Numerical Methods in Engineering, Vol. 92, Issue 12 https://doi.org/10.1002/nme.4368	journal	June 2012
Efficient uncertainty propagation for network multiphysics systems: UNCERTAINTY PROPAGATION FOR NETWORK SYSTEMS Constantine, P. G.; Phipps, E. T.; Wildey, T. M. International Journal for Numerical Methods in Engineering, Vol. 99, Issue 3 https://doi.org/10.1002/nme.4667	journal	April 2014
On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals Halton, J. H. Numerische Mathematik, Vol. 2, Issue 1 https://doi.org/10.1007/BF01386213	journal	December 1960
Sur les formules de quadrature num�rique � nombre minimal de noeuds d'int�gration Gout, J. L.; Guessab, A. Numerische Mathematik, Vol. 49, Issue 4 https://doi.org/10.1007/BF01389541	journal	July 1986
Cubature formulae which are exact on spacesP, intermediate betweenP k andQ k? Guessab, A. Numerische Mathematik, Vol. 49, Issue 5 https://doi.org/10.1007/BF01389706	journal	September 1986
Zur mechanischen Kubatur Radon, Johann Monatshefte f�r Mathematik, Vol. 52, Issue 4 https://doi.org/10.1007/BF01525334	journal	December 1948
Dimension?Adaptive Tensor?Product Quadrature Gerstner, T.; Griebel, M. Computing, Vol. 71, Issue 1 https://doi.org/10.1007/s00607-003-0015-5	journal	August 2003
Extensions of Gauss Quadrature Via Linear Programming Ryu, Ernest K.; Boyd, Stephen P. Foundations of Computational Mathematics, Vol. 15, Issue 4 https://doi.org/10.1007/s10208-014-9197-9	journal	March 2014
An encyclopaedia of cubature formulas Cools, Ronald Journal of Complexity, Vol. 19, Issue 3 https://doi.org/10.1016/S0885-064X(03)00011-6	journal	June 2003
Numerical integration formulas of degree two Xiu, Dongbin Applied Numerical Mathematics, Vol. 58, Issue 10 https://doi.org/10.1016/j.apnum.2007.09.004	journal	October 2008
Dimension-wise integration of high-dimensional functions with applications to finance Griebel, Michael; Holtz, Markus Journal of Complexity, Vol. 26, Issue 5 https://doi.org/10.1016/j.jco.2010.06.001	journal	October 2010
Non-intrusive uncertainty quantification using reduced cubature rules van den Bos, L. M. M.; Koren, B.; Dwight, R. P. Journal of Computational Physics, Vol. 332 https://doi.org/10.1016/j.jcp.2016.12.011	journal	March 2017
Constructing cubature formulae: the science behind the art Cools, Ronald Acta Numerica, Vol. 6 https://doi.org/10.1017/S0962492900002701	journal	January 1997
Monte Carlo and quasi-Monte Carlo methods Caflisch, Russel E. Acta Numerica, Vol. 7 https://doi.org/10.1017/S0962492900002804	journal	January 1998
Sparse grids Bungartz, Hans-Joachim; Griebel, Michael Acta Numerica, Vol. 13 https://doi.org/10.1017/S0962492904000182	journal	May 2004
Inverse problems: A Bayesian perspective Stuart, A. M. Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000061	journal	May 2010
Calculation of Gauss quadrature rules Golub, Gene H.; Welsch, John H. Mathematics of Computation, Vol. 23, Issue 106 https://doi.org/10.1090/S0025-5718-69-99647-1	journal	May 1969
Oscillatory dynamics in a heterogeneous surface reaction: Breakdown of the mean-field approximation Vigil, R. Dennis; Willmore, Frank T. Physical Review E, Vol. 54, Issue 2 https://doi.org/10.1103/PhysRevE.54.1225	journal	August 1996
Quadrature Methods for Functions of more than one Variable Stroud, Arthur H. Annals of the New York Academy of Sciences, Vol. 86, Issue 3 https://doi.org/10.1111/j.1749-6632.1960.tb42842.x	journal	May 1960
Some Modified Matrix Eigenvalue Problems Golub, Gene H. SIAM Review, Vol. 15, Issue 2 https://doi.org/10.1137/1015032	journal	April 1973
Generating Moment Matching Scenarios Using Optimization Techniques Mehrotra, Sanjay; Papp, Dávid SIAM Journal on Optimization, Vol. 23, Issue 2 https://doi.org/10.1137/110858082	journal	January 2013
Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation Narayan, Akil; Jakeman, John D. SIAM Journal on Scientific Computing, Vol. 36, Issue 6 https://doi.org/10.1137/140966368	journal	January 2014
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension? Wang, Xiaoqun; Sloan, Ian H. SIAM Journal on Scientific Computing, Vol. 27, Issue 1 https://doi.org/10.1137/S1064827503429429	journal	January 2005
Integration with Quasirandom Sequences: Numerical Experience Sobol’, Ilya M.; Shukhman, Boris V. International Journal of Modern Physics C, Vol. 06, Issue 02 https://doi.org/10.1142/S0129183195000204	journal	April 1995
Numerical Evaluation of Multiple Integrals II Hammer, Preston C.; Stroud, Arthur H. Mathematical Tables and Other Aids to Computation, Vol. 12, Issue 64 https://doi.org/10.2307/2002370	journal	October 1958

Cited By (2)

Numerical Integration in Multiple Dimensions with Designed Quadrature Keshavarzzadeh, Vahid; Kirby, Robert M.; Narayan, Akil SIAM Journal on Scientific Computing, Vol. 40, Issue 4 https://doi.org/10.1137/17m1137875	journal	January 2018
Weighted approximate Fekete points: Sampling for least-squares polynomial approximation Guo, Ling; Narayan, Akil; Yan, Liang arXiv https://doi.org/10.48550/arxiv.1708.01296	preprint	January 2017

Similar Records

Generation and application of multivariate polynomial quadrature rules

Technical Report · Fri Sep 15 00:00:00 EDT 2017 · OSTI ID:1510651

Quadrature rules for regions having regular hexagonal symmetry

Journal Article · Thu Mar 31 23:00:00 EST 1977 · SIAM J. Numer. Anal.; (United States) · OSTI ID:5438434

Related Subjects

97 MATHEMATICS AND COMPUTING
Bayesian inference
Dimension reduction
Moment-matching quadrature
Multivariate integration
Uncertainty quantification

Generation and application of multivariate polynomial quadrature rules

Citation Formats

References (30)

Cited By (2)

Similar Records

Related Subjects