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2015 | OriginalPaper | Chapter

9. Numerical Integration

Author : T. J. Sullivan

Published in: Introduction to Uncertainty Quantification

Publisher: Springer International Publishing

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Abstract

The topic of this chapter is the numerical evaluation of definite integrals. Many UQ methods have at their core simple probabilistic constructions such as expected values, and expectations are nothing more than Lebesgue integrals. However, while it is mathematically enough to know that the Lebesgue integral of some function exists, practical applications demand the evaluation of such an integral — or, rather, its approximate evaluation. This usually means evaluating the integrand at some finite collection of sample points. It is important to bear in mind, though, that sampling is not free (each sample of the integration domain, or function evaluation, may correspond to a multi-million-dollar experiment) and that practical applications often involve many dependent and independent variables, i.e. high-dimensional domains of integration. Hence, the accurate numerical integration of integrands over high-dimensional spaces using few samples is something of a ‘Holy Grail’ in this area.

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previous chapter Orthogonal Polynomials and Applications
next chapter Sensitivity Analysis and Model Reduction
Footnotes
1
This is exactly the phenomenon that makes car wheels appear to spin backwards instead of forwards in movies. The frame rates in common use are f = 24, 25 and 30 frames per second. A wheel spinning at f revolutions per second will appear to be stationary; one spinning at f + 1 revolutions per second (i.e. \( 1 + \frac{1} {f} \) revolutions per frame) will appear to be spinning at 1 revolution per second; and one spinning at f − 1 revolutions per second will appear to be spinning in reverse at 1 revolution per second.
 
Literature
M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications Inc., New York, 1992. Reprint of the 1972 edition.
I. Bogaert. Iteration-free computation of Gauss–Legendre quadrature nodes and weights. SIAM J. Sci. Comput., 36(3):A1008–A1026, 2014. doi: 10.1137/140954969.CrossRefMathSciNetMATH
S. Byrne and M. Girolami. Geodesic Monte Carlo on embedded manifolds. Scand. J. Stat., 40:825–845, 2013. doi: 10.1111/sjos.12063.CrossRefMathSciNetMATH
J. Charrier, R. Scheichl, and A. L. Teckentrup. Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal., 51(1):322–352, 2013. doi: 10.1137/110853054.CrossRefMathSciNetMATH
K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup. Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci., 14(1):3–15, 2011. doi: 10.1007/ s00791-011-0160-x.CrossRefMathSciNetMATH
C. Derman and H. Robbins. The strong law of large numbers when the first moment does not exist. Proc. Nat. Acad. Sci. U.S.A., 41:586–587, 1955.CrossRefMathSciNetMATH
J. Dick and F. Pillichshammer. Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.CrossRef
S. Duane, A.D. Kennedy, B. J. Pendleton, and D. Roweth. Hybrid Monte Carlo. Phys. Lett. B, 195(2):216–222, 1987. doi: 10.1016/0370-2693(87) 91197-X.CrossRef
V. Eglājs and P. Audze. New approach to the design of multifactor experiments. Prob. Dyn. Strengths, 35:104–107, 1977.
L. Fejér. On the infinite sequences arising in the theories of harmonic analysis, of interpolation, and of mechanical quadratures. Bull. Amer. Math. Soc., 39(8):521–534, 1933. doi: 10.1090/S0002-9904-1933-05677-X.CrossRefMathSciNet
C. F. Gauss. Methodus nova integralium valores per approximationem inveniendi. Comment. Soc. Reg. Scient. Gotting. Recent., pages 39–76, 1814.
W. Gautschi. Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2004.
A. Glaser, X. Liu, and V. Rokhlin. A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput., 29(4):1420–1438, 2007. doi: 10.1137/06067016X.CrossRefMathSciNetMATH
G. H. Golub and J. H. Welsch. Calculation of Gauss quadrature rules. Math. Comp., 23(106):221–230, 1969. doi: 10.1090/S0025-5718-69-99647-1.CrossRefMathSciNetMATH
P. J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4):711–732, 1995. doi: 10.1093/biomet/82.4.711.CrossRefMathSciNetMATH
J. H. Halton. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math., 2:84–90, 1960.CrossRefMathSciNetMATH
W. K. Hastings. Monte Carlo sampling methods using markov chains and their applications. Biometrika, 57(1):97–109, 1970. doi: 10.1093/biomet/ 57.1.97.CrossRefMathSciNetMATH
S. Heinrich. Multilevel Monte Carlo Methods. In S. Margenov, J. Waśniewski, and P. Yalamov, editors, Large-Scale Scientific Computing, volume 2179 of Lecture Notes in Computer Science, pages 58–67. Springer, Berlin Heidelberg, 2001. doi: 10.1007/3-540-45346-6_5.
F. J. Hickernell. A generalized discrepancy and quadrature error bound. Math. Comp., 67(221):299–322, 1998. doi: 10.1090/ S0025-5718-98-00894-1.CrossRefMathSciNetMATH
E. Hlawka. Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura Appl. (4), 54:325–333, 1961.
M. Holtz. Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance, volume 77 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2011. doi: 10.1007/ 978-3-642-16004-2.
R. L. Iman, J. M. Davenport, and D. K. Zeigler. Latin hypercube sampling (program user’s guide). Technical report, Sandia Labs, Albuquerque, NM, 1980.
R. L. Iman, J. C. Helton, and J. E. Campbell. An approach to sensitivity analysis of computer models, Part 1. Introduction, input variable selection and preliminary variable assessment. J. Quality Tech., 13(3):174–183, 1981.
D. Kincaid and W. Cheney. Numerical Analysis: Mathematics of Scientific Computing. Brooks/Cole Publishing Co., Pacific Grove, CA, second edition, 1996.
J. F. Koksma. Een algemeene stelling uit de theorie der gelijkmatige verdeeling modulo 1. Mathematica B (Zutphen), 11:7–11, 1942/1943.
L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. Wiley-Interscience [John Wiley & Sons], New York, 1974. Pure and Applied Mathematics.
M. D. McKay, R. J. Beckman, and W. J. Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(2):239–245, 1979. doi: 10.2307/1268522.MathSciNetMATH
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equation of state calculations by fast computing machines. J. Chem. Phys., 21(6):1087–1092, 1953. doi: 10.1063/1.1699114.CrossRef
R. M. Neal. MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo, Chapman & Hall/CRC Handb. Mod. Stat. Methods, pages 113–162. CRC Press, Boca Raton, FL, 2011.
H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970081.
E. Novak and K. Ritter. The curse of dimension and a universal method for numerical integration. In Multivariate approximation and splines (Mannheim, 1996), volume 125 of Internat. Ser. Numer. Math., pages 177–187. Birkhäuser, Basel, 1997.
A. B. Owen. Monte Carlo Theory, Methods and Examples, 2013. http://​statweb.​stanford.​edu/​~owen/​mc/​.​
C. P. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Texts in Statistics. Springer-Verlag, New York, second edition, 2004.
G. O. Roberts and J. S. Rosenthal. General state space Markov chains and MCMC algorithms. Probab. Surv., 1:20–71, 2004. doi: 10.1214/ 154957804100000024.CrossRefMathSciNetMATH
W. Sickel and T. Ullrich. The Smolyak algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness. East J. Approx., 13 (4):387–425, 2007.MathSciNet
W. Sickel and T. Ullrich. Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross. J. Approx. Theory, 161(2):748–786, 2009. doi: 10.1016/j.jat.2009.01.001.CrossRefMathSciNetMATH
S. A. Smolyak. Quadrature and interpolation formulae on tensor products of certain function classes. Dokl. Akad. Nauk SSSR, 148:1042–1045, 1963.MathSciNetMATH
I. M. Sobol′. Uniformly distributed sequences with an additional property of uniformity. Ž. Vyčisl. Mat. i Mat. Fiz., 16(5):1332–1337, 1375, 1976.
J. Stoer and R. Bulirsch. Introduction to Numerical Analysis, volume 12 of Texts in Applied Mathematics. Springer-Verlag, New York, third edition, 2002. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall.
H. Takahasi and M. Mori. Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci., 9:721–741, 1973/74.
A. L. Teckentrup, R. Scheichl, M. B. Giles, and E. Ullmann. Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer. Math., 125(3):569–600, 2013. doi: 10.1007/ s00211-013-0546-4.CrossRefMathSciNetMATH
A. Townsend. The race to compute high-order Gauss–Legendre quadrature. SIAM News, 48(2):1–3, 2015.MathSciNet
L. N. Trefethen and D. Bau, III. Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. doi: 10.1137/1.9780898719574.
T. Ullrich. Smolyak’s algorithm, sampling on sparse grids and Sobolev spaces of dominating mixed smoothness. East J. Approx., 14(1):1–38, 2008.MathSciNetMATH
J. G. van der Corput. Verteilungsfunktionen. I. Proc. Akad. Wet. Amst., 38:813–821, 1935a.
J. G. van der Corput. Verteilungsfunktionen. II. Proc. Akad. Wet. Amst., 38:1058–1066, 1935b.
Metadata
Title
Numerical Integration
Author
T. J. Sullivan
Copyright Year
2015
Book
Introduction to Uncertainty Quantification

Print ISBN: 978-3-319-23394-9

Electronic ISBN: 978-3-319-23395-6

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-23395-6_9