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Accurate evaluation of Wiener integrals


Author: Alexandre Joel Chorin
Journal: Math. Comp. 27 (1973), 1-15
MSC: Primary 65D30
DOI: https://doi.org/10.1090/S0025-5718-1973-0329205-7
Corrigendum: Math. Comp. 27 (1973), 1011-1012.
MathSciNet review: 0329205
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Abstract: A new quadrature formula for an important class of Wiener integrals is presented, in which the Wiener integrals are approximated by n-fold integrals with an error $ O({n^{ - 2}})$. The resulting n-fold integrals can then be approximated by ordinary finite sums of remarkably simple structure. An example is given.


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  • [1] N. N. Bogoljubov & D. V. Širkov, Introduction to the Theory of Quantized Fields, GITTL, Moscow, 1957; English transl., Interscience, New York, 1959. MR 20 #5047; MR 22 #1349.
  • [2] R. H. Cameron, "A Simpson's rule for the numerical evaluation of Wiener's integrals in function space," Duke Math. J., v. 18, 1951, pp. 111-130. MR 12, 718. MR 0040589 (12:718d)
  • [3] R. H. Cameron & W. T. Martin, "The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals," Ann. of Math. (2), v. 48, 1947, pp. 385-392. MR 8, 523. MR 0020230 (8:523a)
  • [4] A. J. Chorin, "Hermite expansions in Monte-Carlo computation," J. Computational Phys., v. 8, 1971, pp. 472-482. MR 0297092 (45:6150)
  • [5] A. J. Chorin, Determination of the Principal Eigenvalue of Schrödinger Operators. (To appear.)
  • [6] M. D. Donsker & M. Kac, "A sampling method for determining the lowest eigenvalue and the principal eigenfunction of Schrödinger's equation," J. Res. Nat. Bur. Standards, v. 44, 1950, pp. 551-557. MR 13, 590. MR 0045473 (13:590a)
  • [7] R. P. Feynman & A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.
  • [8] L. D. Fosdick, "Approximation of a class of Wiener integrals," Math. Comp., v. 19, 1965, pp. 225-233. MR 31 #4174. MR 0179937 (31:4174)
  • [9] I. M. Gel'fand & A. M. Jaglom, "Integration in functional spaces and its application in quantum physics," Uspehi Mat. Nauk, v. 11, 1956, no. 1 (67), pp. 77-114; English transl., J. Mathematical Phys., v. 1, 1960, pp. 48-69. MR 17, 1261; MR 22 #3455. MR 0112604 (22:3455)
  • [10] A. G. Konheim & W. L. Miranker, "Numerical evaluation of Wiener integrals," Math. Comp., v. 21, 1967, pp. 49-65. MR 36 #4805. MR 0221753 (36:4805)
  • [11] P. Lévy, Le Mouvement Brownien, Mém. Sci. Math., fasc. 126, Gauthier-Villars, Paris, 1954. MR 16, 601. MR 0066588 (16:601b)
  • [12] R. E. A. C. Paley & N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ., vol. 19, Amer. Math. Soc., Providence, R. I., 1934. MR 1451142 (98a:01023)
  • [13] A. H. Stroud & D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, N. J., 1966. MR 34 #2185. MR 0202312 (34:2185)
  • [14] V. S. Vladimirov, "On the approximate calculation of Wiener integrals," Uspehi Mat. Nauk, v. 15, 1960, no. 4 (94), pp. 129-135; English transl., Amer. Math. Soc. Transl. (2), v. 34, 1963, pp. 405-412. MR 23 #A1404. MR 0124087 (23:A1404)
  • [15] N. Wiener, Nonlinear Problems in Random Theory, Wiley, New York, 1958. MR 20 #7337. MR 0100912 (20:7337)

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DOI: https://doi.org/10.1090/S0025-5718-1973-0329205-7
Article copyright: © Copyright 1973 American Mathematical Society

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