Monte Carlo Algorithms for the Parabolic Cauchy Problem
Abstract
:1. Introduction
2. Integral Representation
3. Von-Neumann–Ulam Scheme
Numerical Algorithm
4. Conjugate Scheme
Funding
Conflicts of Interest
References
- Ladyzhenskaya, O.A.; Solonnikov, V.A.; Uraltseva, N.N. Linear and Quasilinear Equations of Parabolic Type; Nauka: Moscow, Russia, 1967. (In Russian) [Google Scholar]
- Wagner, W. Unbiased Monte Carlo estimators for functionals of weak solutions of stochastic diffretial equations. Stoch. Stoch. Rep. 1989, 28, 1–20. [Google Scholar] [CrossRef]
- Ermakov, S.M.; Mikhailov, G.A. Statistical Modeling; Nauka: Moscow, Russia, 1982. (In Russian) [Google Scholar]
- Wagner, W. Unbiased Monte Carlo evaluation of certain functional integrals. J. Comput. Phys. 1987, 71, 21–33. [Google Scholar] [CrossRef]
- Wagner, W. Unbiased Multi-step Estimators for the Monte Carlo Evaluation of Certain Functional Integrals. J. Comput. Phys. 1988, 79, 336–352. [Google Scholar] [CrossRef]
- Wagner, W. Monte Carlo evaluation of functionals of solutions of stochastic differential equations. Variance reduction and numerical examples. Stoch. Anal. Appl. 1988, 6, 447–468. [Google Scholar] [CrossRef]
- Sipin, A.S. Statistical Algorithms for Solving the Cauchy Problem for Second-Order Parabolic Equations. Vestn. Peterburg Univ. Math. 2011, 45, 65–74. [Google Scholar] [CrossRef]
- Sipin, A.S. Statistical Algorithms for Solving the Cauchy Problem for Second-Order Parabolic Equations: The “Dual” Scheme. Vestn. Peterburg Univ. Math. 2012, 45, 57–67. [Google Scholar] [CrossRef]
- Sabelfeld, K.K. Monte Carlo Methods in Boundary Value Problems; Nauka: Novosibirsk, Russia, 1989. (In Russian) [Google Scholar]
- Simonov, N.A. Stochastic iterative methods for solving equations of parabolic type. Sib. Mat. Zhurnal 1997, 38, 1146–1162. [Google Scholar]
- Heinrich, S. Multilevel Monte Carlo Methods; Volume 2179 of Lecture Notes in Computer Science; Springer: Berlin, Germany, 2001; pp. 58–67. [Google Scholar]
- Giles, M.B. Multi-Level Monte Carlo Path Simulation. Oper. Res. 2008, 56, 607–617. [Google Scholar] [CrossRef]
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Sipin, A. Monte Carlo Algorithms for the Parabolic Cauchy Problem. Mathematics 2019, 7, 177. https://doi.org/10.3390/math7020177
Sipin A. Monte Carlo Algorithms for the Parabolic Cauchy Problem. Mathematics. 2019; 7(2):177. https://doi.org/10.3390/math7020177
Chicago/Turabian StyleSipin, Alexander. 2019. "Monte Carlo Algorithms for the Parabolic Cauchy Problem" Mathematics 7, no. 2: 177. https://doi.org/10.3390/math7020177
APA StyleSipin, A. (2019). Monte Carlo Algorithms for the Parabolic Cauchy Problem. Mathematics, 7(2), 177. https://doi.org/10.3390/math7020177