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2024 | OriginalPaper | Chapter

Heuristics for the Probabilistic Solution of BVPs with Mixed Boundary Conditions

Authors : Francisco Bernal, Andrés Berridi

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

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Abstract

The performance of stochastic numerical methods for bounded diffusions (accuracy, speed and weak rate of convergence) is considered. The backdrop is the pointwise solution, via stochastic representations, of boundary value problems with mixed boundary conditions. Three stochastic solvers are tested on application-inspired problems, and their performance is noticeably improved thanks to simple heuristics introduced here.

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previous chapter Error Estimates and Variance Reduction for Nonequilibrium Stochastic Dynamics
next chapter Challenges in Developing Great Quasi-Monte Carlo Software
Literature
1.
Acebrón, J.A., Busico, M.P., Lanucara, P., Spigler, R.: Domain decomposition solution of elliptic problems via probabilistic methods. SIAM J. Sci. Comput. 27, 440–457 (2005)MathSciNetCrossRef
2.
3.
Baldi, P.: Exact asymptotics for the probability of exit from a domain and applications to simulation. Ann. Probab. 23(4), 1644–1670 (1995)MathSciNetCrossRef
4.
Bayer, C., Szepessy, A., Tempone, R.: Adaptive weak approximation of reflected and stopped diffusions. Monte Carlo Methods Appl. 16, 1–67 (2010)MathSciNetCrossRef
5.
Bernal, F.: An implementation of Milstein’s method for general bounded diffusions. J. Sci. Comput. 79(2), 867–890 (2019)MathSciNetCrossRef
6.
Bernal, F., Acebrón, J.A., Anjam, I.: A stochastic algorithm based on fast marching for automatic capacitance extraction in non-manhattan geometries. SIAM J. Imaging Sci. 7(4), 2657–2674 (2014)MathSciNetCrossRef
7.
Bernal, F., Gutiérrez, G., Kindelan, M.: Use of singularity capturing functions in the solution of problems with discontinuous boundary conditions. Eng. Anal. Bound. Elem. 33(2), 200–208 (2009)MathSciNetCrossRef
8.
Bernal, F., Kindelan, M.: On the enriched RBF method for singular potential problems. Eng. Anal. Bound. Elem. 33(8–9), 1062–1073 (2009)MathSciNetCrossRef
9.
Cao N., Ortner M., Nehorai A.: Solutions for diffuse optical tomography using the Feynman-Kac formula and interacting particle method. In: Proceeding of SPIE BIOS’07, San Jose, CA (USA), vol. 6434, pp. 643402–64313 (2007)
10.
Constantini, C., Pacchiarotti, B., Sartoretto, F.: Numerical approximation for functionals of reflecting diffusion processes. SIAM J. Appl. Math. 58, 73–102 (1998)MathSciNetCrossRef
11.
Dzougoutov A., Moon K.S., von Schwerin E., Szepessy A., Tempone R.: Adaptive Monte Carlo algorithms for stopped diffusion. Multiscale methods in science and engineering. Lecture Notes in Computational Science and Engineering 44, pp. 59–88. Springer, Berlin (2005)
12.
Gobet, E.: Advanced Monte Carlo methods for barrier and related exotic options. Handbook of Numerical Analysis, vol. XV. Elsevier. Special Volume: Mathematical Modeling and Numerical Methods in Finance. Editor: P.G. Ciarlet. Guest Editors: Alain Bensoussan and Qiang Zhang, pp. 497–528 (2009)
13.
Gobet, E., Menozzi, S.: Stopped diffusion processes: overshoots and boundary correction. Stoch. Process. Appl. 120, 130–162 (2010)CrossRef
14.
Gobet E.: Euler schemes and half-space approximation for the simulation of diffusions in a domain. ESAIM: Probab. Stat. 5, 261–297 (2001)
15.
Gobet, E.: Monte Carlo Methods and Stochastic Processes. Chapman & Hall/CRC (2016)
16.
Lépingle, D.: Euler scheme for reflected stochastic differential equations. Math. Comput. Simul. 38, 119–126 (1995)MathSciNetCrossRef
17.
Licata, N.A., Grill, S.W.: The first passage problem for diffusion through a cylindrical pore with sticky walls. Euro. Phys. J. E 30(4), 439–447 (2009)CrossRef
18.
Lieberman, G.M.: Mixed boundary value problems for elliptic and parabolic differential equations of second order. J. Math. Anal. App. 113(2), 422–440 (1986)MathSciNetCrossRef
19.
Mairé, S., Simon, M.: A partially reflecting random walk on spheres algorithm for electrical impedance tomography. J. Comput. Phys. 303(C), 413–430 (2015)
20.
Milstein, G.N.: Application of the numerical integration of stochastic equations for the solution of boundary value problems with Neumann boundary conditions. Theor. Prob. Appl. 41, 170–177 (1996)MathSciNet
21.
Milstein, G.N.: Weak approximation of a diffusion process in a bounded domain. Stoch. Stoch. Rep. 62, 147–200 (1997)MathSciNetCrossRef
22.
Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)CrossRef
23.
Mascagni, M., Simonov, N.A.: Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. Sci. Comput. 26, 339–357 (2004)MathSciNetCrossRef
24.
Miranda, C.: Partial Differential Equations of Elliptic Type, 2nd edn. Springer, Berlin/Heidelberg/New York (1970)CrossRef
25.
Rosser, J.B., Papamichael, N.: A power series solution of a harmonic mixed boundary value problem. MRC Technical Summary Report 1405, University of Wisconsin (1975)
26.
Ladyženkaja, O.A., Solonnikov, V.A., Ural’ceva, N.: Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. AMS, Providence, RI (1968)
27.
Wait, R., Mitchell, A.R.: Corner singularities in elliptic problems by finite element methods. J. Comput. Phys. 8, 45–52 (1971)CrossRef
Metadata
Title
Heuristics for the Probabilistic Solution of BVPs with Mixed Boundary Conditions
Authors
Francisco Bernal
Andrés Berridi
Copyright Year
2024
Book
Monte Carlo and Quasi-Monte Carlo Methods

Print ISBN: 978-3-031-59761-9

Electronic ISBN: 978-3-031-59762-6

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-59762-6_8

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