Rate of convergence of a stochastic particle method for the Kolmogorov equation with variable coefficients

Authors:
Pierre Bernard, Denis Talay and Luciano Tubaro

Journal:
Math. Comp. **63** (1994), 555-587

MSC:
Primary 65M12; Secondary 35K57, 60J15, 60J60

DOI:
https://doi.org/10.1090/S0025-5718-1994-1250770-3

MathSciNet review:
1250770

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Abstract: In a recent paper, E. G. Puckett proposed a stochastic particle method for the nonlinear diffusion-reaction PDE in $[0,T] \times \mathbb {R}$ (the so-called "KPP" (Kolmogorov-Petrovskii-Piskunov) equation): \[ \left \{ \begin {array}{*{20}{c}} \frac {{\partial u}}{{\partial t}} = Au = \Delta u + f(u), \hfill \\ u(0, \cdot ) = {u_0}( \cdot ), \hfill \\ \end {array} \right .\] where $1 - {u_0}$ is the cumulative function, supposed to be smooth enough, of a probability distribution, and *f* is a function describing the reaction. His justification of the method and his analysis of the error were based on a splitting of the operator *A*. He proved that, if *h* is the time discretization step and *N* the number of particles used in the algorithm, one can obtain an upper bound of the norm of the random error on $u(T,x)$ in ${L^1}(\Omega \times \mathbb {R})$ of order $1/{N^{1/4}}$, provided $h = \mathcal {O}(1/{N^{1/4}})$, but conjectured, from numerical experiments, that it should be of order $\mathcal {O}h + \mathcal {O}(1/\sqrt N )$, without any relation between *h* and *N*. We prove that conjecture. We also construct a similar stochastic particle method for more general nonlinear diffusion-reaction-convection PDEs \[ \left \{ \begin {array}{*{20}{c}} \frac {{\partial u}}{{\partial t}} = Lu + f(u), \hfill \\ u(0,\cdot ) = {u_0}(\cdot ), \hfill \\ \end {array} \right .\] where *L* is a strongly elliptic second-order operator with smooth coefficients, and prove that the preceding rate of convergence still holds when the coefficients of *L* are constant, and in the other case is $\mathcal {O}(\sqrt h ) + \mathcal {O}(1/\sqrt N )$. The construction of the method and the analysis of the error are based on a stochastic representation formula of the exact solution *u*.

- A. Bensoussan and J.-L. Lions,
*Applications des inéquations variationnelles en contrôle stochastique*, Dunod, Paris, 1978 (French). Méthodes Mathématiques de l’Informatique, No. 6. MR**0513618** - Piermarco Cannarsa and Vincenzo Vespri,
*Generation of analytic semigroups by elliptic operators with unbounded coefficients*, SIAM J. Math. Anal.**18**(1987), no. 3, 857–872. MR**883572**, DOI https://doi.org/10.1137/0518063 - Brigitte Chauvin and Alain Rouault,
*KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees*, Probab. Theory Related Fields**80**(1988), no. 2, 299–314. MR**968823**, DOI https://doi.org/10.1007/BF00356108 - B. Chauvin and A. Rouault,
*A stochastic simulation for solving scalar reaction-diffusion equations*, Adv. in Appl. Probab.**22**(1990), no. 1, 88–100. MR**1039378**, DOI https://doi.org/10.2307/1427598 - B. Chauvin and A. Rouault,
*Supercritical branching Brownian motion and K-P-P equation in the critical speed-area*, Math. Nachr.**149**(1990), 41–59. MR**1124793**, DOI https://doi.org/10.1002/mana.19901490104 - Avner Friedman,
*Stochastic differential equations and applications. Vol. 1*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Probability and Mathematical Statistics, Vol. 28. MR**0494490** - Takeyuki Hida,
*Brownian motion*, Applications of Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1980. Translated from the Japanese by the author and T. P. Speed. MR**562914** - H. Kunita,
*Stochastic differential equations and stochastic flows of diffeomorphisms*, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 143–303. MR**876080**, DOI https://doi.org/10.1007/BFb0099433 - G. N. Mil′šteĭn,
*Approximate integration of stochastic differential equations*, Teor. Verojatnost. i Primenen.**19**(1974), 583–588 (Russian, with English summary). MR**0356225** - A. Pazy,
*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486** - G. Da Prato and E. Sinestrari,
*Differential operators with nondense domain*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**14**(1987), no. 2, 285–344 (1988). MR**939631** - Elbridge Gerry Puckett,
*Convergence of a random particle method to solutions of the Kolmogorov equation $u_t=\nu u_{xx}+u(1-u)$*, Math. Comp.**52**(1989), no. 186, 615–645. MR**964006**, DOI https://doi.org/10.1090/S0025-5718-1989-0964006-X - Franz Rothe,
*Global solutions of reaction-diffusion systems*, Lecture Notes in Mathematics, vol. 1072, Springer-Verlag, Berlin, 1984. MR**755878** - Arthur S. Sherman and Charles S. Peskin,
*A Monte Carlo method for scalar reaction diffusion equations*, SIAM J. Sci. Statist. Comput.**7**(1986), no. 4, 1360–1372. MR**857799**, DOI https://doi.org/10.1137/0907090 - H. Bruce Stewart,
*Generation of analytic semigroups by strongly elliptic operators*, Trans. Amer. Math. Soc.**199**(1974), 141–162. MR**358067**, DOI https://doi.org/10.1090/S0002-9947-1974-0358067-4
D. Talay,

*Simulation and numerical analysis of stochastic differential systems*:

*a review*, Rapport de Recherche INRIA, vol. 1313, 1990 (and to appear in Effective Stochastic Analysis (P. Kree and W. Wedig, eds.), Springer-Verlag).

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© Copyright 1994
American Mathematical Society