Monte Carlo method for parabolic equations involving fractional Laplacian Caiyu Jiao Changpin Li

2025-05-02 0 0 619.27KB 30 页 10玖币

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Monte Carlo method for parabolic equations

involving fractional Laplacian∗

Caiyu Jiao, Changpin Li†

Department of Mathematics, Shanghai University, Shanghai 200444, China

Abstract

We apply the Monte Carlo method to solving the Dirichlet problem of

linear parabolic equations with fractional Laplacian. This method exploits

the idea of weak approximation of related stochastic diﬀerential equations

driven by the symmetric stable L´evy process with jumps. We utilize the

jump-adapted scheme to approximate L´evy process which gives exact exit

time to the boundary. When the solution has low regularity, we establish a

numerical scheme by removing the small jumps of the L´evy process and then

show the convergence order. When the solution has higher regularity, we build

up a higher-order numerical scheme by replacing small jumps with a simple

process and then display the higher convergence order. Finally, numerical

experiments including ten- and one hundred-dimensional cases are presented,

which conﬁrm the theoretical estimates and show the numerical eﬃciency of

the proposed schemes for high dimensional parabolic equations.

Key words: Monte Carlo method; fractional Laplacian; linear parabolic

equation; L´evy process; Jump-adapted scheme

1 Introduction

The fractional Laplacian, (−∆)s, is a prototypical operator for modeling nonlocal

and anomalous phenomenon which incorporates long range interactions [3, 10, 13,

19, 21, 35, 36]. It arises in many areas of applications, including models for turbu-

lent ﬂows, porous media ﬂows, pollutant transport, quantum mechanics, stochastic

dynamics, ﬁnance and so on [8, 11, 12, 16, 26]. Almost known deterministic numer-

ical methods have been proposed for approximating solutions of parabolic problems

with fractional Laplacian in low dimension (less than 3 dimension)[1, 2], while sel-

dom probabilistic approach is taken into account to numerically solve such parabolic

and steady state problems.

∗The work was partially supported by the National Natural Science Foundation of China under

Grant no. 11926319.

†Corresponding author.

arXiv:2210.15192v1 [math.NA] 27 Oct 2022

The aim of this article is to develop a Monte Carlo method for solving the

terminal-boundary value problem in high dimensional cases in the following form











∂u

∂t −(−∆)su+b(t, x) · ∇u+c(t, x)u+f(t, x) = 0,(t, x) ∈[0, T )×D,

u(T, x) = g(x),x∈D,

u(t, x) = χ(t, x),(t, x) ∈[0, T ]×(Rn\D),

(1.1)

where s∈(0,1), T > 0, b(t, x) = (b1(t, x), b2(t, x),··· , bn(t, x)) is an n-dimensional

vector, Dis a bounded region in Rn(n≥3) and the fractional Laplacian is deﬁned

by a singular integral which coincides with Riesz derivative on the whole space

[6, 7, 14, 20, 28, 33],

(−∆)su(t, x) = C(n, s) P.V.ZRn

u(t, x) −u(t, y)

|x−y|n+2sdy.(1.2)

Here P.V.stands for the principle value and the constant C(n, s) is given by [6],

C(n, s) = ZRn

1−cos ζ1

|ζ|n+2sdζ−1

=s22sΓ(n

2+s)

πn

2Γ(1 −s)(1.3)

with ζ1being the ﬁrst component of ζ= (ζ1, ζ2, . . . , ζn)∈Rnand Γ representing the

Gamma function.

If we let v(t, x) = u(T−t, x), then equation (1.1) is changed to the initial-

boundary value problem











∂v

∂t + (−∆)sv+b(t, x) · ∇v+c(t, x)v+f(t, x) = 0,(t, x) ∈(0, T ]×D,

v(0,x) = g(x),x∈D,

v(t, x) = χ(t, x),(t, x) ∈[0, T ]×(Rn\D),

(1.4)

where b(t, x) = −b(T−t, x), c(t, x) = −c(T−t, x), f(t, x) = −f(T−t, x), and

χ(t, x) = χ(T−t, x).

Probabilistic numerical methods (usually implemented with Monte Carlo method)

provide eﬀective approaches for numerically solving partial diﬀerential equation in

high dimension with/without fractional Laplacian as probabilistic methods do not

require any discretization in space [15, 27, 28, 29, 30, 31, 32].

Classical Laplace operator and fractional Laplacian are the inﬁnitesimal gen-

erators of Brownian motion and symmetric 2s-stable process, respectively, which

connect partial diﬀerential equations with stochastic processes. Muller [25] ﬁrst

proposed walk-on-spheres method by simulating the paths of the Brownian motion

in spheres to numerically solve the steady-state equation with classical Laplace oper-

ator. Kyprianou et al. [18] utilized walk-on-spheres method to approximate solution

of the steady-state equation with fractional Laplacian based on the distribution of

symmetric 2s-stable process issued from the origin, when it ﬁrst exits a unit sphere.

However, the walk-on-spheres method generally can not generate the ﬁrst exit time

from the domain D. Therefore, it is diﬃcult for walk-on-spheres method to solve

parabolic problems numerically. Milstein and Tretyakov [23, 24] approximated the

trajectory of diﬀusion process with Brownian motion by Euler scheme which is a

uniformly-time discretization scheme to solve classical parabolic problems with in-

teger order derivative. [9] gave a random walk algorithm for the Dirichlet problem

for parabolic integro-diﬀerential equation where the kernel of the integro-diﬀerential

operator has better regularities than the kernel of fractional Laplacian.

In this article, we can approximate the trajectory of the system of stochastic

diﬀerential equations with symmetric 2s-stable process to numerically solve frac-

tional parabolic problems (1.1). However, simulating this system via Euler scheme

suﬀers from two diﬃculties: First, the jump intensity of the symmetric 2s-stable

process is inﬁnite, which means there is an inﬁnite number of jumps in every inter-

val of nonzero length; Second, the exit time of the process leaving the domain D

is hard to obtain. To overcome the ﬁrst diﬃculty, we utilize the idea of Asmussen

and Rosi´nski [5] and remove small jumps or replace small jumps with correspond-

ing simple processes. Jump-adapted scheme [9, 17, 22] is adapted to go through

the second diﬃculty. Compared with classical Euler scheme, jump-adapted scheme

uses adaptive non-uniform discretization based on the times of jumps of the driving

process.

In this paper, we ﬁrst give the probabilistic representation of the solution to

equation (1.1), which is deeply connected with the system of stochastic diﬀeren-

tial equations with symmetric 2s-stable process. We then consider a simple jump-

adapted Euler scheme to approximate the trajectory of the stochastic process and

obtain the numerical solution to equation (1.1). Furthermore, we propose a high-

order jump-adapted scheme by replacing small jumps with simple process if the

regularity of the solution u(t, x) is good enough. In addition, we give the weak

convergence of the simulated L´evy process and the error estimate , which is related

to the jumping intensity and statistical error. In comparison with [9], we study

the Dirichlet problem for the parabolic problem with fractional Laplacian where the

kernel is more singular and the related symmetric 2s-stable process do not exist any

moments. Based on the diﬀerent regularities of the solution u(t, x), we give the

corresponding numerical schemes. For the optimal error estimate of the numerical

scheme, we require u∈C1,3([0, T ]×Rn), while [9] requires a solution of the auxiliary

Dirichlet problem belonging to C2,4([0, T ]×Rn).

The rest of the paper is organized as follows. In Section 2, the probabilistic

representation of the solution to equation (1.1) is given, which is related to the

system of stochastic diﬀerential equations with 2s-stable process. In Section 3, A

simple jump-adapted Euler scheme is derived. A high-order jump-adapted scheme

is proposed in Section 4. The weak convergence of the simulated process and error

estimates are presented in the corresponding sections. In Section 5, numerical ex-

periments are performed to verify the theoretical analysis. Finally, we summarize

the main work in the last section.

In the following sections, we denote positive constants by C1and C2which may

be dependent of the index s, but not necessarily the same at diﬀerent situations.

2 Probabilistic representation

Let (Ω,F,P) be a complete probability space with a ﬁltration F={Ft}t∈[0,T ]of

σ-algebra satisfying the usual conditions. Consider the symmetric 2s-stable process

[4],

dLη=Z|y|<1

N(dη, dy) + Z|y|≥1

yN(dη, dy),(2.1)

where N(dη, dy) is a Poisson random measure on [0, T ]×Rn

0, (Rn

0=Rn\{0}) with

E[N(dη, dy)] = ν(dy)dη=C(n, s)dy

|y|n+2sdη(2.2)

and

N(dη, dy) = N(dη, dy) −ν(dy)dη(2.3)

is the compensated Poisson random measure. The characteristic function is given

by [34],

Eei(ξ,Lt)= exp tZRnei(ξ,y) −1−i(ξ, y)ν(dy)

= exp tC(n, s)ZRn

cos (ξ, y) −1

|y|n+2sdy

=e−t|ξ|2s.

(2.4)

It can be easily got that the inﬁnitesimal generator of Ltis −(−∆)s[4] . Thus,

fractional Laplacian is closely related to the symmetric 2s-stable process.

Consider the following system of stochastic diﬀerential equations,











dXη=b(η, Xη−)dη+ dLη,Xt= x,

dYη=c(η, Xη−)Yηdη, Yt= 1,

dZη=f(η, Xη−)Yηdη, Zt= 0.

(2.5)

Based on the above system, we give the probabilistic representation of the solution

to equation (1.1) in the following theorem.

Theorem 2.1. Let u(t, x) be the solution of equation (1.1). Then u(t, x) can be

given by

u(t, x) = Ehu(T∧τt,x,Xt,x

T∧τt,x)Yt,x,1

T∧τt,x+Zt,x,1,0

T∧τt,xi

=EnI{τt,x<T }hχ(τt,x,Xt,x

τt,x)Yt,x,1

τt,x+Zt,x,1,0

τt,xio

+EnI{τt,x≥T}hg(Xt,x

T)Yt,x,1

T+Zt,x,1,0

Tio,

(2.6)

where Xt,x

η,Yt,x,y

ηand Zt,x,y,z

η(t≤η≤T)are the solution of Cauchy problem (2.5),

τt,x={η≥t, Xt,x

η/∈D}is the ﬁrst exit time of Xt,x

ηstarting from x∈Rnto the

boundary D.

Proof. From Itˆo formula, we have

du(η, Xη) = ∂u

∂η (η, Xη−)dη+

i=1

bi(η, Xη−)∂u

∂xi

(η, Xη−)dη

+Z|y|≥1

[u(η, Xη−+ y) −u(η, Xη−)] N(dη, dy)

+Z|y|<1

[u(η, Xη−+ y) −u(η, Xη−)] e

N(dη, dy)

+Z|y|<1

[u(η, Xη−+ y) −u(η, Xη−)−(∇u(η, Xη−),y)] ν(dy)dη.

(2.7)

Since

dYη=c(η, Xη−)Yηdηand dYη·du(η, Xη)=0,(2.8)

it follows that

du(η, Xη)Yη=u(η, Xη)dYη+Yηdu(η, Xη).(2.9)

Thus, we obtain

Eu(T∧τt,x,Xt,x

T∧τt,x)Yt,x,1

T∧τt,x+Zt,x,1,0

T∧τt,x−u(t, x)

=E(ZT∧τt,x

t∂u

∂η (η, Xη−) +

i=1

bi(η, Xη−)∂u

∂xi

(η, Xη−)

+Zy∈Rnhu(η, Xη−+ y) −u(η, Xη−)−I{|y|<1}(∇u(η, Xη−),y)iν(dy)

+f(η, Xη−) + c(η, Xη−)u(η, Xη−)exp Zη

c(v, Xv−)dvdη)

=0,

(2.10)

where we have used

Z|y|<1

(∇u(η, Xη−),y)ν(dy) = 0, Yη= exp Zη

c(v, Xv−)dv(2.11)

and equation (1.1). Thus we complete the proof.

At last, we introduce some notations which will be used later.

For β=bβc+{β}+>0, bβc ∈ Z+∪ {0},{β}+∈[0,1), let Cβ([0, T ]×Rn)

denote the space of measurable functions u(t, x) on Rnfor any t∈[0, T ] such that

the norm

|u(t, x)|β=X

|γ|≤bβc

sup

(t,x)∈[0,T ]×Rn|∂γ

xu(t, x)|+ sup

γ=bβc

t,x,h6=0

|∂γ

xu(t, x+h) −∂γ

xu(t, x)|

|h|{β}+(2.12)

is ﬁnite. Cβ(Rn) is just the general H¨older space whose functions are deﬁned on Rn.

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MonteCarlomethodforparabolicequationsinvolvingfractionalLaplacian*CaiyuJiao,ChangpinLiDepartmentofMathematics,ShanghaiUniversity,Shanghai200444,ChinaAbstractWeapplytheMonteCarlomethodtosolvingtheDirichletproblemoflinearparabolicequationswithfractionalLaplacian.Thismethodexploitstheideaofweakapproxi...

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Monte Carlo method for parabolic equations involving fractional Laplacian Caiyu Jiao Changpin Li

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