### Chapter 8. Applications to the Random Walk on Spheres method [chapter]

1997 Spherical Means for PDEs
The integral equations with a convergent Neumann series can be numerically solved by the Monte Carlo method using the Markov chains . To make things clear we first illustrate the situation by a simple evaluation of the spherical mean (in R 3 ) by the Monte Carlo method. Let s be a random vector uniformly distributed on the sphere 5(x, r). Then by definition, the expectation of the random variable u(s) is equal to the spherical mean Nru: Eu(s) = Nru = J-j J u(y) dS(y) . (8.1) S(*,r) By the
more » ... .1) S(*,r) By the law of large numbers, the expectation is approximated by the arithmetic mean: Unauthenticated Download Date | 3/5/20 5:35 PM 8.2. Iterations of the spherical mean operator 151 PI. Put V:=0; i:=l; P2. Sample a random vector s uniformly distributed on the sphere S(x, r): s = x + (r. Here C = (Ci> C2, C3) is a unit isotropic vector which is simulated as follows: 2.1 Ci = 1 -2f *o; 2.2 71 = 1 -2ai; 72 = 1 -2a2; d = 7? + lh 2.3 If d > 1, then go to 2.2, otherwise 2.4 c2 = W(1 -Ct)/d; C3 = 72 -Ci)/d; {Here ao,ai,d2 are independent samples generated by the standard generator of pseudo-random numbers uniformly distributed on [0,1]} P3. V := V + u(s); if i < N go to P2; P4. The arithmetic mean (8.2) is then UN = V/N. Iterations of the spherical mean operator In this section, let us use the following notation for the spherical mean JV®' y [u] where the superscript indicates that the spherical mean is taken over the sphere S(x,r), and y is the variable of integration over S(x,r): h = = J-j J u(y)dS(y) , (8.3) S(r,r) i.e., y -x + rs. Suppose that it is desired to calculate the second iteration of the spherical mean h=Avrt^rMi- a set of Markov chains starting at the point x, with the first random state yi uniformly distributed on S(x, r\) and the second state 2/2 uniformly distributed on S(yi, provided t/i is fixed. On this Markov chain we define the random variable £(x) = u(y2).