A quasi-Monte Carlo method for elliptic boundary value problems (Q2732325)

The essential point of using the Monte Carlo method for calculating linear functionals of the solution of an elliptic boundary problem in a domain \(G\) is to write the solution into a Fredholm integral equation of the second kind with a local probabilistic trasition kernel \(p(x,y)\) on the maximal ball in \(G\) with center \(x\). A Markovian random sample \({\xi_j}\) is then chosen in the maximal ball \(B(\xi_{j-1})\) in \(G\) by the density \(p(\xi_{j-1},\xi_j)\) and this procedure terminates at the boundary of \(G\). This algorithm is known as the random walks on balls. NEWLINENEWLINENEWLINEIn the present paper, for a 3-dimensional domain \(G\), the kernel \(p(\xi_{j-1},\xi_j)\) is represented in spherical coordinates and by using the acceptance-rejection, a Monte Carlo calculation is performed. The method of combining pseudorandom and quasirandom elements constructed in stead of the Markovian random sample is also discussed. Numerical results are compared.