Summary
In this paper we present quasi-Monte Carlo methods for solving elliptic boundary value problems. We consider two types of random walk Monte Carlo approaches: grid-walk and grid-free. The first one uses a discretization of the problem on a mesh and solves the linear algebraic system which approximates the original problem. The second approach uses an integral representation of the solution which leads to a random walk on spheres or to a random walk on balls method. Different strategies for using quasirandom sequences are tested in order to generate quasirandom walks on grids, spheres, and balls. Theoretical error bounds are established, numerical experiments with model elliptic boundary value problems in two and three dimensions are also solved. The quasi-Monte Carlo methods preserve the advantages of ordinary Monte Carlo for solving problems in complicated domains, and show slightly better rates of convergence.
Supported, in part, by the U.S. Army Research Office under Contract # DAAD19- 01-1-0675
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Mascagni, M., Karaivanova, A., Hwang, CO. (2004). Quasi-Monte Carlo Methods for Elliptic BVPs. In: Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2002. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18743-8_21
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DOI: https://doi.org/10.1007/978-3-642-18743-8_21
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