Optimal Korobov Coefficients for Good Lattice Points in Quasi Monte Carlo Algorithms

Abstract

The optimal coefficients in the sense of Korobov serve for obtaining good lattice points sets. By using these sets for multidimensional numerical integration, numerical solution of integral equation and other related applications, the relative error of corresponding Quasi Monte Carlo algorithms can be kept relatively small. This study deals with finding optimal coefficients for good lattice points for high dimensional problems in weighted Sobolev and Korobov spaces. Two Quasi Monte Carlo algorithms for boundary value problems are proposed and analyzed. For the first of them the coefficients that characterize the good lattice points are found “component-by-component”: the (k + l)th coefficient is obtained by one-dimensional search with all previous k coefficients kept unchanged. For the second algorithm, the coefficients depending on single parameter are found in Korobov's form. Some numerical experiments are made to illustrate the obtained results.