Volume 19, Issue 4

Abstract. The generation of appropriate parallel and high-quality quasirandom sequences (low-discrepancy sequences) is crucial to the success of quasi-Monte Carlo methods. The Kronecker sequence is well known to be one of the special types of low-discrepancy sequences, and one of its important advantages is that the Kronecker sequence is easy to implement due to its definition via the fractional parts of multiples of irrationals. However, the original Kronecker sequence suffers from correlations for different dimensions. These correlations result in poorly distributed two-dimensional projections. An approach to this is to find a modified Kronecker sequence via generalizing golden ratio and generate parallel sequences. This paper presents a new algorithm for finding a modified Kronecker sequence within special choices of irrationals. This modified sequence is numerically tested and shown empirically to be superior to the other widely used quasirandom sequences. In addition, based on analysis and insight into the correlations between dimensions of the Kronecker sequence, we illustrate why our algorithm is efficient for breaking these correlations.

Abstract. We propose a new numerical scheme for a class of one-dimensional reflected stochastic differential equations (SDEs) by virtue of their explicit solutions, which enables us to carry out the simulation of this class of reflected SDEs by simulating some related SDEs without reflections. The new scheme yields the same order of convergence as the scheme for SDEs without reflections.

Abstract. In this paper, we make an offer of the lattice approximate method for solving a class of multi-dimensional integro-differential equations with the initial conditions. Also, we analyze the worst case error measured in weighted Korobov spaces for these equations. Finally, numerical examples complete this work.

Abstract. We design a variance reduction method to reduce the estimation error in regression problems. It is based on an appropriate use of other known regression functions. Theoretical estimates are supporting this improvement and numerical experiments are illustrating the efficiency of the method.