Mariya K. Durchova

Generating and Testing the Modified Halton Sequences

The Halton sequences are one of the most popular low-discrepancy sequences, used for calculating multi-dimensional integrals or in quasi-Monte Carlo simulations. Various techniques for their randomization exist. One of the authors proved that for one such modification an estimate of the discrepancy with a very small constant before the leading term can be proved. In this paper we describe an efficient algorithm for generating these sequences on computers and show timing results, demonstrating the efficiency of the algorithm. We also compare the integration error of these sequences with that of the classical Halton sequences on families of functions widely used for such benchmarking purposes. The results demonstrate that the modified Halton sequences can be used successfully in quasi-Monte Carlo methods.

Quasi-Monte Carlo integration on the grid for sensitivity studies

In this paper we present error and performance analysis of quasi-Monte Carlo algorithms for solving multidimensional integrals (up to 100 dimensions) on the grid using MPI. We take into account the fact that the Grid is a potentially heterogeneous computing environment, where the user does not know the specifics of the target architecture. Therefore parallel algorithms should be able to adapt to this heterogeneity, providing automated load-balancing. Monte Carlo algorithms can be tailored to such environments, provided parallel pseudorandom number generators are available. The use of quasi-Monte Carlo algorithms poses more difficulties. In both cases the efficient implementation of the algorithms depends on the functionality of the corresponding packages for generating pseudorandom or quasirandom numbers. We propose efficient parallel implementation of the Sobol sequence for a grid environment and we demonstrate numerical experiments on a heterogeneous grid. To achieve high parallel efficiency we use a newly developed special grid service called Job Track Service which provides efficient management of available computing resources through reservations.

A New Quasi-Monte Carlo Algorithm for Numerical Integration of Smooth Functions

Bachvalov proved that the optimal order of convergence of a Monte Carlo method for numerical integration of functions with bounded kth order derivatives is \(\mathop{O}\left(N^{-\frac{k}{s}-\frac{1}{2}}\right)\), where s is the dimension. We construct a new Monte Carlo algorithm with such rate of convergence, which adapts to the variations of the sub-integral function and gains substantially in accuracy, when a low-discrepancy sequence is used instead of pseudo-random numbers.

Performance Analysis of the Regional Grid Resources for an Environmental Modeling Application

The speed of execution of resource intensive application depends mostly on the performance of the underlying hardware and network infrastructure. The overall performance of complex Grid applications that include different types of processing in the same Grid job is difficult to predict reliably. In this paper we define several key performance indicators and collect data from the execution of a resource intensive environmental modeling application on the regional resources of the European Grid Infrastructure. The application is based on the Models-3 system, consisting of three components: meteorological pre-processor MM5, chemical transport model CMAQ and emission pre-processor SMOKE. The computations are resource intensive with respect to the input and output data which stress both the computational and data capabilities of the resource centers. In the paper we analyze the relative importance of these indicators and draw conclusions, regarding the optimal use of available resource centers.

A quasi-monte carlo method for an elastic electron back-scattering problem

The elastic electron back-scattering is a problem that is important for many theoretical and experimental techniques, especially in the determination of the inelastic mean free paths. This effect arises when a monoenergetic electron beam bombards a solid target and some of the electrons are scattered without energy loss.The description of the flow can be written as an integral equation and may be solved by Monte Carlo methods. In this paper we investigate the possibility of improving the convergence of the Monte Carlo algorithm by using scrambled low-discrepancy sequences. We demonstrate how by taking advantage of the smoothness of the differential elastic-scattering cross-section a significant decrease of the error is achieved. We show how the contribution of the first few collisions to the result can be evaluated by an appropriate integration method instead of direct simulation, which further increases the accuracy of our computations without increase of the computational time. In order to facilitate these techniques, we use spline approximation of the elastic cross-section, which is more accurate than the widely used tables of Jablonski.