Sofiya Ivanovska

Tuning the generation of sobol sequence with owen scrambling

Sobol sequence is the most widely used low discrepancy sequence for numerical solving of multiple integrals and other quasi-Monte Carlo computations Owen first proposed scrambling of this sequence through permutation in a manner that maintained its low discrepancy Scrambling is necessary not only for error analysis but for parallel implementations Good scrambling is especially important for GRID applications However, scrambling is often difficult to implement and time consuming In this paper we propose fast generation of Sobol sequence with Owen scrambling, tuned to specific hardware Numerical and timing results, demonstrating the advantages of our approach are presented and discussed.

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.

Parallel Importance Separation and Adaptive Monte Carlo Algorithms for Multiple Integrals

Monte Carlo Method (MCM) is the only viable method for many high-dimensional problems since its convergence is independent of the dimension. In this paper we develop an adaptive Monte Carlo method based on the ideas and results of the importance separation, a method that combines the idea of separation of the domain into uniformly small subdomains with the Kahn approach of importance sampling. We analyze the error and compare the results with crude Monte Carlo and importance sampling which is the most widely used variance reduction Monte Carlo method. We also propose efficient parallelizations of the importance separation method and the studied adaptive Monte Carlo method. Numerical tests implemented on PowerPC cluster using MPI are provided.

A Quasi-Monte Carlo Method for Integration with Improved Convergence

Quasi-Monte Carlo methods are based on the idea that random Monte Carlo techniques can often be improved by replacing the underlying source of random numbers with a more uniformly distributed deterministic sequence. Quasi-Monte Carlo methods often include standard approaches of variance reduction, although such techniques do not necessarily directly translate. In this paper we present a quasi-Monte Carlo method for integration that combines a separation of the domain into uniformly small subdomains with the approach of importance sampling. Theoretical estimates for the error bounds and the convergence rate are established. A large number of numerical tests of the proposed method are presented and compared with crude Monte Carlo and weighted uniform sampling. All methods are realized using pseudorandom numbers, and Sobol, Halton and Faure quasirandom sequences. The numerical results confirm the improved convergence of the proposed method when the integrand has bounded derivatives.

Importance Separation for Solving Integral Equations

In this paper we describe and study importance separation Monte Carlo method for integral equations. Based on known results for integrals, we extend this method for solving integral equations. The method combines the idea of separation of the domain into uniformly small subdomains (adaptive technique) with the Kahn approach of importance sampling. We analyze the error and compare the results with the crude Monte Carlo method.

Monte Carlo Methods Using New Class of Congruential Generators

In this paper we propose a new class of congruential pseudo random number generator based on sequences generating permutations. These sequences have been developed for other applications but our analysis and experiments show that they are appropriate for approximation of multiple integrals and integral equations.

A superconvergent monte carlo method for multiple integrals on the grid

In this paper we present error and performance analysis of a Monte Carlo variance reduction method for solving multidimensional integrals and integral equations. This method combines the idea of separation of the domain into small subdomains with the approach of importance sampling. The importance separation method is originally described in our previous works [7,9]. Here we present a new variant of this method adding polynomial interpolation in subdomains. We also discuss the performance of the algorithms in comparison with crude Monte Carlo. We propose efficient parallel implementation of the importance separation method for a grid environment and we demonstrate numerical experiments on a heterogeneous grid. Two versions of the algorithm are compared – a Monte Carlo version using pseudorandom numbers and a quasi-Monte Carlo version using the Sobol and Halton low-discrepancy sequences [13,8].

Parallel Importance Separation for Multiple Integrals and Integral Equations

In this paper we present error and performance analysis of a Monte Carlo variance reduction method for solving multidimensional integrals and integral equations. This method, called importance separation, combines the idea of separation of the domain into uniformly small subdomains with the approach of importance sampling. The importance separation method is originally described in our previous works, here we generalize our results and discuss the performance in comparison with crude Monte Carlo and importance sampling. Based on our previous investigation we propose efficient parallelizations of the importance separation method. Numerical tests implemented on PowerPC cluster using MPI are provided. The considered algorithms are carried out using pseudorandom numbers.

On the Monte Carlo Matrix Computations on Intel MIC Architecture

Abstract The tightened energy requirements when designing state-of-the-art high performance computing systems lead to the increased use of computational accelerators. Intel introduced the Many Integrated Core (MIC) architecture for their line of accelerators and successfully competes with NVIDIA on basis of price/performance and ease of development. Although some codes may be ported successfully to Intel MIC architecture without significant modifications, in order to achieve optimal performance one has to make the best use of the vector processing capabilities of the architecture. In this work we present our implementation of Quasi-Monte Carlo methods for matrix computations specifically optimised for the Intel Xeon Phi accelerators. To achieve optimal parallel efficiency we make use of both MPI and OpenMP.

Chapter 3 Efficient Implementation of the Heston Model Using GPGPU

The Heston stochastic volatility model is widely used for modeling of option prices in financial markets. By adding a jump process to the model one can account for large spikes in volatility and achieve better fit of the implied volatility surface. When the parameters of the model have been calibrated to the observed market prices, the model can be used to compute prices of exotic options by Monte Carlo or quasi-Monte Carlo simulations. In our work we concentrate on the efficient implementation of the schemes of Kahl-Jäckel and Andersen while using the scrambled Sobol and Halton sequences. The codes were developed using CUDA for NVIDIA GPUs. We apply our methods to the problem of computing the Sobol sensitivity indices of the option prices as a function of the parameters of the Heston model and present numerical and timing results.