What Monte Carlo models can do and cannot do efficiently?

The question "what Monte Carlo models can do and cannot do efficiently" is discussed for some functional spaces that define the regularity of the input data. Data classes important for practical computations are considered: classes of functions with bounded derivatives and Holder type conditions, as well as Korobov-like spaces. Theoretical performance analysis of some algorithms with unimprovable rate of convergence is given. Estimates of computational complexity of two classes of algorithms - deterministic and randomized for both problems - numerical multidimensional integration and calculation of linear functionals of the solution of a class of integral equations are presented. (c) 2007 Elsevier Inc. All rights reserved.

Monte Carlo methods for matrix computations on the grid

Many scientific and engineering applications involve inverting large matrices or solving systems of linear algebraic equations. Solving these problems with proven algorithms for direct methods can take very long to compute, as they depend on the size of the matrix. The computational complexity of the stochastic Monte Carlo methods depends only on the number of chains and the length of those chains. The computing power needed by inherently parallel Monte Carlo methods can be satisfied very efficiently by distributed computing technologies such as Grid computing. In this paper we show how a load balanced Monte Carlo method for computing the inverse of a dense matrix can be constructed, show how the method can be implemented on the Grid, and demonstrate how efficiently the method scales on multiple processors. (C) 2007 Elsevier B.V. All rights reserved.

A sparse parallel hybrid Monte Carlo algorithm for matrix computations

In this paper we introduce a new algorithm, based on the successful work of Fathi and Alexandrov, on hybrid Monte Carlo algorithms for matrix inversion and solving systems of linear algebraic equations. This algorithm consists of two parts, approximate inversion by Monte Carlo and iterative refinement using a deterministic method. Here we present a parallel hybrid Monte Carlo algorithm, which uses Monte Carlo to generate an approximate inverse and that improves the accuracy of the inverse with an iterative refinement. The new algorithm is applied efficiently to sparse non-singular matrices. When we are solving a system of linear algebraic equations, Bx = b, the inverse matrix is used to compute the solution vector x = B(-1)b. We present results that show the efficiency of the parallel hybrid Monte Carlo algorithm in the case of sparse matrices.

Error analysis of a Monte Carlo algorithm for computing bilinear forms of matrix powers

In this paper we present error analysis for a Monte Carlo algorithm for evaluating bilinear forms of matrix powers. An almost Optimal Monte Carlo (MAO) algorithm for solving this problem is formulated. Results for the structure of the probability error are presented and the construction of robust and interpolation Monte Carlo algorithms are discussed. Results are presented comparing the performance of the Monte Carlo algorithm with that of a corresponding deterministic algorithm. The two algorithms are tested on a well balanced matrix and then the effects of perturbing this matrix, by small and large amounts, is studied.

A Monte Carlo Approach for the Cook -Torrance Model

In this work we consider the rendering equation derived from the illumination model called Cook-Torrance model. A Monte Carlo (MC) estimator for numerical treatment of the this equation, which is the Fredholm integral equation of second kind, is constructed and studied.

Comparison of the computational cost of a Monte Carlo and deterministic algorithm for computing bilinear forms of matrix powers

In this paper we consider bilinear forms of matrix polynomials and show that these polynomials can be used to construct solutions for the problems of solving systems of linear algebraic equations, matrix inversion and finding extremal eigenvalues. An almost Optimal Monte Carlo (MAO) algorithm for computing bilinear forms of matrix polynomials is presented. Results for the computational costs of a balanced algorithm for computing the bilinear form of a matrix power is presented, i.e., an algorithm for which probability and systematic errors are of the same order, and this is compared with the computational cost for a corresponding deterministic method.

A cluster computing implementation of a Monte Carlo simulation of a particle growth mechanism

Clusters of computers can be used together to provide a powerful computing resource. Large Monte Carlo simulations, such as those used to model particle growth, are computationally intensive and take considerable time to execute on conventional workstations. By spreading the work of the simulation across a cluster of computers, the elapsed execution time can be greatly reduced. Thus a user has apparently the performance of a supercomputer by using the spare cycles on other workstations.

Monte Carlo phase diagram for a polydisperse diblock copolymer melt

The phase diagram for an AB diblock copolymer melt with polydisperse A blocks and monodisperse B blocks is evaluated using lattice-based Monte Carlo simulations. Experiments on this system have shown that the A-block polydispersity shifts the order-order transitions (OOTs) towards higher A-monomer content, while the order-disorder transition (ODT) moves towards higher temperatures when the A blocks form the minority domains and lower temperatures when the A blocks form the matrix. Although self-consistent field theory (SCFT) correctly accounts for the change in the OOTs, it incorrectly predicts the ODT to shift towards higher temperatures at all diblock copolymer compositions. In contrast, our simulations predict the correct shifts for both the OOTs and the ODT. This implies that polydispersity amplifies the fluctuation-induced correction to the mean-field ODT, which we attribute to a reduction in packing frustration. Consistent with this explanation, polydispersity is found to enhance the stability of the perforated-lamellar phase.