CCP - lect.09 - Introduction to Percolation

Annals of Computer Science and Information Systems, 2020

An optimal stochastic approach for multidimensional integrals of smooth functions. This is the first time this optimal stochastic approach has been compared with other stochastic approaches for mid and high dimensions. The purpose of the present study is to compare the optimal algorithm with the lattice rules based on the generalized Fibonacci numbers of the corresponding dimension and to discuss the advantages and disadvantages of each method.

2015

The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.

2014

Introduction 2 2 Introducing Quasi-Monte Carlo methods 5 3 Error, discrepancy, and reproducing kernel 32 4 Weighted spaces and tractability 45 5 Lattice rules 59 6 Digital nets and sequences 97 7 Infinite dimensional integration 131 8 Concluding remarks 142 References 143 High dimensional integration-the Quasi-Monte Carlo way 13 Example 2.10. (Component-by-component (CBC) construction) Given n, we construct a generating vector z = (z 1 , z 2 ,. . .) as follows: 1 Set z 1 = 1. 2 With z 1 held fixed, choose z 2 from U n to minimize a desired error criterion in 2 dimensions. 3 With z 1 , z 2 held fixed, choose z 3 from U n to minimize a desired error criterion in 3 dimensions. 4 With z 1 , z 2 , z 3 held fixed, choose z 4 from U n to minimize a desired error criterion in 4 dimensions. 5. . .

Mathematics and Computers in Simulation, 2008

We propose an adaptive Monte Carlo algorithm for estimating multidimensional integrals over a hyper-rectangular region. The algorithm uses iteratively the idea of separating the domain of integration into 2 s subregions. The proposed algorithm can be applied directly to estimate the integral using an efficient way of storage. We test the algorithm for estimating the value of a 30-dimensional integral using a two-division approach. The numerical results show that the proposed algorithm gives better results than using one-division approach.

Lecture Notes in Computer Science, 2011

An efficient Monte Carlo method for multidimensional integration is proposed and studied. The method is based on Sobol's sequences. Each random point in s-dimensional domain of integration is generated in the following way. A Sobol's vector of dimension s (ΛΠ τ point) is considered as a centrum of a sphere with a radius ρ. Then a random point uniformly distributed on the sphere is taken and a random variable is defined as a value of the integrand at that random point. It is proven that the mathematical expectation of the random variable is equal to the desired multidimensional integral. This fact is used to define a Monte Carlo algorithm with a low variance. Numerical experiments are performed in order to study the quality of the algorithm depending of the radius ρ and regularity, i.e. smoothness of the integrand.

Communication Papers of the 17th Conference on Computer Science and Intelligence Systems

We study an optimized Monte Carlo algorithm for solving multidimensional integrals related to intelligent systems. Recently Shaowei Lin consider the difficult task of evaluating multidimensional integrals with very high dimensions which are important to machine learning for intelligent systems. Lin multidimensional integrals with 3 to 30 dimensions, related to applications in machine learning, will be evaluated with the presented optimized Monte Carlo algorithm and some advantages of the method will be analyzed.

The Annals of Applied Probability, 1995

For a wide sense stationary random field (D = {?(x): x E R2}, we investigate the asymptotic errors made in the numerical integration of line intergrals of the form ft f(x)0(x)do(x). It is shown, for example, that if f and F are smooth, and if the spectral density p(A) satisfies p(A) kIAI-4 as A-+ oo, then there is a constant c' with N3EI fr f (x)4(x)dcr(x)-E(p,~(Xj)12 > c'N-3 for all finite sets {xj: 1 < j < N} and all choices of coefficients {/38}. And, if any fixed parameterization x(t) of F is given and the integral fJ f(x(t))0(x(t))Ix'(t)Idt is numerically integrated using the midpoint method, the exact asymptotics of the mean squared error is derived. This leads to asymptotically optimal designs, and generalizes to other power laws and to nonstationary and nonisotropic fields.

arXiv (Cornell University), 2022

The theme of the present paper is numerical integration of C r functions using randomized methods. We consider variance reduction methods that consist in two steps. First the initial interval is partitioned into subintervals and the integrand is approximated by a piecewise polynomial interpolant that is based on the obtained partition. Then a randomized approximation is applied on the difference of the integrand and its interpolant. The final approximation of the integral is the sum of both. The optimal convergence rate is already achieved by uniform (nonadaptive) partition plus the crude Monte Carlo; however, special adaptive techniques can substantially lower the asymptotic factor depending on the integrand. The improvement can be huge in comparison to the nonadaptive method, especially for functions with rapidly varying rth derivatives, which has serious implications for practical computations. In addition, the proposed adaptive methods are easily implementable and can be well used for automatic integration.

2002

Good lattice quadrature rules are known to have O(N −2+ǫ) convergence for periodic integrands with sufficient smoothness. Here it is shown that applying the baker's transformation to lattice rules gives O(N −2+ǫ) convergence for nonperiodic integrands with sufficient smoothness. This approach is philosophically and practically different than making a periodizing transformation of the integrand, and this difference is explained.

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].