scholarly journals Convergence of Monte Carlo algorithm for solving integral equations in light scattering simulations

2020 ◽  
Vol 50 (1) ◽  
Author(s):  
Maciej Kraszewski ◽  
Jerzy Pluciński
Keyword(s):  
Monte Carlo ◽  
Light Scattering ◽  
Integral Equations ◽  
Three Dimensional ◽  
Monte Carlo Algorithm ◽  
Refractive Indices ◽  
Algebraic Equations ◽  
Volume Integral ◽  
Scattering Process ◽  
Convergence Properties

The light scattering process can be modeled mathematically using the Fredholm integral equation. This equation is usually solved after its discretization and transformation into the system of algebraic equations. Volume integral equations can be also solved without discretization using the Monte Carlo algorithm, but its application to the light scattering simulations has not been sufficiently studied. Here we present the implementation of this algorithm for one- and three-dimensional light scattering computations and discuss its applicability in this field. We show that the Monte Carlo algorithm can provide valid and accurate results but, due to its convergence properties, it might be difficult to apply for problems with large volumes or refractive indices of scattering objects.

scholarly journals Application of the Monte Carlo algorithm for solving volume integral equation in light scattering simulations

2020 ◽  
Vol 50 (1) ◽  
Author(s):  
Maciej Kraszewski ◽  
Jerzy Pluciński
Keyword(s):  
Integral Equation ◽  
Monte Carlo ◽  
Light Scattering ◽  
Numerical Methods ◽  
Dipole Approximation ◽  
Discrete Dipole Approximation ◽  
Monte Carlo Algorithm ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Large Particles

Various numerical methods were proposed for analysis of the light scattering phenomenon. An important group of these methods is based on solving the volume integral equation describing the light scattering process. The popular method from this group is the discrete dipole approximation. Discrete dipole approximation uses various numerical algorithms to solve the discretized integral equation. In the recent years, the application of the Monte Carlo algorithm as one of them was proposed. In this research, we analyze the application of the Monte Carlo algorithm for two cases: the light scattering by large particles and by random conglomerates of small particles. We show that if proper preconditioning of the numerical problem is applied, the Monte Carlo algorithm can solve the underlying systems of linear equations. We also show that the efficiency of the Monte Carlo algorithm can be increased by reusing performed computations for various incident electromagnetic waves and the applicability of the Monte Carlo algorithm depends on the particular use case. It is unlikely to be used in the case of light scattering by the large particles due to computational times inferior comparing with the other numerical methods but may become useful in the case of light scattering by the random conglomerates of small scattering particles.

scholarly journals Canonical Least-Squares Monte Carlo Valuation of American Options: Convergence and Empirical Pricing Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Xisheng Yu ◽  
Qiang Liu
Keyword(s):  
Monte Carlo ◽  
Least Squares ◽  
Legendre Polynomials ◽  
American Options ◽  
Computational Cost ◽  
Monte Carlo Algorithm ◽  
Convergence Properties ◽  
Entropy Model ◽  
Least Squares Monte Carlo ◽  
Convergence Results

The paper by Liu (2010) introduces a method termed the canonical least-squares Monte Carlo (CLM) which combines a martingale-constrained entropy model and a least-squares Monte Carlo algorithm to price American options. In this paper, we first provide the convergence results of CLM and numerically examine the convergence properties. Then, the comparative analysis is empirically conducted using a large sample of the S&P 100 Index (OEX) puts and IBM puts. The results on the convergence show that choosing the shifted Legendre polynomials with four regressors is more appropriate considering the pricing accuracy and the computational cost. With this choice, CLM method is empirically demonstrated to be superior to the benchmark methods of binominal tree and finite difference with historical volatilities.

scholarly journals Nonreversible Markov Chain Monte Carlo Algorithm for Efficient Generation of Self-Avoiding Walks

2022 ◽  
Vol 9 ◽  
Author(s):  
Hanqing Zhao ◽  
Marija Vucelja
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Markov Chains ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Monte Carlo Algorithm ◽  
Efficient Generation ◽  
Self Avoiding Walk ◽  
Self Avoiding Walks

We introduce an efficient nonreversible Markov chain Monte Carlo algorithm to generate self-avoiding walks with a variable endpoint. In two dimensions, the new algorithm slightly outperforms the two-move nonreversible Berretti-Sokal algorithm introduced by H. Hu, X. Chen, and Y. Deng, while for three-dimensional walks, it is 3–5 times faster. The new algorithm introduces nonreversible Markov chains that obey global balance and allow for three types of elementary moves on the existing self-avoiding walk: shorten, extend or alter conformation without changing the length of the walk.

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