An Efficient Monte Carlo-Based Solver for Thermal Radiation in Participating Media

Monte Carlo-based solvers, while well-suited for accurate calculation of complex thermal radiation transport problems in participating media, are often deemed computationally unattractive for use in the solution of real-world problems. The main disadvantage of Monte Carlo (MC) solvers is their slow convergence rate and relatively high computational cost. This work presents a novel approach based on a low-discrepancy sequence (LDS) and is proposed for reducing the error bound of a Monte Carlo-based radiation solver. Sobols sequence – an LDS generated with a bit-by-bit exclusive-or operator – is used to develop a quasi-Monte Carlo (QMC) solver for thermal radiation in this work. Preliminary results for simple radiation problems in participating media show that the QMC-based solver has a lower error than the conventional MC-based solver. At the same time, QMC does not add any significant computational overhead. This essentially leads to a lower computational cost to achieve similar error levels from the QMC-based solver than the MC-based solver for thermal radiation

Challenges in Developing Great Quasi-Monte Carlo Software

Quasi-Monte Carlo (QMC) methods have developed over several decades. With the explosion in computational science, there is a need for great software that implements QMC algorithms. We summarize the QMC software that has been developed to date, propose some criteria for developing great QMC software, and suggest some steps toward achieving great software. We illustrate these criteria and steps with the Quasi-Monte Carlo Python library (QMCPy), an open-source community software framework, extensible by design with common programming interfaces to an increasing number of existing or emerging QMC libraries developed by the greater community of QMC researchers

A Study of Biased and Unbiased Stochastic Algorithms for Solving Integral Equations

In this paper we propose and analyse a new unbiased stochastic method for solving a class of integral equations, namely the second kind Fredholm integral equations. We study and compare three possible approaches to compute linear functionals of the integral under consideration: i) biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series, ii) transformation of this problem into the problem of computing a finite number of integrals, and iii) unbiased stochastic approach. Five Monte Carlo algorithms for numerical integration have been applied for approach (ii). Error balancing of both stochastic and systematic errors has been discussed and applied during the numerical implementation of the biased algorithms. Extensive numerical experiments have been performed to support the theoretical studies regarding the convergence rate of Monte Carlo methods for numerical integration done in our previous studies. We compare the results obtained by some of the best biased stochastic approaches with the results obtained by the proposed unbiased approach. Conclusions about the applicability and efficiency of the algorithms have been drawn