When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux radiance because the macroscopic cross-section @math depends on the distance @math to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived and reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
The Monte Carlo and deterministic methods in this paper apply generally to any stochastic processes where a particle moves along straight paths at constant speed between collisions that change the particle's direction and possibly absorb the particle. This include some, but not all, of the stochastic processes that are studied under names such as L 'e vy walks @cite_12 @cite_145 , random evolutions @cite_40 , velocity jump process @cite_22 and semi-Markov random flights @cite_132 @cite_8 @cite_60 @cite_86 @cite_95 @cite_111 @cite_129 @cite_42 @cite_38 .
@cite_38 @cite_22 @cite_8 @cite_132 @cite_60 @cite_145 @cite_42 @cite_129 @cite_95 @cite_40 @cite_86 @cite_111 @cite_12
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We propose a novel tracking algorithm based on the Wang-Landau Monte Carlo sampling method which efficiently deals with the abrupt motions. Abrupt motions could cause conventional tracking methods to fail since they violate the motion smoothness constraint. To address this problem, we introduce the Wang-Landau algorithm that has been recently proposed in statistical physics, and integrate this algorithm into the Markov Chain Monte Carlo based tracking method. Our tracking method alleviates the motion smoothness constraint utilizing both the likelihood term and the density of states term, which is estimated by the Wang-Landau algorithm. The likelihood term helps to improve the accuracy in tracking smooth motions, while the density of states term captures abrupt motions robustly. Experimental results reveal that our approach efficiently samples the object's states even in a whole state space without loss of time. Therefore, it tracks the object of which motion is drastically changing, accurately and robustly. A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form.We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions. This chapter describes a sequence of Monte Carlo methods: importance sampling, rejection sampling, the Metropolis method, and Gibbs sampling. For each method, we discuss whether the method is expected to be useful for high—dimensional problems such as arise in inference with graphical models. After the methods have been described, the terminology of Markov chain Monte Carlo methods is presented. The chapter concludes with a discussion of advanced methods, including methods for reducing random walk behaviour. SUMMARY This paper proposes a post-simulation improvement for two common Monte Carlo methods, the Accept-Reject and Metropolis algorithms. The improvement is based on a Rao-Blackwellisation method that integrates over the uniform random variables involved in the algorithms, and thus post-processes the standard estimators. We show how the Rao-Blackwellised versions of these algorithms can be implemented and, through examples, illustrate the improvement in variance brought by these new procedures. We also compare the improved version of the Metropolis algorithm with ordinary and Rao-Blackwellised importance sampling procedures for independent and general Metropolis set-ups.