Towards Real-Time Monte Carlo for Biomedicine
- 1.3k Downloads
Monte Carlo methods provide the “gold standard” computational technique for solving biomedical problems but their use is hindered by the slow convergence of the sample means. An exponential increase in the convergence rate can be obtained by adaptively modifying the sampling and weighting strategy employed. However, if the radiance is represented globally by a truncated expansion of basis functions, or locally by a region-wise constant or low degree polynomial, a bias is introduced by the truncation and/or the number of subregions. The sheer number of expansion coefficients or geometric subdivisions created by the biased representation then partly or entirely offsets the geometric acceleration of the convergence rate. As well, the (unknown amount of) bias is unacceptable for a gold standard numerical method. We introduce a new unbiased estimator of the solution of radiative transfer equation (RTE) that constrains the radiance to obey the transport equation. We provide numerical evidence of the superiority of this Transport-Constrained Unbiased Radiance Estimator (T-CURE) in various transport problems and indicate its promise for general heterogeneous problems.
KeywordsMonte Carlo simulations Transport-constrained radiance estimators
The third author gratefully acknowledges partial support from award numbers: P41RR001192 from the National Center for Research Resources and P41EB015890 from the National Institute of Biomedical Imaging and Bioengineering.
The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of the National Center For Research Resources, National Institute of Biomedical Imaging and Bioengineering, or the National Institutes of Health.
- 8.Kong, R.: Transport problems and Monte Carlo methods. Ph.D. thesis, Claremont Graduate University (1999)Google Scholar
- 12.Kong, R., Spanier, J.: Transport-constrained extensions of collision and track length estimators for solutions of radiative transport problems. J. Comput. Phys. 242(0), 682–695 (2013). https://doi.org/10.1016/j.jcp.2013.02.023, http://www.sciencedirect.com/science/article/pii/S0021999113001423MathSciNetCrossRefGoogle Scholar
- 16.Scott, D.: Multivariate Density Estimation: Theory. Practice and Visualization. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1990)Google Scholar
- 18.Spanier, J., Gelbard, E.: Monte Carlo Principles and Neutron Transport Problems. Wesley, New York (1969). (reprinted by Dover Publications, Inc. 2008)Google Scholar
- 19.Spanier, J., Kong, R.: A new adaptive method for geometric convergence. Monte Carlo and Quasi-MonteCarlo Methods 2002, pp. 439–449. Springer, Berlin (2004)Google Scholar
- 20.Veach, E.: Robust Monte Carlo methods for light transport simulation. Ph.D. thesis, Stanford, CA, USA (1998). AAI9837162Google Scholar