Early Research Projects
Initial (1998-2001) research projects of the institute concentrated on the development of advanced Monte Carlo and quasi-Monte Carlo methods with these mutually reinforcing objectives in mind:
- improving the analysis of oil well logging problems using nuclear sondes;
- using advanced Monte Carlo methods to accelerate the convergence of MCNP, the world’s most widely used Monte Carlo program (developed at Los Alamos National Laboratory);
- helping to develop transport-based models and computational methods for non-invasive techniques to detect, treat and monitor cancer and other diseases;
- developing improved radiation therapy plans based on Monte Carlo simulation of electron transport for full body dosimetry;
- analyzing advanced algorithms for modeling investment portfolios using the Black-Scholes equation.
Ongoing Methods Development
A great deal of research performed in Claremont for many years has been aimed at the solution of a variety of problems using faster, more efficient Monte Carlo and quasi-Monte Carlo methods.1,2,18
Indeed, hybrid sequence methods (i.e., methods that make use of both pseudorandom and low discrepancy sequences) were developed in Claremont in Mathematics Clinic projects sponsored by Chevron Petroleum Technology Company between 1993 and 1996.3-17,19.
Such methods were developed because of the need for improved convergence in well logging applications. Hybrid sequences accomplish this by utilizing sequences more uniform than conventional pseudo-random ones to generate random walks, with the result that the slow statistical convergence associated with pseudo-randomly-generated walks can be improved by factors of 2-4, or even more in certain problems. Additional gains by factors of perhaps 10 or more can be obtained when more "information-rich" random variables, such as ones relying on an analytic computation of expected next contributions to a detector tally, are utilized in the estimation process.
Another area of recent research in Claremont with very broad applicability is the exploration of learning algorithms, based on either correlated sampling or importance sampling, sequentially applied, that obtain geometric convergence for global solutions of transport problems.20-36
These new adaptive methods have demonstrated the great potential inherent in improving iteratively the Monte Carlo solution of many problems. When applied properly, such methods produce each new decimal digit of precision with only a linear, rather than an exponential, increase in the sample size. The work aimed at achieving geometric convergence was sponsored by Los Alamos National Laboratory between 1996 and 2003. Subsequently, the emphasis has shifted to biomedical applications and this work has continued at the University of California, Irvine.
In addition to these traditional areas of concentration in stochastic transport applications, its connections with the Oil Industry Consortium afford the research institute access to a wide variety of expertise in deterministic solutions of the transport equation.37-41 The intention is to develop a comprehensive research plan in transport applications broadly construed with the objective of understanding better how these two different methodologies can best support and complement each other for a wide variety of practical problems.