Growing out of the long interest and expertise in transport and diffusion applications, students, postdocs and faculty are presently very actively involved, along with scientists at the Beckman Laser Institute and Medical Clinic of the University of California at Irvine, in a research program that seeks to model and understand the responses of human tissue when exposed to laser radiation sources.42-54
The long-range objective of this program is to improve computational methods for solving these problems. The present focus is on the development of perturbation Monte Carlo techniques and adaptive (learning) algorithms for photon transport in turbid media. The virtual tissue simulator, a Monte Carlo program for representing and studying voxelized tissue models, is currently in the beginning implementation phases. This new tool should be of great value in helping with the diagnosis, treatment and monitoring of cancer and other diseases.
An area of longstanding interest and expertise in Claremont is the modeling and analysis of sub-micron semiconductor devices, with emphasis to date on drift-diffusion based models and related models for MOSFETs.55-60 More information on this topic can be found at Engineering and Industrial Applied Mathematics Clinic.
Applications to Finance
Claremont has developed over the course of the past several years a substantial interdisciplinary effort in financial modeling that has led to the establishment of a new MS degree in Financial Engineering jointly offered by the CGU Mathematics Department and CGU’s Peter F. Drucker and Masatoshi Ito Graduate School of Management. The curriculum in this new program draws considerably from the experience with advanced numerical and analytical methods of the faculties of the co-sponsoring academic programs. Monte Carlo and quasi-Monte Carlo methods 61 are in wide use in the financial community and play a prominent role in the new curriculum.
Additional areas of interest and long experience in Claremont include:
- Deterministic numerical analysis, including the numerical approximation of definite integrals, and solutions of ordinary and partial differential equations by finite difference and finite element methods.62-64
- Probabilistic and statistical applications, including the analysis of reliability, the development of artificial neural networks and genetic algorithms and resampling methods for the solution of many scientific and engineering problems.65-67
- Fluid dynamics models for understanding the behavior of nonlinear systems arising especially in the study of fluid transport.68-70