Claremont Research Institute of Applied Mathematical Sciences

 

| Introduction | Mission of the Institute | Organization and Sponsorship |
| Early Research Projects | Ongoing Methods Developement |
| Areas of Current and Future Activity | References |
  

Introduction

 

Claremont Graduate University created the Claremont Research Institute of Applied Mathematical Sciences (CRIAMS) in 1998, matching the needs of industry with research conducted by University faculty and students. This research institute, funded entirely by outside support, developed initially as a natural extension of a quarter century of student/faculty team problem-solving in the Mathematics Clinic, a special course that originated at Claremont Graduate University and Harvey Mudd College in 1973-74.

Since 1974 the Mathematics Clinic has completed nearly 200 project-years of work sponsored by approximately 70 different firms, and involving hundreds of graduate and undergraduate mathematics and science majors. As well, the Mathematics Clinic and CRIAMS programs have attracted more than 50 postdoctoral mathematicians to Claremont for extended periods of participation in its Clinic and CRIAMS research projects. Drawing from this vast reservoir of direct experience with industrial problems, and augmented by a curriculum that emphasizes mathematical modeling of real problems, mathematics faculty at CGU have pioneered the development of advanced mathematical methods and computer algorithms that can be applied across many application areas.

Mission of the institute  

The broad mission of the Institute is to:

  1. conduct research in advanced mathematical, computational and numerical techniques aimed at the solution of complex, real-world problems; 
     
  2. provide a center for graduate and postgraduate education for students in applied mathematics, with emphasis on complex problem-solving; 
      
  3. host a site for continuing education, intensive seminars and sabbatical visits by scientists and applied mathematicians who develop and use these techniques. 

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Organization and Sponsorship

The work of the Institute is performed by project teams, each consisting of a technical Project Director (or co-Directors) and by technical support staff drawn from Claremont faculty and students, a variety of visitors to the Institute , and more advanced post-doctoral researchers. CRIAMS is administered by a Director and an Administrative and Computing Coordinator. The Institute provides a research extension of the School of Mathematical Sciences of Claremont Graduate University.

CRIAMS work is supported financially by consortia of sponsoring corporations, national laboratories and other institutions with common goals and objectives.

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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:

  1. improving the analysis of oil well logging problems using nuclear sondes;
     
     
  2. 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); 
     
  3. helping to develop transport-based models and computational methods for non-invasive techniques to detect, treat and monitor cancer and other diseases; 
     
  4. developing improved radiation therapy plans based on Monte Carlo simulation of electron transport for full body dosimetry;
     
  5. analyzing advanced algorithms for modeling investment portfolios using the Black-Scholes equation.

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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 pseudorandom ones to generate random walks, with the result that the slow statistical convergence associated with pseudorandomly 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. Subsequenty, 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 CRIAMS 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.

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Areas of Current and Future Activity


 

Biomedical Optics

 

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 (VTS), a Monte Carlo program for representing and studying voxeliized 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.

Semiconductor Modeling

An area of long-standing 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.

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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 M.S. 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 M.S. 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 not adequately covered above include

  1. 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];
     
  2. 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];
     
  3. Fluid dynamics models for understanding the behavior of nonlinear systems arising especially in the study of fluid transport [68] - [70].

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References

  1. J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems, Addison- Wesley, 1969.
  2. J. Spanier and E.H. Maize, "Quasi-Random Methods for Estimating Integrals Using Relatively Small Sample Sizes", SIAM Review, 36, (1994).
  3. "Quasi-Monte Carlo Methods Applied to Oil Well Logging Problems'', CGS Mathematics Clinic Interim Report to Chevron Petroleum Technology Company, January, 1994.
  4. "Quasi-Monte Carlo Methods Applied to Oil Well Logging Problems'', CGS Mathematics Clinic Final Report to Chevron Petroleum Technology Company, June, 1994
  5. "Hybrid Monte Carlo Methods Applied to Oil Well Logging Problems'', CGS Mathematics Clinic Final Report to Chevron Petroleum Technology Company, May, 1995
  6. "Hybrid Monte Carlo Methods Applied to Oil Well Logging Problems'', CGS Mathematics Clinic Interim Report to Chevron Petroleum Technology Company, January, 1996.
  7. "Hybrid Monte Carlo Methods Applied to Oil Well Logging Problems'', CGS Mathematics Clinic Final Report to Chevron Petroleum Technology Company, June, 1996.
  8. J. Spanier, Quasi-Monte Carlo Methods for Particle Transport Problems'', Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics #106, 121-148, Springer-Verlag, 1995.
  9. G. Okten, "A Probabilistic Result on the Discrepancy of a Hybrid-Monte Carlo Sequence and Applications", Monte Carlo Methods and Applications, 2, 255-270, (1996).
  10. G. Okten, "Error Estimation for Quasi-Monte Carlo Methods", Monte Carlo and Quasi-Monte Carlo Methods 1996, Lecture Notes in Statistics #127, 353-368, Springer-Verlag, 1998.

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  11.  J. Jiang, M. Maucec, S. Mosher and J. Spanier, "MCNP-CRIAMS 1.0 for the Oil Consortium", CRIAMS Report OC-99008, August, 1999.
  12.  J. Jiang, M. Maucec, S. Mosher and J. Spanier, "Addendum to CRIAMS Report OC-99008", CRIAMS Report OC-99010, October, 1999.
  13.  S. Mosher and J. Spanier, "Final Report to the Oil Consortium", CRIAMS Report OC-00009, Sept., 2000.
  14. Spanier, J., "Random, Quasirandom and Hybrid Methods for Transport Problems", in Advanced  Monte Carlo for Radiation Physics, Particle Transport Simulation and Applications, A. Kling, F. Barao, M. Nakaga, L. Tavora and P. Vaz (eds.), Proceedings of the Monte Carlo 2000 Conference, Lisbon, Portugal, 23-26 October, 2000, Springer, 2000.
  15.  A. Badruzzaman, "Computational Methods in Nuclear Geophysics", Progress in Nuclear Energy, 25, 265-290, (1991).
  16. A. Badruzzaman, P. T. Nguyen, T. Badruzzaman and J. Spanier, "Impact and Efficiency of Monte Carlo Techniques Applied To A New Generation Of Nuclear Logging Measurements", Proc. International Conference on Mathematics and Computation, Reactor Physics, and Environmental Analysis, April 30-May 4, 1995, Portland, OR.
  17. J. Spanier and A. Badruzzaman, "Hybrid Monte Carlo Methods for Particle Transport", Proc. International Conference on Mathematics and Computation, Reactor Physics, and Environmental Analysis, April 30-May 4, 1995, Portland, OR.
  18.  Spanier, J. and E.H. Maize, "Quasi-Random Methods for Estimating Integrals Using Relatively Small Sample Sizes," SIAM Review, p. 36, 1994.
  19. J.A. Christen, P. Jiang, L. Li, J.L. Morales-Perez, and J. Spanier, "Reducing the Cost of Monte Carlo Analysis of Well Logging Data", Proc. Eleventh Workshop on Mathematical Problems in Industry, Albuquerque, New Mexico, June 12-16, 1995.
  20. L. Li , "Quasi-Monte Carlo Methods for Transport Equations", Ph.D. dissertation, The Claremont Graduate School, 1995.

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  21. R. Kong and G.W. McKinney, "Research Notes: Implementing the Monte Carlo Reduced Source Method on a Simple 1-D Problem", LANL-XTM-RN(U)96-010, (September, 1996).
  22. "Adaptive Methods for Accelerating Monte Carlo Convergence", Claremont Graduate University Mathematics Clinic Interim Report to Los Alamos National Laboratory, January, 1997.
  23. "Adaptive Methods for Accelerating Monte Carlo Convergence", The Claremont Graduate School Mathematics Clinic Final Report to Los Alamos National Laboratory, June, 1997.
  24. "Adaptive Methods for Accelerating Monte Carlo Convergence", Claremont Graduate University Mathematics Clinic Report to Los Alamos National Laboratory, September, 1997.
  25. L. Li & J. Spanier, "Approximation of Transport Equations by Matrix Equations and Sequential Sampling", Monte Carlo Methods and Applications, 3, (1997).
  26. J. Spanier & L. Li, "General Sequential Sampling Techniques for Monte Carlo Simulations: Part I - Matrix Problems", Monte Carlo and Quasi-Monte Carlo Methods 1996, Lecture Notes in Statistics #127, Springer-Verlag, 1998.
  27. "Adaptive Methods for Accelerating Monte Carlo Convergence", Claremont Graduate University Mathematics Clinic Interim Report to Los Alamos National Laboratory, January, 1998.
  28. "Adaptive Methods for Accelerating Monte Carlo Convergence", Claremont Graduate University Mathematics Clinic Final Report to Los Alamos National Laboratory, June, 1998.
  29. J. Spanier, "Monte Carlo Methods for Flux Expansion Solutions of Transport Problems", Nuclear Science and Engineering, 133, 1-7, (1999).
  30. J. Spanier, "Geometrically Convergent Learning Algorithms for Global Solutions of Transport Problems'', Monte Carlo and Quasi-Monte Carlo Methods 1998, H. Niederreiter and J.Spanier, Eds., Springer-Verlag, 98-113, 1999.

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  31. R. Kong and J. Spanier, "Sequential Correlated Sampling Methods for Some Transport Problems", Monte Carlo and Quasi-Monte Carlo Methods 1998, H. Niederreiter and J.Spanier, Eds., Springer-Verlag, 238-251, 1999.
  32. Y. Lai and J. Spanier, "Adaptive Importance Sampling Methods for Particle Transport Problems", Monte Carlo and Quasi-Monte Carlo Methods 1998, H. Niederreiter and J.Spanier, Eds., Springer-Verlag, 273-283, 1999.
  33. Carole Hayakawa and Jerome Spanier, "Comparison of Monte Carlo Algorithms for Obtaining Geometric Convergence for Model Transport Problems", Monte Carlo and Quasi-Monte Carlo Methods 1998, H. Niederreiter and J.Spanier, Eds., Springer-Verlag, 214-226, 1999. 
  34. R. Kong and J. Spanier, "Error Analysis of Sequential Monte Carlo Methods for Transport Problems", Monte Carlo and Quasi-Monte Carlo Methods 1998, H. Niederreiter and J.Spanier, Eds., Springer-Verlag, 252-272, 1999..
  35. Spanier, J. and R. Kong, "A New Adaptive Method for Geometric Convergence", Monte Carlo and Quasi-Monte Carlo Methods 2002, (H. Niederreiter ed.), Springer-Verlag, pp. 439-449, 2004.
  36. Kong, R. and J. Spanier, "Residual Versus Error for Transport Problems", in Monte Carlo and Quasi- Monte Carlo Methods 2000, K.T. Fang, F.J. Hickernell and H. Niederreiter (eds.), Proceedings of a conference held at Hong Kong Baptist University, Hong Kong, S.A.R., China, November 27-December 1, 2000, Springer, 2002.
  37. A. Badruzzaman, Z. Xie, J.J. Dorning, and J.J. Ullo, "A Discrete Nodal Transport Method for Three-Dimensional Reactor Physics and Shielding Calculations" Proc. Topical Mtg. On Reactor Physics and Shielding, Chicago, IL., Sept. 17-19, 1984.
  38. A. Badruzzaman, "An Efficient Algorithm for Nodal-Transport Solutions in Multidimensional Geometry", Nucl. Sci. Eng., 89, (1985).
  39. A. Badruzzaman and Z. Xie, "A Three-Dimensional Linear Nodal Transport Method", Trans. Am. Nucl. Soc., 47, (1984).
  40. A. Badruzzaman, "Performance of Three-Dimensional Nodal Discrete Ordinates Method", Prog. In Nucl. Energy, 18, (1986).

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  41. A. Badruzzaman, "Nodal Transport Methods in Transport theory", Advances in Nucl. Sci. and Tech., 21, (1990).
  42. A.K. Dunn, "Light Scattering Properties of Cells", Ph.D. Dissertation, The University of Texas at Austin, 1997.
  43. J.B. Fishkin and E. Gratton, "Propagation of Photon-Density Waves in Strongly Scattering Media Containing an Absorbing Semi-Infinite Plane Bounded by a Straight Edge" J. Opt. Soc. Amer. A, 10, (1993).
  44. Z. Chen, T.E. Milner, D. Dave, and J.S. Nelson, "Optical Doppler Tomographic Imaging of Fluid flow Velocity in Highly Scattering Media", Optics Letters, 22, (1997).
  45. B.J. Tromberg, R.C. Haskell, S.J. Madsen, and L.O. Svaasand, "Characterization of Tissue Optical Properties Using Photon Density Waves", Comments on Molecular and Cellular Biophysics, 8, (1995).
  46. B.J. Tromberg, L.O. Svaasand, M.K. Fehr, S.J. Madsen, P. Wyss, B. P. Sansone and Y. Tadir, "A Mathematical Model for Light Dosimetry in Photodynamic Destruction of Human Endometrium", Phys. Med. Biol. 40, (1995).
  47. B.J. Tromberg, L.O. Svaasand, T.-T. Tsay and R.C. Haskell, "Properties of Photon Density Waves in Multiple Scattering Media", Appl. Opt., 32, (1993).
  48. D.J. Smithies and P.H. Butler, "Modelling the Distribution of Laser Light in Port-Wine Stains with the Monte Carlo Method", Phys. Med. Biol., 40, (1995).
  49. Bruce J. Tromberg, O. Coquoz, J.B. Fishkin, T. Pham, E.R. Anderson, J. Butler, M. Cahn, J.D. Gross, V. Venugopalan, and D. Pham, "Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration", Phil. Trans. Royal Soc. London, 352, (1997).
  50. F. Bevilacqua, D. Piguet, P. Marquet, J.D. Gross, B.J. Tromberg and C. Depeursinge, "In vivo local optical determination of tissue optical properties", Proc. SPIE, 3194, (1998).

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  51. Hayakawa, C., J. Spanier, F. Bevilacqua, A.K. Dunn, J.S. You, B.J. Tromberg and V. Venugopalan, "Use of Perturbation Monte Carlo Techniques to Solve Inverse Problems in Heterogeneous Tissue", Optics Letters, 2001.
  52. Viator, J., B. Choi, M. Ambrose, J. Spanier and J.S. Nelson, "In Vivo Port-Wine Stain Depth Determination with a Photoacoustic Probe", Applied Optics, 42, 3215-3224, (2003. Hayakawa, C. and J. Spanier, "Perturbation Monte Carlo Methods for the Solution of Inverse Problems", Monte Carlo and Quasi-Monte Carlo Methods 2002 , (H. Niederreiter ed.), Springer-Verlag, pp.227-241, 2004.
  53.  Hayakawa, C., B.Y. Hill, J.S. You, F. Bevilacqua, J. Spanier and V. Venugopalan, ¡±Use of the ¥ä-P1approximation for recovery of optical absorption, scattering, and asymmetry coefficients in turbid media¡±, Applied Optics.43, 4677-4684 , (2004).
  54. Tseng, S-H, C. Hayakawa, J. Spanier J., B.J. Tromberg, and A. Durkin, ¡°Quantitative spectroscopy of superficial turbid media¡±, Optics Letters, 30, 3165-3167, (2005).
  55. "Modeling of Short Channel MOSFET Devices for use in VLSI Simulations", The Claremont Graduate School Mathematics Clinic Final Report to USC Information Sciences Institute, June, 1992.
  56. J. Sarvas and J. Spanier, "A Direct Approach to Solving the Drift-Diffusion Model Equations for use in Certain MOSFET Devices", Math. Comp. Modelling, 22, (1995).
  57. E. Cumberbatch, P. Hagan and J. Pimbley, "Analytical Treatment of MOSFET Source-Drain Resistance", IEEE Trans. On Electron Devices, ED-34, (1987).
  58. E. Cumberbatch, "Modeling for Field-Effect Transistors, Proc. Third European Conf. On Mathematics in Industry, B.G. Teubner, Stuttgart, 1990.
  59. E. Cumberbatch and G. Mahinthakumar, "Contact Resistance for Small Contacts", IEEE Trans. On Electron Devices, 38, (1991).
  60. "MOSFET Device Modeling", Claremont Graduate University Mathematics Clinic Interim Report to USC Information Sciences Institute, January, 1998.

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  61. Lai. Y. and J. Spanier, "Applications of Monte Carlo & Quasi-Monte Carlo Methods in Finance: Option Pricing."  Monte Carlo and Quasi Monte Carlo Methods 1998, H. Niederreiter and J. Spanier, Eds., Springer-Verlag, pp. 284--295, 1999.
  62. J. Spanier, "Alternating Direction Methods Applied to Heat Conduction Problems", Mathematical Methods for Digital Computers, Vol. 2, Chap. 11, John Wiley & Sons, Inc., 1967.
  63. C.A. Hall and J. Spanier, "Nested Bounds for the Spectral Radius", SIAM J. Numer. Anal., 5, (1968).
  64. J.P.Lambert, "Some Developments in Optimal and Quasi-Monte Carlo Quadrature and New Outlook on a Classical Chebyshev Problem", Ph.D. Dissertation, The Claremont Graduate School, 1982.
  65. J.E. Angus, "Linear Radar Track Identification from Sequential Cartesian Scan Points in homogeneous Clutter", Hughes aircraft Co. Technical Report, (1991).
  66. J.E. Angus, "On the Connection Between Neural Network Learning and Multivariate Nonlinear Least Squares Estimation", Int. J. Neural Networks, 1, (1989).
  67. J.E. Angus, "Asymptotic Theory for Bootstrapping the Extremes, Comm. in Stat., 22A, (1993).
  68. E. Cumberbatch, "Effects of Viscosity of Ship Waves", J. Fluid Mech., 23, (1965).
  69. E. Cumberbatch, "Control of Second - Order Systems with Large Amplitudes", J. Opt. Theory Appl., 12, (1973).
  70. E. Cumberbatch and E. Varley, "Finite Deformation of Elastic Material Surrounding Cylindrical Holes", J. Elasticity, 10, (1980).

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