Research Projects
Previous Projects/Research Topics
Project Goal : Explore and develop machine learning algorithms for use in improving the efficiency and utility of simulations of physical phenomona, e.g. scientific computing simulations.
Associated Papers/Reports:
C. R. Anderson, Dominant Subspace Preconditioning for Kernel Ridge Regression Problems, November 2016, UCLA CAM Report (1677) (pdf) 
The RayleighChebyshev procedure was developed for the eigenproblems arising in quantum mechanical simulations, however it has found additional utility as an eigensystem routine for the eigenproblems arising in unsupervised learning procedures. 
C. R. Anderson, A Rayleigh–Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matricesm ”, J. Comput. Phys. 229, 74777487, (2010). (https://doi.org/10.1016/j.jcp.2010.06.030) 

Direct Numerical Solution of the Nparticle Schroedinger Equation using Grid Based Methods
Project Goal : Develop a grid based simuluation framework to facilitate the investigation of the solutions of the Nparticle Schroedinger equation
Abstract: The computational capabilities of currently available high performance desktop machines offer the potential of enabling the direct numerical solution of the Nparticle Schroedinger equation for small numbers of particles. This project is devoted to developing efficient grid based numerical techniques that can effectively utilize this computational capability, and ultimately lead to a software framework that will enable the exploration of the mathematical properties associated with solutions of the Nparticle Schroedinger equation and equations currently used to approximate solutions of the Nparticle Schroedinger equation
Associated Papers/Reports:
C. R. Anderson, Grid Based Solutions of the NParticle Schroedinger Equation, February 2015, UCLA CAM Report (1510) (pdf) C. R. Anderson, A Recursive Expanding Domain Method for the Solution of Laplace's Equation In Infinite Domains, May 2014, UCLA CAM Report (1545) (pdf) C. R. Anderson, High order expanding domain methods for the solution of Poisson's equation in infinite domains, Journal of Computational Physics, 14, 2016, Pages 194205, http://dx.doi.org/10.1016/j.jcp.2016.02.074 (C. R. Anderson, Compact Polynomial Mollifiers For Poisson's Equation, May 2014. Revised: November 2015, UCLA CAM Report (1443) (pdf) 
Principal Collaborators :  R. Caflisch (UCLA Mathematics), M. Gyure (HRL), R Ross (HRL), G. Simms (HRL) 
Additional Collaborators :  E. Yablonovich (UCLA EE) and members of the UCLA optoelectronics group 
Project Funding :  DARPA Quantum Information Science & Technology (QuIST) Program 
Associated Papers/Reports:  C. R. Anderson, T. C. Cecil, A FourierWachspress method for solving Helmholtz’s equation in threedimensional
layered domains, Journal of Computational Physics, 205, 2005, 706718. Christopher R. Anderson and Thomas C. Cecil, A FourierWachspress Method for Solving Helmholtz's Equation in Three Dimensional Layered Domains, June 2004 UCLA CAM Report (0434) (pdf) Christopher R. Anderson and Christopher Elion, Accelerated Solutions of Nonlinear Equations Using Stabilized RungeKutta Methods, April 2004, UCLA CAM Report (0426) (pdf) 
Collaborators :  Dr. Marc Reider 
Project Funding :  AFOSR, ONR, NSF 
Associated Papers/Reports:  C.R. Anderson, Numerical Solution of the Incompressible NavierStokes Equations in Infinite Domains, Jan.
2000, CAM Report (0002) (PDF) C.R. Anderson, M.B. Reider, High Order Explicit Method for the Computation of Flow About a Circular Cylinder. Journal of Computational Physics, 125, 1996, 207224. C.R. Anderson, Domain decomposition techniques and the solution of Poisson's equation in infinite domains, in Domain Decomposition Methods, T. Chan, R. Glowinski, J. Periaux and O. Widlund (eds.). SIAM Publications , 1988, 129139. 
Collaborators :  Prof. T. C. Cecil, Dr. M.D. Dahleh 
Project Funding :  DARPA, AFOSR, ONR, NSF 
Associated Papers/Reports:  C. R. Anderson, T. C. Cecil, A Fourier–Wachspress method for solving Helmholtz’s equation in threedimensional
layered domains, Journal of Computational Physics, 205, 2005, 706718. Christopher R. Anderson and Thomas C. Cecil, A FourierWachspress Method for Solving Helmholtz's Equation in Three Dimensional Layered Domains, June 2004, UCLA CAM Report (0434) (PDF) C.R. Anderson, Capacitance Matrix Methods for the Solution of Elliptic Equations Defined by Level Set Functions, May 2000. CAM Report (0015) (PDF) C.R. Anderson, M.D. Dahleh, Rapid Computation of the Discrete Fourier Transform. Siam. J. of Sci. Comput., 17, 1996, 913919. C.R. Anderson, An implementation of the fast multipole method without multipoles. SIAM J. Sci. Stat. Computing, 13, 1992, 923947. C.R. Anderson, Computational aspects of ``fast'' particle simulations, in Comput. Meth. Appl. Sci. \& Engr., Glowinski and Lichnewsky, eds. SIAM Publications, 1990, 123135 C.R. Anderson, A method of local corrections for computing the velocity field due to a distribution of vortex blobs, J. of Computational Physics, 62, 1986, 111123. 
Collaborators :  Dr. Y.C. Chen, Prof. J.S. Gibson, Dr. H.L. Chang, Dr. C. Greengard, Prof. L. Greengard, Prof. V. Rokhlin, Dr. M. Henderson 
Project Funding :  DARPA, AFOSR, ONR, NSF 
Associated Papers/Reports:  C.R. Anderson, Y.C. Chen, J.S. Gibson, Control and Identification of Vortex Wakes. Conference Proceedings
ASME Fluids Engineering Division Summer Meeting, FEDSM98/5299. June 1998. C.R. Anderson, H.L. Chang, J.S. Gibson, Zeros of input/output models for control of vortex dynamics. American Control Conference, Baltimore, MD., June 1994, 15341536. C.R. Anderson, A Finite Vortex Street Model, UCLA Dept. of Mathematics, CAM 9310, 1993. (PDF) C.R. Anderson, H.L. Chang, J.S. Gibson, Adaptive control of vortex dynamics. IEEE Conference on Decision and Control, San Antonio, TX., December 1993, 19091910. C.R. Anderson, C. Greengard, L. Greengard, V. Rokhlin, On the accurate calculation of vortex shedding. Phys. Fluids A. , 2 1990, 883885 C.R. Anderson, C. Greengard, The vortex ring merger problem at infinite Reynolds number. Comm. Pure Appl. Math., 42, 1989, 11231139. C.R. Anderson, Observations on vorticity creation boundary conditions, in Mathematical Aspects of Vortex Dynamics, R.E. Caflisch (ed.). SIAM Publications, 1988, 144160. C.R. Anderson, Vorticity boundary conditions and boundary vorticity generation for 2dimensional viscous incompressible flows. Journal of Computational Physics, 80, 1989, 7297. C.R. Anderson, C. Greengard, Vortex ring interaction by the vortex method, in Vortex Methods, C.R. Anderson and C. Greengard (eds.). SpringerVerlag Lecture Notes in Mathematics Series, 1360, 1988, 2337 C.R. Anderson, C. Greengard, M. Henderson, Instability, vortex shedding and numerical convergence, in Vortex Methods, C.R. Anderson and C. Greengard (eds.), SpringerVerlag Lecture Notes in Mathematical Series, 1360, 1988, 4255. C.R. Anderson, A vortex method for flows with slight density variations, .J. of Computational Physics, 61, 1985, 417444. C.R. Anderson, C. Greengard, On vortex methods. SIAM J. Numer. Analysis, 22, 1985, 413440. C.R. Anderson, C. Greengard, eds.,Vortex Dynamics and Vortex Methods. Lectures in Applied Mathematics, 28, 1991, AMS. C.R. Anderson, C. Greengard, eds., Proceedings of the UCLA Workshop on Vortex Methods. Lecture Notes in Mathematics, 1360, 1988, Springer. 
Collaborators :  Prof. R. Fedkiw, Dr. R. Caiden, Dr. M. B. Reider 
Project Funding :  DARPA, AFOSR, ONR, NSF 
Associated Papers/Reports:  Caiden, R., Fedkiw, R. and Anderson, C., "A Numerical Method for Two Phase Flow Consisting of Separate Compressible
and Incompressible Regions", J. Comput. Phys. 166, 127 (2001). C.R. Anderson, Numerical Solution of the Incompressible NavierStokes Equations in Infinite Domains, Jan. 2000, CAM Report (0002) (PDF) C.R. Anderson, M.B. Reider, High Order Explicit Method for the Computation of Flow About a Circular Cylinder. Journal of Computational Physics, 125, 1996, 207224. C.R. Anderson, M.B. Reider, Investigation of the Use of Prandt/NavierStokes Equation Procedures for Two Dimensional Incompressible Flows. Physics of Fluids, 6, 1994, 23802389 C.R. Anderson, Derivation and solution of the equation of the discrete pressure equation for the incompressibles NavierStokes equations. UCLA report, CAM 8836. (PDF) C.R. Anderson, Manipulating fast solvers  changing their boundary conditions and putting them on multiple processor computers. UCLA report, CAM 8837. (PDF) 