Abstract
In this paper we introduce a new computational method for solving the diffusion equation. In particular, we construct a “generalized” state-space system and compute the impulse response of an equivalent truncated state-space system. In this effort, we use a 3D finite element method (FEM) to obtain the state-space system. We then use the Arnoldi iteration to approximate the state impulse response by projecting on the dominant controllable subspace. The idea exploited here is the approximation of the impulse response of the linear system. We study the homogeneous and heterogeneous cases and discuss the approximation error. Finally, we compare our computational results to our experimental setup.
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References
R. Chattergy, V. L. Syrmos, and P. Misra, Finite modeling of parabolic equations using Galerkin methods and inverse matrix approximations,Circuits Systems Signal Process. vol. 15, no. 5, pp. 631–648, 1996.
R. F. Curtain, Spectral systems,Internat. J. Control, pp. 657–666, 1984.
M. A. Erickson, R. S. Smith, and A. J. Laub, Calculating finite-dimensional approximations of infinite-dimensional linear systems,American Control Conference, WA5, Chicago, Illinois, 1992.
R. A. Friesener, L. S. Tuckerman, B. C. Dornblaser, and T. V. Russo, A method for exponential propagation of large systems of stiff nonlinear differential equations,J. Sci. Comput., no. 4, pp. 327–354, 1989.
E. Gallopoulos and Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods,Siam J. Sci. Statist. Comput., vol. 13, no. 5, pp. 1236–1264, Sept. 1992.
G. H. Golub and C. F. Van Loan,Matrix Computations, The Johns Hopkins Press, Baltimore, MD, 1989.
A. S. Hodel and K. Poolla, Numerical solution of very large, sparse Lyapunov equations through approximate power iteration,Proc. 29th Conference on Decision and Control, Hawaii, Dec. 1990.
C. Johnson,Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, U.K., 1987.
T. Kailath,Linear Systems, Prentice Hall, Englewood Cliffs, NJ, 1980.
M. V. Klibanov, T. R. Lucas, and R. M. Frank, New imaging algorithm in diffusion tomography, preprint.
R. Model, R. Hunlich, D. Richter, H. Rinneberg, H. Wabnitz, and M. Walzel Imaging in random media: Simulating light transport by numerical integration of the diffusion equation,SPIE, vol. 2336, pp. 11–22, Feb. 1995.
C. Moler and C. V. Loan, Nineteen dubious ways to compute the exponential of a matrix,SIAM Rev., vol. 20, pp. 801–836, 1978.
B. Nour-Omid, Applications of the Lanczos algorithm,Comput. Phys. Comm., no. 53, pp. 157–168, 1989.
Y. Saad,Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, England, 1992.
Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator,Siam J. Numer. Anal., vol. 29, no. 1, pp. 209–228, Feb. 1992.
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This research was supported by the Central Research Laboratory, Hamamatsu Photonics K.K., 5000 Hirakuchi, Hamakita 434, Japan.
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Su, Q., Syrmos, V.L. & Yun, D.Y.Y. A numerical algorithm for the diffusion equation using 3D FEM and the Arnoldi method. Circuits Systems and Signal Process 18, 291–314 (1999). https://doi.org/10.1007/BF01225699
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DOI: https://doi.org/10.1007/BF01225699