Abstract
The purpose of this paper is to survey different approaches to utilize parallel computers in the numerical treatment of large-scale regularization problems. Four possible “levels” of parallelization are discussed, each with its own granularity and application to regularization problems.
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E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov & D. Sorensen, LAPACK Users' Guide, SIAM, Philadelphia, 1992.
S. M. Balle, P. C. Hansen & N. J. Higham, A Strassen-type matrix inversion algorithm for the Connection Machine, APPARC PaA2 Deliverable, Esprit BRA III Contrat # 6634; Report UNIC-93-11, UNI.C, October 1993.
R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine & H. A. van der Vorst, Templates for the Solution of Linear Systems, SIAM, Philadelphia, 1994.
A. Basermann, P. Weidner, P. C. Hansen, Tz. Ostromsky & Z. Zlatev, Reordering of sparse matrices for parallel processing, APPARC PaA3a Deliverable, Esprit BRA III Contract # 6634; Report UNIC-94-03, UNI.C, February 1994.
Å. Björck, Least Squares Methods; in P. G. Ciarlet & J. L. Lions, Handbook of Numerical Analysis, Vol. I, North-Holland, Amsterdam, 1990.
T. F. Chan, J. Olkin & D. W. Cooley, Solving quadratically constrained least squares using black box unconstrained solvers, BIT 32 (1992), 481–495.
I. J. D. Craig & J. C. Brown, Inverse Problems in Astronomy, Adam Hilger, Bristol, 1986.
J. W. Demmel, Trading off parallelism and numerical stability; in M. S. Moonen, G. H. Golub & B. L. R. De Moor, Linear Algebra for Large Scale and Real-Time Applications, NATO ASI Series, Kluwer, Dordrecht, 1993.
J. Demmel, M. T. Heath & H. A. van der Vorst, Parallel numerical linear algebra, Acta Numerica 2 (1993), 111–199.
E. D. Deprettere (Ed.), SVD and Signal Processing, North-Holland, Amsterdam, 1988.
J. J. Dongarra, I. S. Duff, D. C. Sorensen & H. A. van der Vorst, Solving Linear Systems of Vector and Shared Memory Computers, SIAM, Philadelphia, 1991.
J. J. Dongarra & D. C. Sorensen, A fully parallel algorithm for the symmetric eigenvalue problems, SIAM J. Sci. Stat. Comput. 8 (1987), s139–s154.
L. Eidén, Algorithms for the regularization of ill-conditioned least squares problems, BIT 17 (1977), 134–145.
H. W. Engl, Regularization methods for the stable solution of inverse problems, Surv. Math. Ind. 3 (1993), 71–143.
R. D. Fierro, G. H. Golub, P. C. Hansen & D. P. O'Leary, Regularization by truncated total least squares, Report UNIC-93-14, December 1993 (20 pages); SIAM J. Sci. Comput., to appear.
G. H. Golub & C. F. Van Loan, Matrix Computations, 2. Ed., Johns Hopkins University Press, Baltimore, 1989.
C. W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg, Wiesbaden, 1993.
M. Hanke & P. C. Hansen, Regularization methods for large-scale problems, Surv. Math. Ind. 3 (1993), 253–315.
P. C. Hansen, Computation of the singular value expansion, Computing 40 (1988), 185–199.
P. C. Hansen, Regularization, GSVD, and truncated GSVD, BIT 29 (1989), 491–504.
P. C. Hansen, Truncated SVD solutions to discrete ill-posed problems with ill-determined numerical rank, SIAM J. Sci. Stat. Comput. 11 (1990), 503–518.
P. C. Hansen, Numerical tools for analysis and solution of Fredholm integral equations of the first kind, Inverse Problems 8 (1992), 849–872.
P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review 34 (1992), 561–580.
P. C. Hansen, Regularization algorithms for MPP; in K. Mosegaard (Ed.), Proceedings of Interdisciplinary Inversion Workshop 2, Niels Bohr Institute, Copenhagen University, 1993.
P. C. Hansen, Regularization Tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numerical Algorithms 6 (1994), 1–35.
P. C. Hansen, Experience with regularizing CG iterations, Report UNIC-94-02, May 1994; submitted to BIT.
P. C. Hansen & D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput. 14 (1993), 1487–1503.
P. C. Hansen, T. Sekii & H. Shibahashi, The modified truncated-SVD method for regularization in general form, SIAM J. Sci. Stat. Comput. 13 (1992), 1142–1150.
E. R. Jessup & D. C. Sorensen, A parallel algorithm for computing the singular value decomposition of a matrix, SIAM J. Matrix Anal. Appl. 15 (1994), 530–548.
J. T. King, Multilevel algorithms for ill-posed problems, Numer. Math. 61 (1992), 311–334.
A. K. Louis, Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989.
C. C. Paige & M. A. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software 8 (1982), 43–71.
E. de Sturler & H. A. van der Vorst, Communication cost reduction for Krylov methods on parallel computers;, in W. Gentzsch & U. Harms (Eds.), High-Performance Computing and Networking, Lecture Notes in Computer Science 797, Springer-Verlag, Berlin, 1994.
V. S. Sunderam, PVM: A framework for parallel distributed computing, Report ORNL/TM-11375, Oak Ridge National Laboratory.
R. Vaccaro (Ed.), SVD and Signal Processing II, Elsevier, Amsterdam, 1991.
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© 1994 Springer-Verlag Berlin Heidelberg
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Hansen, P.C. (1994). Parallel issues of regularization problems. In: Dongarra, J., Waśniewski, J. (eds) Parallel Scientific Computing. PARA 1994. Lecture Notes in Computer Science, vol 879. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0030156
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DOI: https://doi.org/10.1007/BFb0030156
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