Skip to main content
SpringerLink
Log in

Scientific applications of iterative Toeplitz solvers

  • Published:
CALCOLO Aims and scope Submit manuscript

Abstract

Recent research on using the preconditioned conjugate gradient method as an iterative method for solving Toeplitz systems has brought much attention. One of the main important results of this methodology is that the complexity of solving a large class of Toeplitz systems can be reduced toO (n logn) operations as compared to theO(n log2 n) operations required by fast direct Toeplitz solvers, provided that a suitable preconditioner is chosen under certain conditions on the Toeplitz operator. In this paper, we survery some applications of iterative Toeplitz solvers to Toeplitz-related problems arising from scientific applications. These applications include partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Abbiss, P. Earwicker,Compact Operator Equations, Regularization and Super-resolution, in Mathematics in Signal Processing, T. Durrani, J. Abbiss, J. Hudson, R. Madan, J. Macwhirter and T. Moore, eds., Clarendon Press, Oxford, 1987, pp. 225–236.

    Google Scholar 

  2. T. Alexander, Adaptive Signal Processing, Theory and Applications, Springer-Verlag, New York, 1986.

    MATH  Google Scholar 

  3. J. Bunch,Stability of Methods for Solving Toeplitz Systems of Equations, SIAM J. Sci. Stat. Comput.6, (1985) 349–364.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Chan,Iterative Methods for Overflow Queueing Models I, Numerische Mathematik51, (1987) 143–180.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Chan,Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions, IMA J. Numer. Anal.11, (1991) 333–345.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Chan,Numerical Solutions for the Inverse Heat Problems in ℝN, SEA Bull. Math.16, (1992) 97–105.

    MATH  Google Scholar 

  7. R. Chan, T. Chan,Circulant Preconditioners for Elliptic Problems, Numer. Linear Algebra Appl.1, (1992) 77–101.

    MathSciNet  Google Scholar 

  8. R. Chan, T. Chan, C. Wong,Cosine Transform Based Preconditioners for Total Variation Minimization Problems in Image Processing, Iterative Methods in Linear Algebra, II, V3, IMACS Series in Computational and Applied Mathematics, Proceedings of the Second IMACS International Symposium on Iterative Methods in Linear Algebra, Bulgaria, June, 1995, pp. 311–329, Ed: S. Margenov and P. Vassilevski.

  9. R. Chan, W. Ching,Toeplitz-Circulant Preconditioners for Toeplitz Systems and Their Applications to Queueing Networks with Batch Arrivals, SIAM J. Sci. Comput.17, (1996) 762–772.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Chan, W. Ching, C. Wong,Optimal Trigonometric Preconditioners for Elliptic Problems and Queueing Problems, SEA Bull. Math.20, (1996), to appear.

  11. R. Chan, J. Nagy, R. Plemmons,Circulant Preconditioned Toeplitz Least Squares Iterations, SIAM J. Matrix Anal. Appl.15, (1994) 80–97.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Chan, J. Nagy, R. Plemmons,FFT-Based Preconditioners for Toeplitz-Block Least Squares Problems, SIAM J. Numer. Anal.30, (1993) 1740–1768.

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Chan, M. Ng,Conjugate Gradient Methods for Toeplitz Systems, SIAM Review38, No. 3 (1996) 427–482.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Chan, M. Ng, R. Plemmons,Generalization of Strang’s Preconditioner with Applications to Toeplitz Least Squares Problems, Numer. Linear Algebra Appl.3, (1996) 45–64.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Chan, H. Sun, W. Ng,Circulant Preconditioners for Ill-Conditioned Boundary Integral Equations from Potential Equations, Tech. Rept. 96-20, CUHK, Department of Mathematics, 1996.

  16. R. Chan, C. Wong,Sine Transform Based Preconditioners for Elliptic Problems, Tech. Rept. 93-12-4, CUHK, Department of Mathematics, 1993.

  17. R. Chan, M. Yeung,Circulant Preconditioners for Toeplitz Matrices with Positive Cintinuous Generating Functions, Math. Comp.58, (1992) 233–240.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Chan,An Optimal circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Stat. Comput.9, (1988) 766–771.

    Article  MATH  Google Scholar 

  19. T. Chan, J. Olkin,Circulant Preconditioners for Toeplitz-block Matrices, Numer. Algo.6, (1994) 89–101.

    Article  MATH  MathSciNet  Google Scholar 

  20. G. Chen, J. Zhou, Boundary Element Methods, Academic Press, London, 1992.

    MATH  Google Scholar 

  21. W. Ching, R. Chan, X. Zhou,Circulant Preconditioners for Markov Modulated Poisson Processes and Their Applications to Manufacturing Systems, SIAM J. Matrix. Anal. Appl., to appear.

  22. P. Davis, Circulant Matrices, John Wiley & Sons, New York, 1979.

    MATH  Google Scholar 

  23. H. Feichtinger, K. Gröchenig, T. Strohmer,Efficient Numerical Methods in Non-Uniform Sampling Theory, Numerische Mathematik69, (1995) 401–422.

    Article  MathSciNet  Google Scholar 

  24. G. Fiorentino, S. Serra,Fast parallel solvers for elliptic problems, Comp. Math. Applic.32, (1996) 61–68.

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Gilliam, C. Martin, J. Lund,Analytic and Numerical Aspects of the Observation of the Heat Equation, in Proc. 26th IEEE Conf. on Decision and Control2, (1987) 975–976.

  26. I. Gohberg, M. Fel’dman,Convolution Equations and Projection Methods for Their Solution, Transl. Math. Monographs, Vol. 41, Amer. Math. Soc., Providence, RI, 1974.

    MATH  Google Scholar 

  27. I. Gohberg, M. Hanke, I. Koltracht,Fast Preconditioned Conjugate Gradient Algorithms for Wiener-Hopf Integral Equations, SIAM J. Numer. Anal.31, (1994) 429–443.

    Article  MATH  MathSciNet  Google Scholar 

  28. U. Grenander, G. Szegö, Toeplitz Forms and Their Applications, 2nd ed., Chelsea Publishing, New York, 1984.

    MATH  Google Scholar 

  29. C. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman Publishing, London, 1984.

    MATH  Google Scholar 

  30. I. Gustafsson,A Class of First Order Factorization Methods, BIT18, (1978) 142–156.

    Article  MATH  MathSciNet  Google Scholar 

  31. L. Hemmingsson,Toplitz Preconditioners with Block Structure for First-order PDEs, Numer. Linear Algebra Appl.3, (1996) 21–44.

    Article  MATH  MathSciNet  Google Scholar 

  32. L. Hemmingsson, K. Otto,Analysis of Semi-Toeplitz Preconditioners for Firstorder PDE, SIAM J. Sci. Comput.17, (1996) 47–64.

    Article  MATH  MathSciNet  Google Scholar 

  33. S. Holmgren, K. Otto,A Comparison of Preconditioned Iterative Methods for Nonsymmetric Block-tridiagonal Systems of Equations, Report No. 123, (revised), Dept. of Scientific Computing, Uppsala Univ., Uppsala, Sweden, 1990.

    Google Scholar 

  34. S. Holmgren, K. Otto,Iterative Solution Methods and Preconditioners for Block-tridiagonal Systems of Equations, SIAM J. Matrix Anal. Appl.13, (1992) 863–886.

    Article  MATH  MathSciNet  Google Scholar 

  35. S. Holmgren, K. Otto,Semicirculant Preconditioners for First-order Partial Differential Equations, SIAM J. Sci. Comput.15, (1994) 385–407.

    Article  MATH  MathSciNet  Google Scholar 

  36. S. Holmgren, K. Otto,Semicirculant Solvers and Boundary Corrections for First-order Partial Differential Equations, SIAM J. Sci. Comput.17, (1996) 613–630.

    Article  MATH  MathSciNet  Google Scholar 

  37. T. Huckle,A Note on Skew Circulant Preconditioners for Elliptic Problems, Numer. Algo.2, (1992) 279–286.

    Article  MathSciNet  Google Scholar 

  38. IMSL, User’s Manual, IMSL Math/Library: FORTRAN Subroutines for Mathematical Applications, International Mathematical and Statistical Libraries (U.S.), Houston, 1990.

    Google Scholar 

  39. A. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989.

    MATH  Google Scholar 

  40. X. Jin,A Note on Circulant Preconditioners for Hyperbolic and Parabolic Equations, Chinese J. Math.21, (1993) 129–142.

    MATH  MathSciNet  Google Scholar 

  41. X. Jin, R. Chan,Circulant Preconditioners for Second Order Hyperbolic Equations, BIT32, (1992) 650–664.

    Article  MATH  MathSciNet  Google Scholar 

  42. T. Kailath, A. Sayed,Displacement Structure: Theory and Applications, SIAM Review37, (1995) 297–386.

    Article  MATH  MathSciNet  Google Scholar 

  43. L. Kaufman,Matric Methods for Queueing Problems, SIAM J. Sci. Stat. Comput.4, (1983) 525–552.

    Article  MATH  MathSciNet  Google Scholar 

  44. B. Kiss, G. Molnárka,A Preconditioned Domain Decomposition Algorithm for the Solution of the Elliptic Neumann Problem, Period. Math. Hungar.24, (1992) 151–165.

    Article  MATH  MathSciNet  Google Scholar 

  45. B. Kiss, G. Molnárka, N. Rahman,A Circulant Preconditioner for Domain Decomposition Algorithm for the Solution of the Elliptic Problem, Period. Math. Hungar.29, (1994) 67–80.

    Article  MATH  MathSciNet  Google Scholar 

  46. F. Lin, M. Ng, R. Chan,Precoditioners for Wiener-Hopf Equations with Higher Order Quadrature Rules, SIAM J. Numer. Anal., (1997), to appear.

  47. F. Lin, M. Ng,Higher-order Quadratures for Circulant Preconditioned Wiener-Hopf Equations, BIT36, (1996) 110–121.

    Article  MATH  MathSciNet  Google Scholar 

  48. I. Lirkov, S. Margenov, P. Vassilevski,Circulant Block-factorization Preconditioners for Elliptic Problems, Computing53, (1994) 59–74.

    Article  MATH  MathSciNet  Google Scholar 

  49. K. Meier-Hellstern,The Analysis of a Queue Arising in Overflow Models, IEEE Trans. Commun.37, (1989), pp. 367–372.

    Article  MathSciNet  Google Scholar 

  50. J. Nagy, R. Plemmons, T. Torgersen,Iterative Image Restoration Using Approximate Inverse, IEEE Trans. Image Process.5, (1996) 1151–1162.

    Article  Google Scholar 

  51. M. Ng,Fast Interative Methods for Solving Toeplitz-plus-Hankel Least Squares Problems, Elec. Trans. Numer. Anal.2, (1994) 154–170.

    MATH  Google Scholar 

  52. M. Ng, R. Chan,Fast Iterative Methods for Least Squares Estimations, Numer. Algo.6, (1994) 353–378.

    Article  MATH  MathSciNet  Google Scholar 

  53. M. Ng, R. Plemmons,Fast RLS Adaptive Filtering by FFT-based Conjugate Gradient Interations, SIAM J. Sci. Comput.17 (1996) 920–941.

    Article  MATH  MathSciNet  Google Scholar 

  54. M. Ng, R. Plemmons,LMS-Newton Adaptive Filtering Using FFT-based Conjugate Gradient Iterations, Electronic Transactions on Numerical Analysis4, (1996) 14–36.

    MATH  MathSciNet  Google Scholar 

  55. T. Oda,Moment Analysis for Traffic Associated with Markovian Queueing Systems, IEEE Trans. Comm.39, (1991) 737–745.

    Article  Google Scholar 

  56. K. Otto,Analysis of Preconditioners for Hyperbolic Partial Differential Equations, SIAM J. Numer. Anal.33, (1996) 2131–2165.

    Article  MATH  MathSciNet  Google Scholar 

  57. L. Rudin, S. Osher, E. Fatemi,Nonlinears Total Variation Based Noise Removal Algorithms, Physica D.60, (1992) 259–268.

    Article  MATH  Google Scholar 

  58. J. Roux, P. Lise, E. Zerbib, M. Foguet,A Formulation in Concordance with the Sampling Theorem for Band-Limited Images Reconstruction from Projections, Multidimensional Systems and Signal Processing7, (1996) 27–52.

    Article  MATH  Google Scholar 

  59. A. Seila,Multivariate Estimation of Conditional Performance Measure in Regenerative Simulation, Amer. J. Math. Manag. Sci.10, (1990) 17–50.

    Google Scholar 

  60. O. Toeplitz,Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veränderlichen. I. Teil: Theorie der L-Formen., Math. Annal.70, (1911) 351–376.

    Article  MathSciNet  Google Scholar 

  61. C. Van Loan Computational Frameworks for the Fast Fourier Transform, SIAM, Philadelphia, PA, 1992.

    MATH  Google Scholar 

  62. C. Vogel, M. Oman,Iterative Methods for Total Variation Denoising, SIAM J. Sci. Comput.17, (1996) 227–238.

    Article  MATH  MathSciNet  Google Scholar 

  63. O. Widlund,Iterative Substructuring Methods: Algorithms and Theory for Elliptic Problems in the Plane, in First Intl. Symp. on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. Golub, G. Meurant and J. Périaux, eds., SIAM, Philadelphia, PA, 1988.

    Google Scholar 

  64. U. Yule,On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer’s Sunspot Numbers, Phil. Trans. Roy. Soc. London,A 226, (1927) 267–298.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ACSys, Computer Sciences Laboratory, Research School of Information Sciences and Engineering, The Australian National University, 0200, Canberra, ACT, Australia

    Michael K. Ng

  2. Deparment of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Raymond H. Chan

Authors
  1. Michael K. Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raymond H. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by the Cooperative Research Centre for Advanced Computational Systems.

Research supported in part by HKRGC grants no. CUHK 316/94E.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ng, M.K., Chan, R.H. Scientific applications of iterative Toeplitz solvers. Calcolo 33, 249–267 (1996). https://doi.org/10.1007/BF02576004

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02576004

Keu words

AMS Subject classifications

Search

Navigation