Advertisement

Prediction Bands for ILL-Posed Problems

  • Andrzej Jonca
Part of the Progress in Systems and Control Theory book series (PSCT, volume 1)

Abstract

Consider the linear ill-posed operator equation
$$Kx = z$$
(1.1)
where K: XYis an operator between two Hilbert spaces with the inner product denoted by
$$\left\langle {x,y} \right\rangle ,\,x,y\, \in \mathcal{X}\,or\,x,y\, \in \,\mathcal{Y}\,$$
The ill-posedness of the problem (see [1]) means that a small perturbation in the data zmay result in large changes in the solution to (1.1). This discontinuous dependence of the solution on the data requires regularization in order to approximately solve the ill-posed equation.

Keywords

Cumulative Distribution Function True Solution Singular System Truncation Level Spectral Filter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. W. Groetsch, The Theory of Tikhonov Regulation for Fred-holm Equations of the First Kind, Pitman Boston, 1984.Google Scholar
  2. [2]
    C. W. Groetsch, C. R. Vogel, “Asymptotic Theory of Filtering for Linear Operator Equations with Discrete Noisy Data”, Mathematics of Computation 49, No. 180 (1987) pp. 499–506.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    J. P. Imhof ,“Computing the Distribution of Quadratic Forms in Normal Variables”, Biometrica 48, 3 and 4 (1961) pp. 419–426.MathSciNetzbMATHGoogle Scholar
  4. [4]
    E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York, 1978.zbMATHGoogle Scholar
  5. [5]
    O.N. Strand, E. R. Westwater, “Statistical Estimation of the Numerical Solution of a Fredholm Integral Equation of the First Kind”, Journal of the Association for Computing Machinery 15, No. 1 (1968) pp. 100–114MathSciNetzbMATHGoogle Scholar
  6. [6]
    A. Taylor, D. Lay, Introduction to Functional Analysis, John Wiley, New York, 1980.zbMATHGoogle Scholar
  7. [7]
    C. R. Vogel, “Optimal Choice of a Truncation Level for the Truncated SVD Solution of Linear First Kind Integral Equations when Data are Noisy”, SIAM J. Numer. Anal. 23 (1986) pp. 109–117.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    G. Wahba, “Practical Approximate Solutions to Linear Operator Equations when the Data are Noisy”, SIAM J. Numer. Anal. 14 (1977) pp. 651–667.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • Andrzej Jonca
    • 1
  1. 1.Department of MathematicsCalifornia State University at Long BeachLong BeachUSA

Personalised recommendations