Ill-Posed Problems, Regularization and Singular Value Decompositions.

Abstract

Consider ill-posed problems of the form g(t) =Integral from 0 to 1 of K(t,s)f(s)ds and their discrete approximations obtained by quadrature, Ax=b. Assume that our desired solution f is smooth and that our data g is measured experimently and contains highly oscillatory noise. These theorems and examples demonstrate the effect of each of these procedures, the singular value decomposition with truncation, (SVDT) a Hankel transformation with damping, and the Tikhonov regularization procedure, on such noise in the data. It is demonstrated that in general, regularization is the most natural setting for mollifying the effects of such noise. However, for certain problems SVDT is equally suitable and in fact may be better if the rate of convergence of the regularization procedure is too slow.

Open PDF

Document Details

Document Type
Technical Report
Publication Date
Apr 05, 1977
Accession Number
ADA042789

Entities

People

  • Jane Cullum

Organizations

  • IBM Thomas J. Watson Research Center

Tags

Communities of Interest

  • Materials and Manufacturing Processes
  • Space

DTIC Thesaurus Topics

  • Air Force
  • Auger Electron Spectroscopy
  • Auger Electrons
  • Computations
  • Differential Equations
  • Eigenvalues
  • Eigenvectors
  • Electron Spectroscopy
  • Equations
  • Frequency
  • Functional Analysis
  • Integral Equations
  • Linear Algebra
  • Linear Algebraic Equations
  • New York
  • Numbers
  • Statistics

Fields of Study

  • Mathematics

Readers

  • Approximation Theory.
  • Linear Algebra
  • Wave Propagation and Nonlinear Chaotic Dynamics.