Multilevel algorithms for ill-posed problems

SummaryIn this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations. Convergence estimates are established for multilevel algorithms applied to Tikhonov type regularization methods. Our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter. The algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations.

[1]  Robert Plato,et al.  On the regularization of projection methods for solving III-posed problems , 1990 .

[2]  Piet Hemker,et al.  Multiple grid methods for the solution of Fredholm integral equations of the second kind , 1979 .

[3]  A. Carasso.  Determining Surface Temperatures from Interior Observations , 1982 .

[4]  H. Engl,et al.  A posteriori parameter choice for general regularization methods for solving linear ill-posed problems , 1988 .

[5]  Lars Eldén,et al.  The Numerical Solution of a Non-Characteristic Cauchy Probelm for a Parabolic Equation , 1983 .

[6]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[7]  J. King.  On the construction of preconditioners by subspace decomposition , 1990 .

[8]  Heinz W. Engl,et al.  Stability estimates and regularization for an inverse heat conduction prolem in semi - infinite and finite time intervals , 1989 .

[9]  J. Pasciak,et al.  New convergence estimates for multigrid algorithms , 1987 .

[10]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[11]  J. King,et al.  A minimal error conjugate gradient method for ill-posed problems , 1989 .

[12]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[13]  D. Murio.  The Mollification Method and the Numerical Solution of an Inverse Heat Conduction Problem , 1981 .

[14]  L. Lardy.  A class of iterative methods of conjugate gradient type , 1990 .

[15]  Jan Mandel,et al.  On multilevel iterative methods for integral equations of the second kind and related problems , 1985 .

[16]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[17]  H. Gfrerer.  An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates , 1987 .

[18]  P. M. van den Berg,et al.  Iterative Methods for Solving Integral Equations , 1991, Progress In Electromagnetics Research.

[19]  Diego A. Murio,et al.  A stable space marching finite differences algorithm for the inverse heat conduction problem with no initial filtering procedure , 1990 .