The approximate inverse in action II: convergence and stability

The approximate inverse is a scheme for constructing stable inversion formulas for operator equations. Originally, it is defined on L2-spaces. In the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of the approximate inverse which actually underlies numerical computations. Indeed, we show convergence if the discretization parameter tends to zero. Further, we prove stability, that is, we show the regularization property. Finally we apply the results to the filtered backprojection algorithm in 2D-tomography to obtain convergence rates.

[1]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[2]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[3]  A. Louis.  Inverse und schlecht gestellte Probleme , 1989 .

[4]  B. F. Logan,et al.  The Fourier reconstruction of a head section , 1974 .

[5]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[6]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[7]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[8]  D. A. Popov.  On convergence of a class of algorithms for the inversion of the numerical Radon transform , 1990 .

[9]  J. Wloka,et al.  Partial differential equations: Strongly elliptic differential operators and the method of variations , 1987 .

[10]  Alfred K. Louis,et al.  A unified approach to regularization methods for linear ill-posed problems , 1999 .

[11]  Alfred K. Louis.  Approximate inverse for linear and some nonlinear problems , 1995 .

[12]  Peter Maass,et al.  A mollifier method for linear operator equations of the first kind , 1990 .

[13]  Thomas Schuster,et al.  The Approximate Inverse in Action with an Application to Computerized Tomography , 2000, SIAM J. Numer. Anal..

[14]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[15]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

[16]  J. Bramble,et al.  Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation , 1970 .