|
The approximate inverse in action II: convergence and stability
Author(s):
Andreas
Rieder;
Thomas
Schuster.
Journal:
Math. Comp.
72
(2003),
1399-1415.
MSC (2000):
Primary 65J10, 65R10
Posted:
March 26, 2003
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
The approximate inverse is a scheme for constructing stable inversion formulas for operator equations. Originally, it is defined on  -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.
References:
-
- 1.
- J. H. BRAMBLE AND S. R. HILBERT, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7 (1970), pp. 112-124. MR 41:7819
- 2.
- A. COHEN, I. DAUBECHIES, AND J.-C. FEAUVEAU, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45 (1992), pp. 485-560. MR 93e:42044
- 3.
- W. DAHMEN, A. KUNOTH, AND K. URBAN, Biorthogonal spline wavelets on the interval - stability and moment conditions, Appl. Comp. Harm. Anal., 6 (1999), pp. 132-196. MR 99m:42046
- 4.
- H. W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of Inverse Problems, vol. 375 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1996. MR 97k:65145
- 5.
- J. L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications, Vol.
 , Springer-Verlag, New York, 1972. MR 50:2670 - 6.
- A. K. LOUIS, Inverse und schlecht gestellte Probleme, Studienbücher Mathematik, B. G. Teubner, Stuttgart, Germany, 1989. MR 90g:65075
- 7.
- -, Corrigendum: Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), pp. 175-190. MR 96m:65063
- 8.
- -, A unified approach to regularization methods for linear ill-posed problems, Inverse Problems, 15 (1999), pp. 489-498. MR 2000b:65111
- 9.
- A. K. LOUIS AND P. MAASS, A mollifier method for linear operator equations of the first kind, Inverse Problems, 6 (1990), pp. 427-440. MR 91g:65130
- 10.
- F. NATTERER, The Mathematics of Computerized Tomography, Wiley, Chichester, 1986. MR 88m:44008
- 11.
- P. OSWALD, Multilevel Finite Element Approximation: Theory and Applications, Teubner Skripten zur Numerik, B. G. Teubner, Stuttgart, Germany, 1994. MR 95k:65110
- 12.
- D. A. POPOV, On convergence of a class of algorithms for the inversion of the numerical Radon transform, in Mathematical Problems of Tomography, I. M. Gelfand and S. G. Gindikin, eds., vol. 81 of Translations of Mathematical Monographs, AMS, Providence, R.I., 1990, pp. 7-65. MR 92g:92010
- 13.
- A. RIEDER, Principles of reconstruction filter design in
 D-computerized tomography, in Radon Transforms and Tomography, T. Quinto, L. Ehrenpreis, A. Faridani, F. Gonzales, and E. Grinberg, eds., vol. 278 of Contemporary Mathematics, AMS, Providence, RI, 2001. MR 2002f:65188 - 14.
- A. RIEDER AND TH. SCHUSTER, The approximate inverse in action with an application to computerized tomography, SIAM J. Numer. Anal., 37 (2000), pp. 1909-1929. MR 2001a:65072
- 15.
- L. L. SCHUMAKER, Spline Functions: Basic Theory, Pure & Applied Mathematics, John Wiley & Sons, New York, 1981. MR 82j:41001
- 16.
- L. A. SHEPP AND B. F. LOGAN, The Fourier reconstruction of a head section, IEEE Trans. Nuc. Sci., 21 (1974), pp. 21-43.
- 17.
- J. WLOKA, Partial Differential Equations, Cambridge University Press, Cambridge, U. K., 1987. MR 88d:35004
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(2000):
65J10, 65R10
Retrieve articles in all Journals with MSC
(2000):
65J10, 65R10
Additional Information:
Andreas
Rieder
Affiliation:
Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung (IWRMM), Universität Karlsruhe, 76128 Karlsruhe, Germany
Email:
andreas.rieder@math.uni-karlsruhe.de
Thomas
Schuster
Affiliation:
Fachbereich Mathematik, Geb.~36, Universität des Saarlandes, 66041 Saarbrücken, Germany
Email:
thomas.schuster@num.uni-sb.de
DOI:
10.1090/S0025-5718-03-01526-6
PII:
S 0025-5718(03)01526-6
Keywords:
Approximate inverse,
mollification,
Radon transform,
filtered backprojection
Received by editor(s):
September 21, 2001
Posted:
March 26, 2003
Additional Notes:
The second author was supported by Deutsche Forschungsgemeinschaft under grant Lo310/4-1
Copyright of article:
Copyright
2003,
American Mathematical Society
|
Comments: webmaster@ams.org