## The approximate inverse in action II: convergence and stability

HTML articles powered by AMS MathViewer

- by Andreas Rieder and Thomas Schuster PDF
- Math. Comp.
**72**(2003), 1399-1415 Request permission

## Abstract:

The approximate inverse is a scheme for constructing stable inversion formulas for operator equations. Originally, it is defined on $L^2$-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

- 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), 112–124. MR**263214**, DOI 10.1137/0707006 - A. Cohen, Ingrid Daubechies, and J.-C. Feauveau,
*Biorthogonal bases of compactly supported wavelets*, Comm. Pure Appl. Math.**45**(1992), no. 5, 485–560. MR**1162365**, DOI 10.1002/cpa.3160450502 - Wolfgang Dahmen, Angela Kunoth, and Karsten Urban,
*Biorthogonal spline wavelets on the interval—stability and moment conditions*, Appl. Comput. Harmon. Anal.**6**(1999), no. 2, 132–196. MR**1676771**, DOI 10.1006/acha.1998.0247 - Heinz W. Engl, Martin Hanke, and Andreas Neubauer,
*Regularization of inverse problems*, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR**1408680** - J.-L. Lions and E. Magenes,
*Non-homogeneous boundary value problems and applications. Vol. I*, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR**0350177** - Alfred Karl Louis,
*Inverse und schlecht gestellte Probleme*, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1989 (German). MR**1002946**, DOI 10.1007/978-3-322-84808-6 - A. K. Louis,
*Corrigendum: “Approximate inverse for linear and some nonlinear problems” [Inverse Problems 11 (1995), no. 6, 1211–1223; MR1361769 (96f:65068)]*, Inverse Problems**12**(1996), no. 2, 175–190. MR**1382237**, DOI 10.1088/0266-5611/12/2/005 - A. K. Louis,
*A unified approach to regularization methods for linear ill-posed problems*, Inverse Problems**15**(1999), no. 2, 489–498. MR**1684469**, DOI 10.1088/0266-5611/15/2/009 - A. K. Louis and P. Maass,
*A mollifier method for linear operator equations of the first kind*, Inverse Problems**6**(1990), no. 3, 427–440. MR**1057035** - F. Natterer,
*The mathematics of computerized tomography*, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR**856916** - Peter Oswald,
*Multilevel finite element approximation*, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. MR**1312165**, DOI 10.1007/978-3-322-91215-2 - D. A. Popov,
*On convergence of a class of algorithms for the inversion of the numerical Radon transform*, Mathematical problems of tomography, Transl. Math. Monogr., vol. 81, Amer. Math. Soc., Providence, RI, 1990, pp. 7–65. MR**1104013** - Andreas Rieder,
*Principles of reconstruction filter design in 2D-computerized tomography*, Radon transforms and tomography (South Hadley, MA, 2000) Contemp. Math., vol. 278, Amer. Math. Soc., Providence, RI, 2001, pp. 207–226. MR**1851489**, DOI 10.1090/conm/278/04606 - Andreas Rieder and Thomas Schuster,
*The approximate inverse in action with an application to computerized tomography*, SIAM J. Numer. Anal.**37**(2000), no. 6, 1909–1929. MR**1766853**, DOI 10.1137/S0036142998347619 - Larry L. Schumaker,
*Spline functions: basic theory*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR**606200** - L. A. Shepp and B. F. Logan,
*The Fourier reconstruction of a head section*, IEEE Trans. Nuc. Sci., 21 (1974), pp. 21–43. - J. Wloka,
*Partial differential equations*, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR**895589**, DOI 10.1017/CBO9781139171755

## 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
- Received by editor(s): September 21, 2001
- Published electronically: March 26, 2003
- Additional Notes: The second author was supported by Deutsche Forschungsgemeinschaft under grant Lo310/4-1
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp.
**72**(2003), 1399-1415 - MSC (2000): Primary 65J10, 65R10
- DOI: https://doi.org/10.1090/S0025-5718-03-01526-6
- MathSciNet review: 1972743