The approximate inverse in action II: convergence and stability. (English) Zbl 1022.65066
[For part I see SIAM J. Numer. Anal. 37, 1909-1929 (2000; Zbl 0961.65112).]
The approximate inverse is a numerical scheme for solving operator equations of the first kind in Hilbert spaces. In this paper, the authors further develop its analytic foundation, relying on their previous results. This general setting allow them to investigate the discrete version of the approximation inverse which actually under lies numerical computations. The convergence is shown where the discretization parameter tends to zero. The authors also prove stability by showing the regularization property. Finally they apply the results to the filtered back projection algorithm in 2D-tomography to obtain convergence rates. One of the theorems is given below:
Theorem. Let \(\alpha>\frac 12\) and let \(f\) be in \(H_0^{\alpha}(\Omega)\) with supp \(f\Subset\Omega\). The radially symmetric mollifier \(e\) is assumed to be in \(H_0^{\alpha+1}(\Omega)\). Let \(\tilde d=\tilde d(f)\) be the smallest positive integer such that supp \(f\) is contained in \(B_{1-\frac 1{\tilde d}}(0)\), the ball about the origin with radius \(1-\frac 1{\tilde d}\). If \(d\geq \tilde d\), then \[ ||\tilde R_{nl,d}^{(l)} \psi_{q,p}^{(l)} Rf||_{L^2(\Omega)}\lesssim(d^{-\min\{2,\alpha\}}+h^{\min\{\alpha,\alpha+\frac 12\}}d^{\alpha+2})||f||_{H^{\alpha}(\Omega)}. \]
The approximate inverse is a numerical scheme for solving operator equations of the first kind in Hilbert spaces. In this paper, the authors further develop its analytic foundation, relying on their previous results. This general setting allow them to investigate the discrete version of the approximation inverse which actually under lies numerical computations. The convergence is shown where the discretization parameter tends to zero. The authors also prove stability by showing the regularization property. Finally they apply the results to the filtered back projection algorithm in 2D-tomography to obtain convergence rates. One of the theorems is given below:
Theorem. Let \(\alpha>\frac 12\) and let \(f\) be in \(H_0^{\alpha}(\Omega)\) with supp \(f\Subset\Omega\). The radially symmetric mollifier \(e\) is assumed to be in \(H_0^{\alpha+1}(\Omega)\). Let \(\tilde d=\tilde d(f)\) be the smallest positive integer such that supp \(f\) is contained in \(B_{1-\frac 1{\tilde d}}(0)\), the ball about the origin with radius \(1-\frac 1{\tilde d}\). If \(d\geq \tilde d\), then \[ ||\tilde R_{nl,d}^{(l)} \psi_{q,p}^{(l)} Rf||_{L^2(\Omega)}\lesssim(d^{-\min\{2,\alpha\}}+h^{\min\{\alpha,\alpha+\frac 12\}}d^{\alpha+2})||f||_{H^{\alpha}(\Omega)}. \]
Reviewer: R.S.Dahiya (Ames)
MSC:
65J10 | Numerical solutions to equations with linear operators |
65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
65R10 | Numerical methods for integral transforms |
44A12 | Radon transform |
47A50 | Equations and inequalities involving linear operators, with vector unknowns |
47A52 | Linear operators and ill-posed problems, regularization |
92C55 | Biomedical imaging and signal processing |
65R30 | Numerical methods for ill-posed problems for integral equations |
Keywords:
approximate inverse; mollification; radon transform; operator equations of the first kind; Hilbert spaces; convergence; stability; regularization; filtered back projection algorithm; tomographyCitations:
Zbl 0961.65112References:
[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), 112 – 124. · Zbl 0201.07803 · doi:10.1137/0707006 |
[2] | A. Cohen, Ingrid Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485 – 560. · Zbl 0776.42020 · doi:10.1002/cpa.3160450502 |
[3] | 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. · Zbl 0922.42021 · doi:10.1006/acha.1998.0247 |
[4] | Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. · Zbl 0859.65054 |
[5] | J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. · Zbl 0223.35039 |
[6] | Alfred Karl Louis, Inverse und schlecht gestellte Probleme, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1989 (German). · Zbl 0667.65045 |
[7] | 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. · Zbl 0851.65036 · doi:10.1088/0266-5611/12/2/005 |
[8] | A. K. Louis, A unified approach to regularization methods for linear ill-posed problems, Inverse Problems 15 (1999), no. 2, 489 – 498. · Zbl 0933.65060 · doi:10.1088/0266-5611/15/2/009 |
[9] | 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. · Zbl 0713.65040 |
[10] | F. Natterer, The mathematics of computerized tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. · Zbl 0617.92001 |
[11] | Peter Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. · Zbl 0830.65107 |
[12] | 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. |
[13] | 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. · Zbl 0986.65133 · doi:10.1090/conm/278/04606 |
[14] | 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. · Zbl 0961.65112 · doi:10.1137/S0036142998347619 |
[15] | Larry L. Schumaker, Spline functions: basic theory, John Wiley & Sons, Inc., New York, 1981. Pure and Applied Mathematics; A Wiley-Interscience Publication. · Zbl 0449.41004 |
[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, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. · Zbl 0623.35006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.