The approximate inverse in action II: convergence and stability. (English) Zbl 1022.65066

Rieder, Andreas; Schuster, Thomas

Math. Comput. 72, No. 243, 1399-1415 (2003).

[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 \[ ||\tilde R_{nl,d}^{(l)} \psi_{q,p}^{(l)} Rf||_{L^2(\Omega)}\lesssim(d^{-\min\{2,\alpha\}}+h^{\min\{\alpha,\alpha+\frac 12\}}d^{\alpha+2})||f||_{H^{\alpha}(\Omega)}. \]

Reviewer: R.S.Dahiya (Ames)

MSC:

65J10	Numerical solutions to equations with linear operators (do not use 65Fxx)
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; tomography

References:

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