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Two-stage approximation of nonsmooth solutions and restoration of noised images. (English. Russian original) Zbl 1072.65076
Autom. Remote Control 65, No. 2, 270-279 (2004); translation from Avtom. Telemekh. 2004, No. 2, 126-135 (2004).
The paper is concerned with the approximation of nonsmooth solutions to linear ill-posed equations \(Au=f\) with noisy data. Let \(A\) be a linear bounded operator from \(L_p(D)\) into \(L_q(S)\), where \(1<p, q <\infty\) and \(D \subset \mathbb R^m\), \(S \subset \mathbb R^k\) are domains with piecewise-smooth boundaries. Suppose the original equation has a solution \({\hat u}\) in the space \(U=\{ u \in L_p(D): J(u) < \infty \}\), \(J(u)=\sup\{ \int_D u(x) \text{ {div}} v(x) dx: v \in C_0^1(D,\mathbb R^m), | v(x)| \leq 1 \}\), and instead of true elements \((A,f)\) their approximations \((A_h,f_{\delta})\) are available. Let \(u^0\) be an approximation to \({\hat u}\). The authors present convergence results for the Tikhonov regularization method \(\min \{ \| A_h u-f_{\delta}\|^q_{L_q}+\alpha (\| u-u^0 \|^p_{L_p}+ J(u)): u \in U \}\) and discuss approaches to its numerical implementation.

65J10 Numerical solutions to equations with linear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47A52 Linear operators and ill-posed problems, regularization
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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