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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.

##### MSC:
 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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