An optimal stopping rule for the \(v\)-method for solving ill-posed problems, using Christoffel functions
The iterates \(x_ k\) of the \(\nu\)-method are defined by \(x^ \varepsilon_ 0= 0\), \(z^ \varepsilon_ 0= 0\) and \[ z^ \varepsilon_ k= \mu_ k z^ \varepsilon_{k-1}+ (1- \mu_ k) z^ \varepsilon_{k-2}+ \omega_ k(g^ \varepsilon- Kx^ \varepsilon_{k-1}),\;x^ \varepsilon_ k= K^* z^ \varepsilon_ k\quad\text{for }k=1,2,\dots, \] where scalar parameters \(\mu_ 1=1\), \(\omega_ 1=(4\nu+2)/(4\nu+ 1)\), and for \(k>1\), \[ \mu_ k= 1+ (k-1)(2k- 3)(2k+ 2\nu-1)/(k+ 2\nu- 1)(2k+4\nu-1)(2k+ 2\nu- 3), \] \[ \omega_ k= 4(2k+ 2\nu-1)(k+ \nu-1)/(k+ 2\nu-1)(2k+ 4\nu- 1). \] In this paper the authors design an order-optimal stopping rule for the \(\nu\)-method for solving ill-posed linear equations \(Kx= g\), where \(K\) is a bounded linear operator between two Hilbert spaces. They give an algorithm as follows, and prove that the \(\nu\)-method with the stopping rule proposed in this algorithm is order-optimal in all sets \(X_ \mu= R((K^* K)^ \mu)\), the range of \((K^* K)^ \mu\), with \(0< \mu\leq \nu\). ``Algorithm'': Let \(\eta_ 0= \| g^ \varepsilon\|^ 2\), where \(g^ \varepsilon\) is noisy data with \(\| g^ \varepsilon- g\|\leq \varepsilon\), and \(\tau> 1\). Within the \((k+1)\)st iterative step compute \(\eta_ k= (1- \alpha_ k) \eta_{k-1}+ \beta_ k\| z^ \varepsilon_{k+1}- z^ \varepsilon_ k\|^ 2\), where the parameters \(\alpha_ k\) and \(\beta_ k\) are determined recursively by \(\alpha_ 0= 1\) and \[ \alpha^{-1}_ k= 1+ {k(2k- 1)(2k+ 2\nu- 1)\over (k+ 2\nu)(2k+ 4\nu+1)(2k+ 2\nu+1)} \alpha^{-1}_{k-1},\;\beta_ k= {(2\nu+ {1\over 2})^ 2\over (2k+ 2\nu+ 1)^ 2} \alpha_ k. \] Then, as stopping index \(k= k(\varepsilon)\), choose the smallest integer \(k>0\) such that \(\eta_ k\leq \tau\varepsilon^ 2\).
scientific article; zbMATH DE number 645059