Simple-iteration method with alternating step size for solving operator equations in Hilbert space

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• 作者：Oleg Matysik^a ; ^{matysikoleg@mail.ru" class="auth_mail" title="E-mail the corresponding author} ; Marc M. Van Hulle^b ; ^{marc.vanhulle@med.kuleuven.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Regularization ; Ill-posed problems ; Hilbert space ; Operator equation of first kind ; Self-conjugate operator ; Stopping criterion selection
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：300
• 期：Complete
• 页码：290-299
• 全文大小：442 K

文摘

We introduce a new explicit iterative method with alternating step size for solving ill-posed operator equations of the first kind: class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716000030&_mathId=si11.gif&_user=111111111&_pii=S0377042716000030&_rdoc=1&_issn=03770427&md5=4cf50208a58d73868c4c0178bd813c12" title="Click to view the MathML source">Ax=yclass="mathContainer hidden">class="mathCode">$Ax=y$. We investigate the basic properties of the method for a positive bounded self-conjugate operator class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716000030&_mathId=si12.gif&_user=111111111&_pii=S0377042716000030&_rdoc=1&_issn=03770427&md5=668e3f0d41b5781f7e5ad1ca3affb497">class="imgLazyJSB inlineImage" height="11" width="80" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716000030-si12.gif">class="mathContainer hidden">class="mathCode">$A:H\to H$ in Hilbert space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716000030&_mathId=si13.gif&_user=111111111&_pii=S0377042716000030&_rdoc=1&_issn=03770427&md5=4e2a200dfb035479e2f180f8d65d966e" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$H$ under the assumption that the error for the right part of the equation is available. We discuss the convergence of the method, for a given number of iterations, in the original Hilbert space norm, estimate its precision and formulate recommendations for choosing the stopping criterion. Furthermore, we prove the convergence of the method with respect to the stopping criterion and estimate the remaining error. In case the equation has multiple solutions, we prove that the method converges to the minimum norm solution.