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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Landweber iteration ： ウィキペディア英語版 Landweber iteration The Landweber iteration or Landweber algorithm is an algorithm to solve ill-posed linear inverse problems, and it has been extended to solve non-linear problems that involve constraints. The method was first proposed in the 1950s,〔 and it can be now viewed as a special case of many other more general methods.〔 == Basic algorithm == The original Landweber algorithm 〔 attempts to recover a signal ''x'' from measurements ''y''. The linear version assumes that $y=Ax$ for a linear operator ''A''. When the problem is in finite dimensions, ''A'' is just a matrix. When ''A'' is nonsingular, then an explicit solution is $x\; =\; A^\; y$. However, if ''A'' is ill-conditioned, the explicit solution is a poor choice since it is sensitive to any errors made on ''y''. If ''A'' is singular, this explicit solution doesn't even exist. The Landweber algorithm is an attempt to regularize the problem, and is one of the alternatives to Tikhonov regularization. We may view the Landweber algorithm as solving: : $\backslash min\_x\; \backslash |Ax-y\backslash |\_2^2/2$ using an iterative method. For ill-posed problems, the iterative method may be purposefully stopped before convergence. The algorithm is given by the update : $x\_\; =\; x\_\; -\; \backslash omega\; A^$ *(Ax_k - y). where the relaxation factor $\backslash omega$ satisfies . Here $\backslash sigma\_1$ is the largest singular value of $A$. If we write $f(x)\; =\; \backslash |Ax-y\backslash |\_2^2\; /2$, then the update can be written in terms of the gradient : $x\_\; =\; x\_k\; -\; \backslash omega\; \backslash nabla\; f(x\_k)$ and hence the algorithm is a special case of gradient descent. Discussion of the Landweber iteration as a regularization algorithm can be found in.〔Louis, A.K. (1989): Inverse und schlecht gestellte Probleme. Stuttgart, Teubner〕〔Vainikko, G.M., Veretennikov, A.Y. (1986): Iteration Procedures in Ill-Posed Problems. Moscow, Nauka (in Russian)〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Landweber iteration」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.