Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems
Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems
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DOI：
10.1137/120871547
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发表时间：
2013-03
期刊：
影响因子：
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通讯作者：
B. F. Nielsen;K. Mardal
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文献类型：
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作者：
B. F. Nielsen;K. Mardal
我们分析了适用于线性karush-kuhn--tucker系统的最小残差方法（微小）方法的性能，该系统与反问题有关。这样的最佳系统通常具有鞍点结构，并且具有所有$ \ alpha> 0 $的独特解决方案，其中$ \ alpha $是Tikhonov正则化中使用的参数。不幸的是，相关的光谱条件数对于$ \ alpha $的小值非常大，这强烈表明它们的数值处理很困难。我们的主要结果表明，可以通过微小方法有效地解决广泛的线性不良最佳系统。通过仔细分析相关的鞍点算子的频谱来获得该结果：除了一些孤立的特征值，该频谱包括三个有界间隔。 Krylov子空间方法很好地解决了此类问题。对于严重不良的案例，采用基于Chebyshev多项式的技术来证明...
We analyze the performance of the minimal residual (MINRES) method applied to linear Karush--Kuhn--Tucker systems arising in connection with inverse problems. Such optimality systems typically have a saddle point structure and have unique solutions for all $\alpha>0$, where $\alpha$ is the parameter employed in the Tikhonov regularization. Unfortunately, the associated spectral condition number is very large for small values of $\alpha$, which strongly indicates that their numerical treatment is difficult. Our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the MINRES method. This result is obtained by carefully analyzing the spectrum of the associated saddle point operator: Except for a few isolated eigenvalues, the spectrum consists of three bounded intervals. Krylov subspace methods handle such problems very well. For severely ill posed cases, techniques based on Chebyshev polynomials are applied to prove that the number of iterations needed by...