Case Studies and a Pitfall for Nonlinear Variational Regularization Under Conditional Stability
Case Studies and a Pitfall for Nonlinear Variational Regularization Under Conditional Stability
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DOI：
10.1007/978-981-15-1592-7_9
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发表时间：
2018-10
期刊：
影响因子：
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通讯作者：
D. Gerth;B. Hofmann;Christopher Hofmann
中科院分区：
文献类型：
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作者：
D. Gerth;B. Hofmann;Christopher Hofmann
条件稳定性 估计是不适定问题正规化的流行工具。特别是在非线性算子下的一个缺点是，如果这种估计的有效区域不完全已知，则需要额外的正则化来获得稳定的近似解。本文考虑Hilbert尺度下非线性不适定算子方程的Tikhonov正则化条件稳定性估计。我们总结断言的收敛性和收敛速度在三种情况下描述的相对光滑的罚款的吉洪诺夫功能和精确的解决方案。对于过平滑的处罚，真正的解决方案不再达到一个有限的值，我们提出了一个结果与修改后的假设先验选择的正则化参数产生的最佳顺序的噪声数据的收敛速度。我们强烈强调的条件稳定性估计的局部特征，并表明，陷阱可能会发生通过不正确的稳定性估计。然后收敛可能完全失败，条件稳定性的稳定效果可能会丢失。一些非线性例子的综合数值案例研究说明了这种影响。
Conditional stability estimates are a popular tool for the regularization of ill-posed problems. A drawback in particular under nonlinear operators is that additional regularization is needed for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this paper we consider Tikhonov regularization under conditional stability estimates for nonlinear ill-posed operator equations in Hilbert scales. We summarize assertions on convergence and convergence rate in three cases describing the relative smoothness of the penalty in the Tikhonov functional and of the exact solution. For oversmoothing penalties, for which the true solution no longer attains a finite value, we present a result with modified assumptions for a priori choices of the regularization parameter yielding convergence rates of optimal order for noisy data. We strongly highlight the local character of the conditional stability estimate and demonstrate that pitfalls may occur through incorrect stability estimates. Then convergence can completely fail and the stabilizing effect of conditional stability may be lost. Comprehensive numerical case studies for some nonlinear examples illustrate such effects.