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Arcangeli's discrepancy principle for a modified projection scheme for ill-posed problems

机译：不适定问题的修正投影方案的Arcangeli差异原理

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摘要

Pereverzev (1995) considered Tikhonov regularization combined with a modified form of a projection method for obtaining stable approximate solutions for ill-posed operator equations. He showed, under a certain a priori choice of the regularization parameter and a specific smoothness assumption on the solution, that the method yields the optimal order with less computational information, in the sense of complexity, than the projection method considered by Plato and Vainikko (1990). In this paper we apply a modified form of the Arcangeli's discrepancy principle for choosing the regularization parameter, and show that the conclusions of Pereverzev still hold. In fact, we do the analysis using a modified form of the generalized Arcangeli's method suggested by Schock (1984) under more flexible smoothness assumption on the solution, as has been done by George and Nair (1998), and derive the optimal result as a special case. Moreover, we compare the computational complexity of the present method with two traditional projection methods in the case of a priori parameter choice, and also discuss the computational complexity required to implement the suggested discrepancy principle.
机译：Pereverzev（1995）考虑将Tikhonov正则化与投影方法的改进形式相结合，以获得不适定算子方程的稳定近似解。他表示，在某种先验性的正则化参数选择和对解决方案的特定平滑度假设下，与Plato和Vainikko所考虑的投影方法相比，从复杂性上讲，该方法产生的最优阶具有较少的计算信息。 1990）。在本文中，我们将Arcangeli差异原理的改进形式用于选择正则化参数，并表明Pereverzev的结论仍然成立。实际上，正如乔治和奈尔（George and Nair，1998）所做的那样，我们在更灵活的光滑度假设下，使用了由修克（1984）建议的广义Arcangeli方法的改进形式进行分析，并得出了最优结果。特殊情况。此外，在先验参数选择的情况下，我们将本方法的计算复杂性与两种传统的投影方法进行了比较，并讨论了实现建议的差异原理所需的计算复杂性。

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