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首页> 外文期刊>Journal of complexity >Regularized collocation method for Fredholm integral equations of the first kind
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Regularized collocation method for Fredholm integral equations of the first kind

机译：第一类Fredholm积分方程的正则配置方法。

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摘要

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An "unregularized" use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.
机译：在本文中，我们考虑一种用于求解第一类Fredholm积分方程的搭配方法，这是一个不适定问题。在事先已知搭配矩阵的最小奇异值减小的速率的情况下，此方法的“非正规”使用可以提供可靠的结果。在这种情况下，并置点的数量起着正则化参数的作用。如果上述先验信息不可用，则可以选择搭配搭配使用Tikhonov正则化。当使用操作符单调索引函数将解决方案平滑度作为源条件给出时，我们将在相当笼统的环境中分析这种正规化搭配。此设置涵盖了迄今为止在Tikhonov正则化理论中研究的所有类型的平滑度。本文讨论的另一个问题是规则化参数的后验选择，这使我们可以在不确定解决方案平滑度的情况下获得确定性噪声模型的最佳精度。

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