第一类Fredholm积分方程的正则化
On the regularization of Fredholm integral equations of the first kind
AI总结：
针对第一类Fredholm积分方程，提出一种截断特征函数展开的正则化方法，并引入概率正则化框架，通过时间序列分析工具克服变分正则化的内在局限。
摘要：
本文考虑恢复第一类Fredholm积分方程的正则化解的问题，其中核为Hermitian且平方可积，数据被加性噪声污染。我们不使用基于先验全局界的Tikhonov型变分正则化，而是提出一种截断特征函数展开的方法，该方法在噪声趋于零时，在$L^2$范数意义下渐近收敛。我们通过构造概率正则化解来扩展该过程的概率对应部分，而不假设数据的Fourier系数序列具有任何阶结构。这种概率方法允许我们使用时间序列分析中的统计工具，从而得到一种新的正则化算法，并通过数值例子加以说明。最后，与变分正则化方法所得解的比较表明，基于变分的技术的一些内在局限性可以被克服。
英文摘要：
In this paper the problem of recovering a regularized solution of the Fredholm integral equations of the first kind with Hermitian and square-integrable kernels, and with data corrupted by additive noise, is considered. Instead of using a variational regularization of Tikhonov type, based on a priori global bounds, we propose a method of truncation of eigenfunction expansions that can be proved to converge asymptotically, in the sense of the $L^2$--norm, in the limit of noise vanishing. Here we extend the probabilistic counterpart of this procedure by constructing a probabilistically regularized solution without assuming any structure of order on the sequence of the Fourier coefficients of the data. This probabilistic approach allows us to use the statistical tools proper of time-series analysis, and in this way we attain a new regularizing algorithm, which is illustrated by some numerical examples. Finally, a comparison with solutions obtained by the means of the variational regularization exhibits how some intrinsic limits of the variational-based techniques can be overcome.
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