Hilbert尺度下正则化求解Abel积分方程
Solving Abel integral equations by regularisation in Hilbert scales
摘要：
阶数a > 0的Abel型积分算子自然地出现在大量的物理过程中。他们的反演需要小心，因为由此产生的逆问题是不适定的。这项工作的目的是设计和分析一个家庭的适当的希尔伯特尺度，使运营商是不适定的秩序的规模。我们提供弱正则性假设的核心基础上的运营商上述成立。我们的建设导致一个定义良好的正则化策略Tikhonov正则化希尔伯特尺度。因此，我们概括了Gorenflo和Yamamoto的结果为< 1 to arbitrary a >0和更一般的内核。由于插值理论的工具，我们还表明，先验的希尔伯特尺度制定在通常的索伯列夫空间的边界条件下的光滑性，以及正规化项实际上相当于惩罚衍生物。最后，根据理论建设，我们开发了一个全面的数值方法，先验编码在一个单一的参数，而不是在一个完整的运营商。几个数值例子，证实了理论收敛速度，并显示了该方法的普遍适用性。
英文摘要：
Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a family of appropriate Hilbert scales so that the operator is ill-posed of order a in the scale. We provide weak regularity assumptions on the kernel underlying the operator for the above to hold true. Our construction leads to a well-defined regularisation strategy by Tikhonov regularisation in Hilbert scales. We thereby generalise the results of Gorenflo and Yamamoto for a < 1 to arbitrary a > 0 and more general kernels. Thanks to tools from interpolation theory, we also show that the a priori associated to the Hilbert scale formulates in terms of smoothness in usual Sobolev spaces up to boundary conditions, and that the regularisation term actually amounts to penalising derivatives. Finally, following the theoretical construction, we develop a comprehensive numerical approach, where the a priori is encoded in a single parameter rather than in a full operator. Several numerical examples are shown, both confirming the theoretical convergence rates and showing the general applicability of the method.
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