Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces
Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces
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DOI：
10.1088/1361-6420/ab5314
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发表时间：
2019-07
期刊：
影响因子：
2.1
通讯作者：
Y. Grabovsky;N. Hovsepyan
中科院分区：
文献类型：
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作者：
Y. Grabovsky;N. Hovsepyan
在反问题的背景下，经常出现分析延拓的需要。尽管存在唯一性定理，但在没有附加的正则化约束的情况下，此类问题是出了名的不适定。我们考虑了几个具有典型的全局有界性约束的解析连续问题，它们恢复了适定性。我们表明，当一个人离开数据来源时，所有这些问题都表现出幂定律精度的恶化。在这篇文章中，我们证明了我们的一般基于希尔伯特空间的方法在确定这些指数方面的有效性。该方法将“最坏情况”函数识别为Fredholm型线性积分方程解。在特殊的几何形状中，如圆环或上半平面，这个方程可以显式求解。然后利用在环空中得到的解来确定从椭圆焦点到椭圆内任意点的解析延拓的精确的幂指数。在指数确定的特殊情况下，我们的公式与前人的结果是一致的。
The need for analytic continuation arises frequently in the context of inverse problems. Notwithstanding the uniqueness theorems, such problems are notoriously ill-posed without additional regularizing constraints. We consider several analytic continuation problems with typical global boundedness constraints that restore well-posedness. We show that all such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the ‘worst case’ function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus or upper half-plane this equation can be solved explicitly. The obtained solution in the annulus is then used to determine the exact power law exponent for the analytic continuation from an interval between the foci of an ellipse to an arbitrary point inside the ellipse. Our formulas are consistent with results obtained in prior work in those special cases when such exponents have been determined.