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Abel type integral equations in stereology: I. General discussion

机译:阿贝尔型积分方程在立体学中的应用: 一、一般性讨论

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SUMMARYIn developing numerical methods for the solution of computational problems of an involved and pragmatic nature, as occur for example in stereology, the approach should be first to classify the distinct types of numerical problems which arise. If available, appropriate results from numerical analysis can then be used to develop stable numerical processes for their solution. In fact, many of the results relating to the construction of basic numerical methods, such as spectral methods for the stable differentiation of experimental data and product integration methods for the evaluation of integrals with oscillatory and singular integrands, have been resolved. Thus, we are now in a position to apply the above approach to the more complex computational problems which arise in stereology.Many of these problems belong to one of the following two classes: (i) the solution of integral equations of Abel type; and (ii) the solution of some numerical differentiation problem.In Part I of this paper, we illustrate the generality in stereology of the Abel type integral equation and numerical differentiation formulations. As well, we derive the specific properties required in Part II which is mainly concerned with the construction of stable computational methods for (i). Included among these properties are the explicit inversion formulae which are known for general Abel type integral equations. It is these inversion formulae which we use to construct stable computational methods.Often, when estimates for linear properties of the solutions of (i) and (ii) are required, the numerical solution of (i) and (ii) can be circumvented by estimating the linear properties directly from the given observational data. In deriving such estimates, use of the properties of the Abel type integral equation and differentiation formulations plays an essential role. Because of the close connection between such estimation problems and (i) and (ii), the estimation of linear properties from truncated observational data is also examined in Part I.
机译:摘要在开发数值方法来解决涉及和实用性质的计算问题时,例如在立体学中发生的问题,该方法应首先对出现的不同类型的数值问题进行分类。如果可用,则可以使用数值分析的适当结果来开发其解决方案的稳定数值过程。事实上,许多与构建基本数值方法有关的结果,例如用于实验数据稳定微分的谱方法和用于评估振荡积分和奇异积分积分的乘积积分方法,已经得到解决。因此,我们现在能够将上述方法应用于方体学中出现的更复杂的计算问题。其中许多问题属于以下两类之一:(i)阿贝尔型积分方程的解;(ii)一些数值微分问题的解。在本文的第一部分中,我们说明了阿贝尔型积分方程和数值微分公式在空间学中的普遍性。此外,我们推导出了第二部分所需的特定性质,该部分主要涉及构建(i)的稳定计算方法。这些属性包括显式反演公式,该公式以一般 Abel 型积分方程而闻名。我们正是用这些反演公式来构建稳定的计算方法。通常,当需要估计 (i) 和 (ii) 解的线性性质时,可以通过直接从给定的观测数据估计线性性质来规避 (i) 和 (ii) 的数值解。在推导此类估计时,使用阿贝尔型积分方程的性质和微分公式起着至关重要的作用。由于这些估计问题与(i)和(ii)之间有着密切的联系,因此第一部分还研究了从截断的观测数据中估计线性性质的问题。

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