It is known that dual representation of problems (through the main function and its adjoint in the Lagrange sense) makes it possible to formulate an effective perturbation theory on which the successive approximation method in the inverse problem theory relies. Let us suppose that according to preliminary predictions, a solution to the inverse problem (for example, the structure of medium of interest) belongs to a certain set A. Then selecting a suitable (trial, reference) element a₀ as an unperturbed one and applying the perturbation theory, one can approximately describe the behavior of a solution to the forward problem in this domain and find a subset A₀ that matches the measurement data best. However, as the accuracy requirements increase, the domain of applicability of the first approximation A₀ is rapidly narrowing, and its expansion via addition of higher terms of the expansion complicates the solving procedure. For this reason, a number of works have searched for unperturbed approaches, including the method of variational interpolation (VI method). In this method, not one but several reference problems a₁, a₂,...,a_n are selected, from which a linear superposition of the principal function and the adjoint one is constructed, followed by determination of coefficients from the condition of stationarity of the form of the desired functional representation. This paper demonstrates application of the VI method to solving inverse problems of cosmic rays astrophysics in the simplest statement.

中文翻译：

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运输理论反问题中泛函的变分插值

众所周知，问题的双重表示（通过拉格朗日意义上的主函数及其伴随函数）使得有可能制定一种有效的扰动理论，逆问题理论中的逐次逼近方法所基于。让我们假设，根据初步预测，对逆问题的解决方案（例如，所关注介质的结构）属于一组特定的甲。然后选择一个合适的（试验，参考）元素a ₀作为无扰动元素，并应用扰动理论，可以大致描述该域中前向问题的解的行为并找到子集A ₀最匹配测量数据。但是，随着精度要求的提高，一阶近似值A ₀的适用范围正在迅速缩小，并且通过添加更高的扩展项来扩展其求解复杂度。因此，许多工作都在寻找不受干扰的方法，包括变分插值方法（VI方法）。在这种方法中，不是一个参考问题，而是几个参考问题a ₁，a ₂，...，a _n从中选择一个，从中构造出主要函数和伴随函数的线性叠加，然后根据所需函数表示形式的平稳性条件确定系数。本文以最简单的方式演示了VI方法在解决宇宙射线天体物理学反问题中的应用。