This paper outlines a new approach to identify a source term of a [Formula: see text]D elliptic equation for anisotropic nonhomogenous media. The proposed methodology is based on the minimization of an objective function representing differences between the measured potential and those calculated by using the discontinuous dual reciprocity boundary element method, the measurements are required to render a unique solution and supposed to be pointwise in the problem domain. Since the additional data may be contaminated by measurement noises or the numerical computing errors, we adopt a regularizing Levenberg–Marquardt method to solve the nonlinear least-squares problem attained from the inverse source problem. The numerical performance of the proposed approach is studied at the end for both geometries: smooth and piecewise smooth one. The results show a very good agreement with the analytical solutions under exact and noisy data.

中文翻译：

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不连续对偶互易法与正则化 Levenberg-Marquardt 方法结合用于非均匀各向异性材料中的源项恢复

本文概述了一种识别各向异性非均匀介质的 [公式：见正文]D 椭圆方程的源项的新方法。所提出的方法是基于目标函数的最小化，该目标函数表示测量的电位与使用不连续双互易边界元方法计算的电位之间的差异，测量需要提供唯一的解决方案，并且应该在问题域中是逐点的。由于附加数据可能受到测量噪声或数值计算误差的污染，我们采用正则化 Levenberg-Marquardt 方法来解决从逆源问题得到的非线性最小二乘问题。最后研究了两种几何形状的所提出方法的数值性能：平滑和分段平滑。