SIAM Journal on Imaging Sciences, Volume 13, Issue 4, Page 1781-1833, January 2020.
In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and inverse problems, such as quantitative photoacoustic tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.

中文翻译：

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高阶椭圆方程的渐近展开及其在定量光声层析成像中的应用

SIAM影像科学杂志，第13卷，第4期，第1781-1833页，2020
年1月。在存在小夹杂物的情况下，我们为高阶椭圆方程的解导出新的渐近展开。作为副产品，我们导出了基于拓扑导数的算法，用于重建分段平滑函数。该算法可用于成像，拓扑优化和逆问题（例如定量光声层析成像）等问题的边缘检测，为此，我们通过数值方法证明了渐近展开方法的有效性。