Recently an algorithm was given in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any $L^2$ perturbation from linearised data in the two-dimensional linearised Calder\'on problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calder\'on problem in a ball, and use a 3D Zernike basis to obtain a method for exact direct reconstruction of any $L^3$ perturbation from linearised data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the 3D method only makes use of a relatively small subset of boundary measurements for exact reconstruction, compared to a full $L^2$ basis of current densities.

翻译：最近，[Garde & Hyvönen, SIAM J. Math. Anal., 2024] 给出了一种算法，用于从二维线性化Calderón问题的线性化数据中精确直接重建任意 $L^2$ 扰动。该算法是一种基于二维Zernike基的简单前向替换方法。现在，我们考虑球体内的三维线性化Calderón问题，并利用三维Zernike基获得一种从线性化数据中精确直接重建任意 $L^3$ 扰动的方法。该方法同样是前向替换，因此数值实现非常高效。此外，与基于电流密度的完整 $L^2$ 基相比，该三维方法仅使用相对较小的一部分边界测量即可实现精确重建。