In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for developing numerical reconstruction algorithm for this inverse problem. For this purpose, we develop a new method, based on different properties of solutions of elliptic equations, for treating the determination of the semilinear term as a source term from a point measurement of the solutions. This approach not only allows us to make important relaxations on the data used so far for solving this class of inverse problems, including general Dirichlet excitation lying in a space of dimension one and measurements located at one point on the boundary of the domain, but it also allows us to derive a novel algorithm for the reconstruction of the semilinear term. The effectiveness of our algorithm is corroborated by extensive numerical experiments. Notably, as demonstrated by the theoretical analysis, we are able to effectively reconstruct the unknown nonlinear source term by utilizing solely the information provided by the measurement data at a single point.

翻译：本文研究通过边界测量确定椭圆方程中出现的半线性项的逆问题。我们的主要目标是建立灵活且普适的理论结果，以用于开发该逆问题的数值重构算法。为此，我们基于椭圆方程解的不同性质，提出了一种新方法，将半线性项的确定视为从解的单个点测量中恢复源项的问题。该方法不仅使我们能够显著放宽目前用于求解此类逆问题所需的数据条件——包括使用一维空间中的一般狄利克雷激励以及仅在区域边界单一点处进行测量——还使我们能够推导出一种重构半线性项的新算法。大量数值实验验证了我们算法的有效性。值得注意的是，如理论分析所示，我们能够仅利用单一点处的测量数据信息，有效地重构未知的非线性源项。