基于牛顿型算法的椭圆方程多参数识别
Multi-Parameter Identification of Elliptic Equations Based on Newton-Type Algorithms
摘要: 针对椭圆型偏微分方程(PDE) Neumann边值问题中的多参数联合识别问题展开研究，重点解决在状态测量数据精度不足的条件下，同时辨识扩散矩阵、源项、边界条件及物理状态的挑战。建立了一个结合能量泛函与Tikhonov正则化的变分框架，推导出Karush-Kuhn-Tucker (KKT)系统，并采用有限元方法对PDE进行空间离散，发展牛顿型算法以实现数值求解。我们的算法在不同测量数据条件下能够有效地辨识多个参数，最终的数值算例验证了该方法的有效性。
Abstract: The study focuses on the multi-parameter joint identification problem in the Neumann boundary value problem for elliptic partial differential equations (PDEs). It specifically addresses the challenge of simultaneously identifying the diffusion matrix, source term, boundary conditions, and physical states under conditions of insufficient accuracy in state measurement data. We establish a variational framework that combines an energy functional with Tikhonov regularization and derive the Karush-Kuhn-Tucker (KKT) system. We also employ the finite element method for spatial discretization of the PDE and develop a Newton-type algorithm for numerical solution. Our algorithm effectively identifies multiple parameters under different measurement data conditions, and the final numerical examples validate the effectiveness of the method.
文章引用：吴宇航, 刘欢. 基于牛顿型算法的椭圆方程多参数识别[J]. 统计学与应用, 2026, 15(2): 163-178. https://doi.org/10.12677/sa.2026.152044

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