一维反应扩散方程的参数辨识和源项反演

Simultaneous inversion of coefficient and source term of a one-dimensional reaction-diffusion equation

中文关键词  反应扩散方程  反问题  参数辨识  开关切换控制  矩阵束方法  最优扰动正则化

英文关键词  reaction-diffusion equation  inverse problems  parameter identification  switch on/off boundary control  matrix pencil method  optimal perturbation regularization

中文摘要

      反应扩散方程在许多物理、化学以及生物过程的建模中都有着广泛的应用. 本文研究基于边界控制、边界观测的一维反应扩散方程反应项系数和源项的同时反演问题. 首先, 通过设计边界切换开关控制, 并借助于Dirichlet级数表示的唯一性以及逆谱理论, 证明了反应项系数和源项可以由边界观测唯一确定. 在可辨识性基础上, 结合矩阵束方法和最优扰动正则化算法, 提出了一种稳定的重构系统参数和源项的算法. 最后, 通过数值算例验证了辨识算法的有效性.

英文摘要

      Reaction-diffusion equations are widely used in the modeling of many physical, chemical and biological processes. This paper investigates simultaneous inversion of the reaction coefficient and source term for a one-dimensional reaction-diffusion equation based on boundary control and boundary observation. To begin with, by virtue of the uniqueness of the Dirichlet series representation and inverse spectral theory, it is shown that the reaction coefficient and source term can be uniquely determined from the boundary observation by designing a switch on/off boundary control. Next, a stable numerical algorithm is proposed for the simultaneous reconstruction of the reaction coefficient and source term by combining the matrix pencil method and optimal perturbation regularization technique. Finally, the effectiveness of the identification algorithm is verified by a numerical example.