基于分段常值水平集的参数识别算法

A New Algorithm for Parameter Identification Based on Piecewise Constant Level Set Method

研究了椭圆方程不连续参数的识别算法.根据原有算法计算效率较低、抗噪性较差、可识别区域数较少的不足,本文基于分段常值水平集方法,根据水平集函数和优化过程的特点,修正原有Uzawa型算法中的带有总变差(TV)正则化的极小化模型和对常值向量的极小化模型,并且结合Barzilai-Borwein方法和预处理共轭梯度算法(PCG)构造一种新的参数识别算法格式.数值实验结果显示,新算法具有计算时间短、精度高、抗噪性强的优点,并且可以识别较复杂的几何区域.

We apply a piecewise constant level set method to elliptic inverse problems. The discontinuity of the co- efficients is represented implicitly by a piecewise constant level set function, which allows to use one level set function to represent multiple phases. The inverse problem is solved using a variational penalization method with the total variation regularization of the level set function. According to the feature of level set function and optimization process, the existing minimization functional in Uzawa algorithm and original algorithm are modified, and a Barzilai- Borwein gradient method and preconditioned conjugate gradient algorithm are used to get efficient and robust numer- ical schemes for solving the obtained problem. Numerical experiments show that the new method can recover coefficients with rather complicated geometry of discontinuities and it has advantages of high precision and strong performance of anti-noise.