二维Poisson方程中不完备边界及未知参数的反演

Inverse problem of incomplete boundaries and unknown parameters in two-dimensional Poisson equation

采用基本解方法和Kalman滤波技术对含有不完备边界或未知参数的Poisson方程的反问题进行研究。通过有限观测点的测量值重建了方程中的不完备边界,得到较为准确的边界值;根据测量值及参数预测值,经过有限次滤波反演得到方程中未知参数值。结果表明:基本解方法和Kalman滤波技术相结合在处理二维Poisson方程的反问题时具有较强的鲁棒性,是解决反问题的有力工具。

The inverse problem of Poisson equation with incomplete boundary or unknown parameters have been solved by using the method of fundamental solutions (MFS) and Kalman filtering technique. According to Kalman filtering technique, the incomplete boundaries have been reconstructed by finite known value of the observation points.The unknown parameters in the equation have also been solved by several filtering with initial values. The results show that MFS and Kalman filtering technique have strong robustness in dealing with the inverse problems of two-dimensional Poisson equation, which can be used in many inverse problems.