Poisson方程系数识别的有限元反演方法

Finite Element Inversion Method for CoefficientIdentification of Poisson Equation

Poisson方程的系数识别问题在医学成像和遥测勘探等许多领域有重要应用,由于该问题的不适定性且实际应用中采样含噪音,往往导致求解困难,为此提出一种有限元反演方法进行求解.首先,给出Poisson方程边值问题的弱解形式,在有限元空间对边值问题进行逼近;接着,基于三角剖分网格建立其系数识别问题的优化模型,得到系数识别问题的线性方程组;进而,利用高阶Tikhonov正则化模型降低反问题不适定性的影响;最后分别对光滑和分片光滑的扩散系数进行数值反演实验,验证该方法的有效性.

The coefficient identification problem of the Poisson equation has important applications in many fields such as medical imaging and telemetry exploration.Due to its ill-posed nature and noisy sampling in practical applications,it is often difficult to solve.Therefore,a finite element inversion method is proposed to solve this problem.Firstly,the weak solution form of the boundary value problem for the Poisson equation is given,and the boundary value problem is approximated in finite element space.Based on the triangulation grid,an optimization model for coefficient identification problem is established,and a linear equation system for coefficient identification problem is obtained.Then,a higher-order Tikhonov regularization model is used to reduce the ill-posed effect of the inverse problem.Finally,numerical inversion experiments are conducted on the diffusion coefficients with smooth and piecewise smooth surfaces,and the effectiveness of this method is verified.