山体表面重构数值反演的同伦算法

Numerical inversion of reconstruction of mountain surface by homotopy method

本文主要研究山体表面重构过程.山体表面重构主要包括地球内部热流变化和地球表面运动规律两个过程,是三维对流扩散方程和二维山体表面运动方程的耦合,求解十分困难.为了模拟山体表面重构过程,还要对耦合方程进行反演研究,是一个大尺度非线性优化问题.为了克服对于初值的强烈依赖和非线性优化中存在的多极值难点,我们将同伦反演算法应用到山体表面重构模型,通过数值算例可以看出,本算法具有大范围收敛和较好的稳定特性.

In this paper, we focus on the reconstruction of a mountain surface with time span of millions of years. Reconstruction of a mountain surface consists of two processes. One is the heat transport process model in three dimensional spatial domain, which describes the change of temperature of rocks, and the other is the surface process model in two dimensional spatial domain, which explains the evolution of mountain surface. This is a large scale nonlinear optimization for reconstruction of a mountain surface back to millions of years. To overcome the strong dependency on initial guess value and multiple local minima, we apply a homotopy method to restore the mountain surface. Numerical results show that our algorithm has a good convergent oropertv for given initial guess values and is robust.