二维椭圆型方程反问题中优化算法的比较

The Comparison of Optimization Techniques for 2-D Elliptic Inverse Problem

近年来,决定椭圆型方程系数反问题在地磁、地球物理、冶金和生物等实际问题上有着广泛的应用．本文讨论了二维的决定椭圆型方程系数反问题的数值求解方法．由误差平方和最小原则,这个反问题可化为一个变分问题,并进一步离散化为一个最优化问题,其目标函数依赖于要决定的方程系数．本文着重考察非线性共轭梯度法在此最优化问题数值计算中的表现,并与拟牛顿法作为对比．为了提高算法的效率我们适当选择加快收敛速度的预处理矩阵．同时还考察了线搜索方法的不同对优化算法的影响．数值实验的结果表明,非线性共轭梯度法在这类大规模优化问题中相对于拟牛顿法更有效．

The inverse problem of determining coefficients in elliptic equations has applied to a variety of industrial fields such as geomagnetism, geophysics, metallurgy and biology, etc., in the past years. A numerical method of solving 2-D inverse problem of deter- mining coefficients in elliptic equations is studied. By the least-squares technique, the inverse problem can be transformed into a variational problem and discretized into a non- linear optimization problem with the objective function depending on the coefficients to be determined. Nonlinear conjugate gradients （NLCG） algorithm is mainly investigated in numerical computations and is compared with the quasi-Newton method. Additional efficiencies in the scheme are sought by incorporating preconditioning to accelerate so- lution convergence. The impact on the efficiencies of these two algorithms for different line search methods is also considered. Numerical experiments indicate that the method using nonlinear conjugate gradients is more efficient than the quasi-Newton method in these large-scale optimization problems.