基于预处理共轭梯度法的高精度曲面模型解算

Solving High Accuracy Surface Modelling Based on Preconditioning Conjugate Gradient

为了提高高精度曲面模型(HASM)的计算速度,以HASM对角线元素为预处理算子,充分考虑HASM系数矩阵的稀疏性,基于预处理共轭梯度法(PCG)实现了HASM解算,并发展了HASM-PCG.以高斯合成曲面作为研究对象,用两组模拟试验表明,采用相同的网格数和采样数,达到相同的模拟精度,HASM-PCG收敛速度比MATLAB提供的方法至少快一个数量级;固定采样间隔,改变模拟区域网格数,完成相同的迭代次数,HASM-PCG计算时间与模拟区域网格数呈线性相关,比采用直接法时间复杂度降低两个数量级.以山东省济南市某测区全站仪实测高程数据为研究对象,模拟该测区DEM表明,以中误差和平均绝对误差为误差指标,HASM-PCG精度比SPLINE,KRIGING和IDW至少高7倍.

In order to improve the computational speed of high accuracy surface modelling （HASM）, the preconditioning conjugate gradient （PCG） was employed to solye the system equations of HASM, and the HASM-PCG has been developed, which considers the diagonal elements of HASM as the preconditioning operator and makes full use of the sparseness of HASM coefficient matrix. The Gaussian synthetic surface was used to test the efficiency of HASM-PCG. Two simulation results indicate that under the same lattice numbers and uniform sampling, HASM-PCG has a higher computational speed than the classical iterative methods provided by MATLAB when the simulation results reach the same accuracy; under the fixed sampling interval and different lattice numbers, the computing time of HASM-PCG is in proportion to the first power of the total number of grid cells in the computational domain, which is much faster than the direct methods for solving HASM. Taking the elevation data captured from total station instrument as the database, the real world example, located at Jinan, Shandong province, China, demonstrates that HASM-PCG is more than seven times as accurate as the classical interpolation methods including SPLINE, KRIGING and IDW.