高精度曲面建模(HASM)是一种全新的曲面建模方法,其整个过程可分为偏微分方程的离散、采样方程建立和代数方程组求解3个阶段。目前所采用的求解对称正定方程组的方法主要是共轭梯度法。为了解决HASM的计算速度问题,本文给出了2种新的预处理共轭梯度算法,分别为不完全Cholesky分解共轭梯度法和对称逐步超松弛预处理共轭梯度法。实验表明,不完全Cholesky分解共轭梯度法收敛速度最快,且这2种预处理方法均比其他方法具有更快的收敛速度。

As a novel surface modeling method, the whole computing process of HASM can be divided into three parts: deriving finite difference approximations to differential equations, establishing the sampling point equations and solving the algebra equations. The well known conjugate gradient method has been used extensively to solve this symmetric positive definite system instead of Gaussian's elimination based direct methods, especially for very large systems when parallel solution environment is preferred. However, a difficulty associated with the method of conjugate gradients is that it works well on matrices that are either well conditioned or have just a few distinct eigenvalues and the coefficient matrix of the algebra equation in HASM is ill-conditioned. In this paper, we show how to preprocess a linear system so that the matrix of coefficients assumes one of these nice forms. We give two other preconditioners, i.e. incomplete Cholesky decomposition conjugate gradient method (ICCG) and symmetric successive over relaxation-preconditioned conjugate gradient method (SSORCG), so as to improve the convergence rate of HASM. Furthermore, we give adequate consideration in storage scheme of the large sparse matrix and optimize the performance of sparse matrix-vector multiplication. The cost of the computation is also considered in each iteration. We implement and test the proposed method on a Dell OptiPlex 990MT machine. Numerical tests show that ICCG has the fastest convergence rate of HASM. We also find that both ICCG and SSORCG have much faster convergence rates than others.