关键词: 不完全cholesky分解, HASM, 预处理共轭梯度法, 对称逐步超松弛算法

Abstract:

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.

Key words: preconditioned conjugate gradients method, HASM, successive over-relaxation algorithm, incomplete Cholesky factorization