【作者】 张海燕；

【导师】 闵涛；

【作者基本信息】 西安理工大学 ， 计算数学， 2008， 硕士

【摘要】 小波作为一个新兴的数学分支,起始于S.Mallat和Y.Meyer在八十年代中后期所作的工作,即构造小波基的通用方法,多分辨分析MRA。此后小波得到了迅猛的发展,在应用方面更是掀起了一股应用小波的热潮,如信号处理、图像分析、奇性检测、边缘分析、微分方程数值求解等。随着小波在数值分析领域的快速发展,越来越多的数学工作者关注小波在求解不适定问题的应用。本文阐述了小波分析基本理论,并将它和GMRES算法引入到求解不适定问题中,展开了一系列收敛速度快,求解精度高的数值算法的研究。首先,GMRES算法与处理不适定问题的Tikhonov正则化方法相结合,分析它们之间相互联系,对大规模不适定问题进行数值求解,并给出了求解算法,数值模拟结果表明了该算法的有效性。此方法和小波变换方法求解不适定问题的相对误差和最大误差进行比较,数值模拟表明该方法的精度较高,但是其运行时间比小波变换方法的运行时间长。然后,在两重网格迭代法的基础上,深入研究了Jacobi预优法,对称Gauss-Seidel预优法和Schur补共轭梯度法,给出了对称Gauss-Seidel预优法和Schur补共轭梯度法的算法。结合小波基本理论,改进Schur补共轭梯度法的算法,提出了小波变换在求解不适定问题中应用的算法,并将其应用于小波变换方法求解重力测定反问题和黑体辐射反问题,数值模拟中将多种方法进行比较,从不同角度验证了该算法的有效性和可行性。更多还原

【Abstract】 As a new mathematics embranchment, wavelet started at S.Mallat and Y.Meyer’s work in the middle and latter half of the 1980s, which was to construct the wavelet base in a general way (multiresolution analysis).Later the wavelet got a drastic development and raised an upsurge in applications, such as signal processing, image analysis, singular check, marginal analysis, numerical solution of differential equations and so on. With the rapid development of wavelet in the field of numerical analysis, an increasing number of mathematical researchers pay attention to the applications of wavelet in solving ill-posed problems.In this thesis, we describes the basic theory of wavelet in detail, and draws the theory of wavelet and GMRES algorithm into solving ill-posed problems to carry out a series of studies in numerical algorithms, which have the abilities of fast convergence and high precision of the procedure. Then, we combine GMRES algorithm with Tikhonov regularization method to solve ill-posed problems, analyze their mutual relationship, and then propose the algorithm of solving the large scale ill-posed problems. Numerical simulations show that the algorithm is effective in programming. Comparing the relative error and absolute error of GMRES method with the wavelet transform method to solve ill-posed problems, numerical simulations indicate the former has higher precision while longer running time than the latter.Based on the two-grid iterative method, the theoretical analysis of Jacobi precondition, symmetric Gauss-Seidel precondition and Schur complement conjugate gradient are studied thoroughly, then we propose the algorithms of symmetric Gauss-Seidel precondition and Schur complement conjugate gradient.So we improve Schur complement conjugate gradient algorithm with the basic theory of wavelet, and then propose the algorithm of wavelet transform applied to solve ill-posed problems. Also, the algorithm is applied to solve the inverse gravity problem and the inverse blackbody radiation problem and compared with some other methods. At last, numerical simulations from different aspects illustrate that, the wavelet transform method adopted in this thesis is effective and feasible in solving ill-posed problems.更多还原