偏微分方程的小波分析方法
摘要
工程中的许多问题可归类于偏微分方程的求解。传统的各种分析与计算方法在具
有其特定优点的同时，也均具有其不足之处。例如有限元方法生成稀疏但却病态的刚
度矩阵；积分方程法需处理满阵，要解决大量的积分，因而计算量大等。如果应用小
波则上述缺点均可克服。这就是基于小波的算法的优越性。然而在具体应用中仍有许
多问题需要解决。由于小波定义的开放性，使得首先要处理的是针对不同的问题构造
或选择合适的小波，使问题的表述变得容易和简单；其次要对算法进行设计和改造，
以充分利用所选小波基的特点。本文在对小波变换及小波基理论进行研究的基础上，
主要提出并解决了如下问题：
1.提出了框架算子族和小波算子族的新概念，它是小波变换的推广。我们讨论了
框架算子族和小波算子族的性质，证明了连续小波变换归类于小波算子族。在此基础
上，讨论了电磁小波和通常的小波函数对应的算子族之间的关系，得出了它们之间相
差一个共轭运算的结论。我们还以声波为例，给出了几种不同参数下声波的共轭小波
算子族的图形。
2.我们在小波Galerkin法中用区间小波替代J.Xu用的I.Daubechies的紧支
正交小波来生成Sobolev空间H_O~1（I）及H~1（I）的小波基，使基的相关性大大减小，进
一步使刚度矩阵的条件数减少，从而提高了线性方程组求解的速度和精度。在
Galerkin法基础上，我们利用小波多分辨分析的特性，给出了小波有限元逐层校正
算法成立的条件，并提出了条件不满足时的全局修正算法。对两点边值问题的计算验
证了我们的结论。
3.讨论了利用周期小波进行积分方程矩量法求解的解空间的两种生成方法，在
此基础上给出了积分方程小波矩量法解空间的收敛性及误差估计。我们提出了多函数
小波概念，并按照单函数的多分辨分析方法，讨论了多函数小波的一些性质。通过酉
阵的方式建立了Multiwavelet，并将其成功地应用于二维声波散射问题和非线性
Hammerstein积分方程。
4.利用样条函数理论证明了区间样条小波插值是在三次样条函数空间中的最佳
逼近函数，并在此基础上给出了用尺度函数和用小波函数为基插值的一致性和插值函
数的误差估计式。我们提出了一种用两个一阶导算子矩阵的乘积替换二阶导算子矩阵
的替代算法。用两种不同的插值算法对含不同参数的Burgers方程进行了验算，结果
表明对小参数的Burgers方程，两种算法结果相当；而对稍大的粘性系数，替代算法
的震荡明显小于原算法，因而替代算法有较大的稳定性范围。
As is well known, many problems in science can be eventually presented in the form of partial differential equation (PDE). While the methods now we using to analysis and compute PDE have their own advantages, they also have some shortcomings, such as the ill condition matrix by FEM and a lot of integrals needed by the BIE. But when wavelets are used, we will not meet the disadvantages mentioned above. Usually applying wavelet to the PDE, we need to solving such problems as follows (1) choose a suitable wavelet basis according to the problem we want to deal with (2) design or modify a good algorithm in order to make use of the characters of the wavelet we choose, etc.
In this dissertation, based on the research of wavelet theory, the following problems are discussed:
1. A new conception. wavelet operator group, is presented, which is the generalization of the wavelet transformation. Having discussed the properties, we prove that the continuous wavelet transformation belongs to the wavelet operator group. We come to the conclusion that the physical wavelet is the conjugate of the wavelet operator group.
2. The substitution is made of the interval wavelet basis for the compact one in order to produce the basis of Sobolev space with the lower condition number, which will improve the speed and the accuracy of the linear system made by wavelet Galerkin method. Also the algorithms based on the local or global correction are presented and their properties are discussed in detail. The numerical results on two point boundary problems test and verify the algorithms.
3. Two ways are discussed to produce solution spaces when the BIE is used with the basis of period wavelet. Based on these, the convergence and the error estimation are given. The MRA with more then one-wavelet function is presented and the multi-wavelet basis is produced by unitary matrix. This basis is used successfully in scattering problem of sonic wave and nonlinear Hanimersterin integral equation.
4. The optimal approximation property is shown when the cubic spline interval wavelet is used in the interpolation with some special points. This property ensures the identical of the interpolation by use of scale basis and of wavelet basis. The interpolation error is also discussed. A new algorithm is presented in which two one-order differential matrixes are used to substitute for the two-order one. This way produces less vibration for the numerical solution of Burgers equation when the coefficient is somewhat larger.
有其特定优点的同时，也均具有其不足之处。例如有限元方法生成稀疏但却病态的刚
度矩阵；积分方程法需处理满阵，要解决大量的积分，因而计算量大等。如果应用小
波则上述缺点均可克服。这就是基于小波的算法的优越性。然而在具体应用中仍有许
多问题需要解决。由于小波定义的开放性，使得首先要处理的是针对不同的问题构造
或选择合适的小波，使问题的表述变得容易和简单；其次要对算法进行设计和改造，
以充分利用所选小波基的特点。本文在对小波变换及小波基理论进行研究的基础上，
主要提出并解决了如下问题：
1.提出了框架算子族和小波算子族的新概念，它是小波变换的推广。我们讨论了
框架算子族和小波算子族的性质，证明了连续小波变换归类于小波算子族。在此基础
上，讨论了电磁小波和通常的小波函数对应的算子族之间的关系，得出了它们之间相
差一个共轭运算的结论。我们还以声波为例，给出了几种不同参数下声波的共轭小波
算子族的图形。
2.我们在小波Galerkin法中用区间小波替代J.Xu用的I.Daubechies的紧支
正交小波来生成Sobolev空间H_O~1（I）及H~1（I）的小波基，使基的相关性大大减小，进
一步使刚度矩阵的条件数减少，从而提高了线性方程组求解的速度和精度。在
Galerkin法基础上，我们利用小波多分辨分析的特性，给出了小波有限元逐层校正
算法成立的条件，并提出了条件不满足时的全局修正算法。对两点边值问题的计算验
证了我们的结论。
3.讨论了利用周期小波进行积分方程矩量法求解的解空间的两种生成方法，在
此基础上给出了积分方程小波矩量法解空间的收敛性及误差估计。我们提出了多函数
小波概念，并按照单函数的多分辨分析方法，讨论了多函数小波的一些性质。通过酉
阵的方式建立了Multiwavelet，并将其成功地应用于二维声波散射问题和非线性
Hammerstein积分方程。
4.利用样条函数理论证明了区间样条小波插值是在三次样条函数空间中的最佳
逼近函数，并在此基础上给出了用尺度函数和用小波函数为基插值的一致性和插值函
数的误差估计式。我们提出了一种用两个一阶导算子矩阵的乘积替换二阶导算子矩阵
的替代算法。用两种不同的插值算法对含不同参数的Burgers方程进行了验算，结果
表明对小参数的Burgers方程，两种算法结果相当；而对稍大的粘性系数，替代算法
的震荡明显小于原算法，因而替代算法有较大的稳定性范围。
As is well known, many problems in science can be eventually presented in the form of partial differential equation (PDE). While the methods now we using to analysis and compute PDE have their own advantages, they also have some shortcomings, such as the ill condition matrix by FEM and a lot of integrals needed by the BIE. But when wavelets are used, we will not meet the disadvantages mentioned above. Usually applying wavelet to the PDE, we need to solving such problems as follows (1) choose a suitable wavelet basis according to the problem we want to deal with (2) design or modify a good algorithm in order to make use of the characters of the wavelet we choose, etc.
In this dissertation, based on the research of wavelet theory, the following problems are discussed:
1. A new conception. wavelet operator group, is presented, which is the generalization of the wavelet transformation. Having discussed the properties, we prove that the continuous wavelet transformation belongs to the wavelet operator group. We come to the conclusion that the physical wavelet is the conjugate of the wavelet operator group.
2. The substitution is made of the interval wavelet basis for the compact one in order to produce the basis of Sobolev space with the lower condition number, which will improve the speed and the accuracy of the linear system made by wavelet Galerkin method. Also the algorithms based on the local or global correction are presented and their properties are discussed in detail. The numerical results on two point boundary problems test and verify the algorithms.
3. Two ways are discussed to produce solution spaces when the BIE is used with the basis of period wavelet. Based on these, the convergence and the error estimation are given. The MRA with more then one-wavelet function is presented and the multi-wavelet basis is produced by unitary matrix. This basis is used successfully in scattering problem of sonic wave and nonlinear Hanimersterin integral equation.
4. The optimal approximation property is shown when the cubic spline interval wavelet is used in the interpolation with some special points. This property ensures the identical of the interpolation by use of scale basis and of wavelet basis. The interpolation error is also discussed. A new algorithm is presented in which two one-order differential matrixes are used to substitute for the two-order one. This way produces less vibration for the numerical solution of Burgers equation when the coefficient is somewhat larger.
引文
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