Mathematical Sciences: Construction of Wavelets on Finite Domans and Applications to Boundary Integral Equations

摘要提出了在有限域上构造小波并利用小波求解边界积分方程的方法。边界积分方程是平面或空间上偏微分方程边值问题的重新表述。求解边界积分方程的传统方法有伽辽金法、配点法和以分段多项式、b样条或三角多项式为标准基的积积分法。这些方法通常导致离散方程的满矩阵，这是计算昂贵的。小波基为改进这些方法提供了一种替代方法。它们通常在分段多项式或其他函数类的空间中形成更好的基，因为从这些基中得到的线性或非线性系统的系数矩阵是稀疏且条件良好的。这导致了数字快速算法，既保留了传统方法所具有的良好特征，同时又对其进行了改进。提出的研究项目包括利用矩阵精化方程在多维空间有限域上构造正交小波、预小波和双正交小波，根据精化方程中的矩阵设计分解和重构算法，利用所构造的小波基开发边界积分方程的快速数值算法。长期以来，偏微分方程一直被用于对流体流动、电化学、材料应力分析和许多类似实际问题中发生的各种物理现象进行数学建模。将这些偏微分方程转化为边界积分方程所产生的问题可以从理论上和数值上进行分析。由于产生的非线性方程很难（如果不是不可能）从理论上完全求解，因此有必要采用数值方法来解决这些问题。小波基提供了一种替代经典方法的方法，并导致可以获得快速数值算法的数学问题。这种替代的小波方法保留了传统方法的大部分优点，并在许多情况下进行了实质性的改进。

Constructions of Wavelets on Finite Domains and Applications to Boundary Integral Equations Proposed by Charles A. Micchelli and Yuesheng Xu Abstract: It is proposed to construct wavelets on finite domains and use them to solve boundary integral equations which are reformulations of boundary value problems of partial differential equations in the plane or space. Traditional methods employed to solve the boundary integral equations are Galerkin methods, collocation methods, and product integration methods by using a standard basis of piecewise polynomials, B-splines or trigonometric polynomials. These methods usually lead to full matrices for the discrete equations, which are computationally expensive. Wavelet bases provide an alternative approach for the possibility of improving these methods. They often form better bases in the space of piecewise polynomials or other function classes in the sense that the coefficient matrices of the linear or nonlinear systems obtained from these bases are sparse and well-conditioned. This leads to numerically fast algorithms that preserve the nice features that the traditional methods possess and at the same time improve upon them. The proposed projects include constructing orthogonal wavelets, pre-wavelets and biorthogonal wavelets on finite domains in multidimensional spaces using the matrix refinement equations, designing decomposition and reconstruction algorithms in terms of the matrices in the refinement equations, and developing numerically fast algorithms for the boundary integral equations using the wavelet bases constructed. Partial differential equations have long been used to mathematically model a wide variety of physical phenomena that occur in fluid flows, electro-chemistry, stress analysis in materials and a host of similar practical problems. Transforming these partial differential equations into boundary integral equations gives rise to problems that can be analyzed both theoretically and nu merically. Since the nonlinear equations arising are difficult if not impossible to completely solve theoretically, a numerical method to these problems is necessary. Wavelet bases provide an alternative approach to classical methods and result in mathematical problems for which numerically fast algorithms can be obtained. This alternative wavelet method preserves most nice features of traditional methods and in many instances gives substantial improvements.