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

有限域上的小波构造及其在边界积分方程中的应用由Charles A.Micchelli和徐月生提出。摘要：提出了在有限域上构造小波，并将其用于求解边界积分方程解边界积分方程组，这是对平面或空间中偏微分方程组边值问题的重新表述。传统的求解边界积分方程组的方法有Galerkin法、配置法和积分法，这些方法都是利用分段多项式、B-样条或三角多项式的标准基。这些方法通常导致离散方程的全矩阵，这是计算昂贵的。小波基为改进这些方法的可能性提供了另一种途径。它们通常在分段多项式或其他函数类空间中形成更好的基，从这个意义上说，从这些基得到的线性或非线性系统的系数矩阵是稀疏的和条件良好的。这导致了数值上的快速算法，它们保留了传统方法所具有的良好特征，同时又对它们进行了改进。所提出的项目包括利用矩阵精化方程在多维空间的有限区域上构造正交小波、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，以及利用所构造的小波基开发边界积分方程组的数值快速算法。长期以来，偏微分方程一直被用来对流体流动、电化学、材料中的应力分析和许多类似的实际问题中发生的各种物理现象进行数学建模。将这些偏微分方程化成边界积分方程解出的问题既有理论上的，也有数值上的。由于所产生的非线性方程在理论上即使不是不可能完全求解，也是困难的，因此有必要对这些问题进行数值求解。小波基提供了一种替代经典方法的方法，并产生了可以获得数值快速算法的数学问题。这种替代的小波方法保留了传统方法的最好的特征，并且在许多情况下有了实质性的改进。

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.