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

数学科学：有限域上的小波构造及其在边界积分方程中的应用

• 批准号: 9504780
• 负责人: Yuesheng Xu
• 金额: $ 7.49万
• 依托单位: North Dakota State University Fargo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-08-01 至 1998-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504780&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Construction; Wavelets; Finite

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.

有限域上的小波构造及其在边界积分方程中的应用 Charles A. Micchelli 和 Yuesheng Xu 摘要：提出了在有限域上构造小波并用它们来求解边界积分方程，该方程是偏微分边值问题的重新表述平面或空间中的方程。 用于求解边界积分方程的传统方法是伽辽金法、配置法和使用分段多项式、B样条或三角多项式的标准基的乘积积分法。这些方法通常会产生离散方程的完整矩阵，这在计算上是昂贵的。小波基为改进这些方法的可能性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，因为从这些基获得的线性或非线性系统的系数矩阵是稀疏的且条件良好的。这导致了数值快速算法，保留了传统方法所拥有的良好特征，同时对其进行了改进。提出的项目包括使用矩阵细化方程在多维空间的有限域上构造正交小波、预小波和双正交小波，根据细化方程中的矩阵设计分解和重构算法，以及开发边界积分的数值快速算法使用构建的小波基的方程。 偏微分方程长期以来一直被用来对流体流动、电化学、材料应力分析以及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换为边界积分方程会产生可以从理论上和数值上分析的问题。由于所产生的非线性方程即使不是不可能在理论上完全求解也是很困难的，因此有必要采用数值方法来解决这些问题。小波基提供了经典方法的替代方法，并导致可以获得数值快速算法的数学问题。这种替代小波方法保留了传统方法的最佳特征，并且在许多情况下给出了实质性改进。