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

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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.