Wavelets and Applications (Mathematics)

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Wavelets are families of functions generated from one single function (or, in higher dimensions, from a finite set of functions) by dilations and translations. Particularly interesting are discrete families of wavelets that span all of L2(Rn), and lead to numerically stable decomposition + reconstruction algorithms. Such families can either be redundant (frames) or not (bases). For certain frames or bases of wavelets, there exist easily implementable, elegant tree- algorithms for the numerical computation of decomposition coefficients. Dr. Daubechies will investigate how these can be used to solve wave equations numerically. Wavelets seem very promising for this purpose for both theoretical and practical reasons, since they are inherently a tool for microlocalization and have been used as such in singular integral problems, while they also incorporate the idea of different scales and "zoom-in" on singularities, similar to refining grid techniques in numerical analysis. A successful use of wavelets for this purpose would lead to better orientation specificity than is available in presently used numerical techniques. The interactive activities involve teaching a graduate course and participating in seminars. This project furthers VPW program objectives which are (1) to provide opportunities for women to advance their careers in engineering and in the disciplines of science supported by NSF and (2) to encourage women to pursue careers in science and engineering by providing greater visibility for women scientists and engineers employed in industry, government, and academic institutions. By encouraging the participation of women in science, it is a valuable investment in the Nation's future scientific vitality.