Wavelets and Applications (Mathematics)

波函数是由一个函数生成的函数族 (or，在更高的维度中，从有限的函数集合）， 扩张和翻译。 特别有趣的是离散的 小波族，跨越所有的L2（Rn），并导致数值 稳定分解+重构算法。 这样的家庭可以 或者是冗余的（帧）或者不是冗余的（基）。 对于某些帧或 小波基，存在容易实现的，优雅的树， 分解的数值计算算法 系数 多贝西博士将研究如何使用这些 来数值求解波动方程 Wavelet似乎很有希望， 这是出于理论和实践的原因，因为它们 本质上是一种微定位工具， 在奇异积分问题中，虽然它们也包含了 不同的尺度和“放大”奇点，类似于细化 数值分析中的网格技术。 小波的成功应用 为此目的，将导致更好的定向特异性， 在目前使用的数值技术中可用。 互动活动包括教授研究生课程， 参加研讨会。 该项目进一步促进了VPW计划的目标，即（1）提供 为妇女提供在工程和技术领域发展职业生涯的机会 由NSF支持的科学学科和（2）鼓励 妇女追求科学和工程事业， * 提高妇女科学家和工程师的知名度， 行业、政府和学术机构。 通过鼓励 妇女参与科学，这是一项宝贵的投资， 国家未来的科学活力。

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.