A New Initiative in Computational Mathematics at Princeton

该奖项由2009年美国复苏和再投资法案(公共法律111-5)资助。自20世纪70年代中期应用和计算数学(PACM)项目成立以来，计算数学一直是该项目的核心。这一传统植根于计算流体力学、控制理论和运筹学等传统、创新和强大的领域，这些领域是PACM致力于继续投入能源和资源的领域，在这些领域，动态和顶级研究人员扩展到量子化学、材料科学和纳米技术，是这一传统的良好代表。与这些传统的计算数学领域平行的是，近年来数学和计算机科学取得了令人振奋的新发展，为计算数学开辟了新的应用领域。这些都伴随着它们自己的挑战，必须并正在为此开发新的方法和工具。机器学习和压缩感知是两个典型的例子；它们不仅借鉴了传统的基于线性代数的数值分析或近似理论，而且还借鉴了信息论、图论、Banach空间的几何、概率论等。这项提议旨在资助三名被这些新的计算挑战所吸引的PACM教员的研究，他们还越来越多地发现，他们的不同专业领域都有助于开发更有效的工具。这种利益的连贯性，以及相信联合他们的努力将产生一个超过其各部分之和的整体的信念，构成了推动这里提出的方法的引擎。PI将利用调和分析、组合群论和统计数据分析方法来构建算法，以解决传统方法尚未解决的大规模计算问题。过去获取足够数量的数据曾是科学家和工程师的主要任务，但现在他们经常发现自己淹没在大量往往是非结构化的数据中，这些数据可以比以前存储的更多。面对大量无结构、有噪音的数据，挑战是识别和研究数据中隐藏的结构(通常比数据集本身低得多的维度)。这就像大海捞针，你甚至不知道有多少针(如果有的话)。针藏在干草中。在非常真实的意义上，这类似于婴儿完成的学习任务，在几年的时间里，他们学会了理解感官提供给他们的大量信息，并通过识别这些数据中的结构成功地学习了包括语言在内的许多技能。PI将研究如何在几个应用程序中发现这种隐藏结构。这些包括确定生物分子的几何结构，因为这种物质不能结晶，所以标准的X射线晶体学不起作用；研究大师的画作，以学习如何更好地区分原作、赝品或复制品；在只知道一小部分条目的超大型数据阵列中识别结构，以便可以推断出其他条目；检测互联网或其他网络流量中的异常。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Computational Mathematics has been central to the Program in Applied and Computational Mathematics(PACM) since its inception in the mid 1970s. This tradition is rooted in the traditional, inuential and powerful fields of computational fluid dynamics, control theory and operations research, fields in which PACM is committed to continue to invest energy and resources, and in which it is well represented by dynamic and top-level researchers branching out into quantum chemistry, materials science and nanotechnology. Parallel to these traditional computational mathematics elds, recent years have seen exciting new developments in mathematics and computer science, which have opened up new domains of application for computational mathematics. These come with their own challenges, for which new approaches and tools must be and are being developed. Machine learning and compressive sensing are two typical examples; they draw not only from traditional linear-algebra-based numerical analysis or approximation theory, but also from information theory, graph theory, the geometry of Banach spaces, probability theory, and more. This proposal seeks to fund the research of three PACM faculty drawn to these new computational challenges, who are also finding increasingly that their different fields of expertise all contribute to the development of dramatically more effective tools. This contiuence of interests, and the conviction that joining their efforts will produce a whole that exceeds the sum of its parts, constitute the engine that drives the approaches proposed here. The PIs will bring to bear harmonic analysis, combinatorial group theory and statistical data analysis approaches on the construction of algorithms that address large-scale computational problems not yet solved by traditional approaches.Whereas acquiring a sufficient amount of data used to be the main preoccupation of scientists and engineers, they now often find themselves deluged with massive amounts of often unstructured data, of which much more can be saved than was possible before. Faced with an enormous mass of unstructured, noisy data, the challenge is then to identify and study the hidden structures within the data (often much lower dimensional than the data set itself). This is like searching for needles in haystacks, when one doesn't even know how many (if any!) needles are hidden among the hay. In a very real sense, this is similar to the learning task accomplished by babies, who, in a few years' time, learn to make sense of the overwhelming amount of information provided to them by their senses, and succeed in learning many skills, including language, from identifying the structure in these data. The PIs will study how to discover such hidden structure in several applications. These include the determination of the geometric structure of biological molecules for which standard X-ray Crystallography doesn't work because the substance cannot be crystallized; the study of masters' paintings to learn how to better distinguish original from fakes or copies; the identification of the structure in very large data arrays for which only a small percentage of the entries are known, so that the others can be inferred; the detection of anomalies in internet or other network traffic.