A New Initiative in Computational Mathematics at Princeton

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。自20世纪70年代中期成立以来，计算数学一直是应用与计算数学（PACM）计划的核心。这一传统根植于计算流体动力学、控制理论和运筹学等传统的、有影响力的和强大的领域，PACM致力于继续在这些领域投入精力和资源，在这些领域中，PACM的动态和顶级研究人员在量子化学、材料科学和纳米技术方面有很好的代表。与这些传统的计算数学领域并行，近年来数学和计算机科学也有了令人兴奋的新发展，为计算数学开辟了新的应用领域。这些都带来了自己的挑战，必须并且正在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；他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，还借鉴了信息论、图论、巴拿赫空间几何、概率论等。该提案旨在资助三位被这些新的计算挑战所吸引的PACM教师的研究，他们也越来越多地发现他们不同的专业领域都有助于开发更有效的工具。这种利益的连续性，以及他们共同努力将产生一个超过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将把谐波分析、组合群论和统计数据分析方法引入到算法的构建中，以解决传统方法尚未解决的大规模计算问题。获取足够的数据曾经是科学家和工程师的主要任务，但现在他们经常发现自己被大量的非结构化数据所淹没，而这些数据比以前可以保存的要多得多。面对大量的非结构化、有噪声的数据，接下来的挑战是识别和研究数据中的隐藏结构（通常比数据集本身低得多）。这就像在干草堆里找针一样，当一个人甚至不知道有多少针（如果有的话！）藏在干草中。在一个非常真实的意义上，这类似于婴儿完成的学习任务，他们在几年的时间里，学会理解通过他们的感官提供给他们的大量信息，并成功地学习许多技能，包括语言，通过识别这些数据的结构。pi将研究如何在几种应用中发现这种隐藏结构。其中包括确定生物分子的几何结构，因为物质不能结晶，标准x射线晶体学无法工作；研究大师的画作，学习如何更好地区分真伪或复制品；识别非常大的数据数组中只有一小部分条目已知的结构，以便推断其他条目；对Internet或其他网络流量的异常检测。

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.