A New Initiative in Computational Mathematics at Princeton

Princeton University, Princeton NJ

Investigators

Abstract

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.