CAREER: Foundations for Understanding and Reaching the Limits of Standard Numerical Linear Algebra

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Faster numerical simulations are critical, for example, in basic science for enabling novel scientific discoveries, or in engineering for developing revolutionary new products. Most research in this vein aims to develop new algorithmitic implementations to accelerate computations. Rather than moving down that path, this research first answers the question: "by how much can we accelerate a given computation?" The second component, then, is to develop highly-efficient algorithms able to reach these minimum time limits. Our research leads to a better understanding of our algorithms and their intrinsic limitations, and eventually results in better, near-optimal algorithms. Both components of this research represent tremendous challenges given the complexity of the current computing architecture and the problem to be solved.The results of this research will be communicated in an integrated education component of this project in which previous and recent work and methodologies are disseminated through classes, the publication of a book, the distribution of source codes and the development of web documents. Important taxonomy and bibliographic work will be performed during interdisciplinary math/CS reading classes. Broader outreach activities are conducted in K-12 classrooms and through public STEM- related events in the metropolitan Denver area.The research in the project includes three components. Given a numerical problem to be solved, the investigator (1) develops an ad-hoc model detailing the theoretical limitations of the computational machine, (2) simulates the execution of an algorithm based on the ad-hoc model, and (3) conducts numerical experiments on the targeted architecture. The investigator's methodology applies iterate improvements between these three components until they match. Each improvement requires answering several open questions in terms of lower bounds and algorithms.