Mathematics

Modern engineering designs require fast and high credible scientific computations which usually run in a parallel way. The proposed research focuses on the development of the parallel preconditioning technology used in large scale scientific computations. A multi-level recursive strategy is developed to improve the parallel computing performance when a large number of processors (up to at least 5000) are used. An existing Newton-Krylov flow solver will be improved by coupling with this multi-level preconditioner.

The theoretical and practical aspects of manipulating mathematical expressions on computers are usually referred to as computer algebra or symbolic computation. In this field, calculations are designed to yield exact and complete results, by opposition to numerical analysis which is meant to handle approximate values, potentially producing incomplete results. Exactness and completeness have some significant computational overhead. Computer algebra software is highly demanding in CPU time and memory.

Protecting copyright is one of the hottest topics in information and media technology at the moment. Digital technology enables perfect copying on amateur equipment. Digital Fingerprinting is an emerging technology to protect multimedia from unauthorized redistribution. It embeds a unique ID into each user's copy, which can be extracted to help identify culprits when an unauthorized leak is found. Thereby any emerging illegitimate copy can be traced back to the guilty party. A major challenge is to make this system secure against coalitions of pirates.

This research develops a comprehensive methodology for the integrated optimization and control of human-friendly robotic technology that will be applied for the advanced healthcare and biomedical manipulation. Some original ideas, methods and algorithms are proposed in this research based on several novel mathematical models, which will benefit the development of general robotics in the direction of safety with high performance to human beings.

Statistical machine learning is an endeavor in which statisticians and computer scientists use computation to identify useful information from large amounts of data. Telecommunications, insurance and pharmaceutical companies use the team’s machine learning and data mining techniques to determine customer patterns, predict future customer behavior and better understand their needs. The project addresses some of the main practical and theoretical difficulties encountered when dealing with large datasets.

Computer algebra systems such as Maple compute using mathematical formulae as well as numbers, mechanizing the mathematics used in education and research labs. This project focuses on the design and implementation of algorithms for these systems. Emphasis is placed on efficiency that allows large and complex problems of the type encountered in industrial settings to be solved. In the past year the team has made major advances in the core tools that are needed to solve these complex problems.

Efficiency in modern industrial operations requires that available resources are deployed in an optimal manner. The study of facility location is concerned with the placement of one of more facilities in a way that meets a particular objective, such as minimizing transportation costs, providing a high level of service to customer or capturing market share. This project, by exploiting the mathematics of computational geometry and algorithmic graph theory, develops new tools to aid in the location of facilities to optimally serve the demands of customers.

Engineers use mathematical models to describe the production of plastics and other chemicals. The models contain unknown parameters that are estimated from plant data. In the past year, the research team analyzed several criteria that modelers use to decide how complex or how simplified their models should be. They showed that one popular model-selection criterion, the corrected Akaike Information Criterion, tends to select very simple models, and that another, the adjusted coefficient of determination, tends to select models with many parameters.