Integral equation-based design and simulation tools: Fast, accurate, and robust

Mechanical and Aerospace Engineering
NYU Community Event

Mechanical and Aerospace Engineering
Department Seminar Series

02/24 (Monday)    Noon – 1:00 pm     RH 500

Integral equation-based design and simulation tools: Fast, accurate, and robust

Dr. Mike O’Neil
Courant Instructor / Assistant Professor
Courant Institute, NYU

If the future of scientific computing and computational engineering is declared to be "design by simulation", then not only must a single problem be solved accurately and rapidly, but rather an entire sequence of problems with varying geometry and parameter perturbations must be addressed. Recent advances in high-performance computing environments (e.g. distributed systems, multi-core or GPU-based architectures) have allowed problems in science and engineering to be solved which were previously thought intractable. Simultaneous with the growth of computing power in the past 25 years, several new numerical methods have been developed which have enabled an even more rapid increase in the size of calculations that can be performed. In particular, fast multipole methods and related hierarchical algorithms have led to fast O(N) integral equation solvers for many PDEs of classical mathematical physics.

In this talk, I will give an overview of the analytical techniques and numerical methods that are needed in order to build a high-order accurate, fast, and robust integral equation solver for use in fields such as physics, engineering, chemistry, and medicine. Aside from the mathematical formulation of the PDE as an integral equation, problems in numerical quadrature, geometric discretization, and fast solvers must also be addressed. The topics and examples that will be discussed are applicable to many problems in acoustics, elasticity, fluid dynamics, electromagnetics, heat flow, etc.


Mike O’Neil is pursuing post-doctoral work as a Courant Instructor at the Courant Institute at NYU. His current research focuses on problems in computational physics, in particular, constructing fast high-order computational methods to solve integral equation formulations of PDEs occurring in, among other areas, acoustics, heat flow, electromagnetics, fluid mechanics, and magneto-hydrodynamics. He received his B.A. in Mathematics from Cornell University in 2003, and his Ph.D. in Applied Mathematics from Yale University in 2007. In his doctoral thesis, he developed a new class of fast, analysis-based numerical algorithms that enable the rapid evaluation of many classical special function integral transforms. These integral transforms frequently occur in physics, signal processing, and numerical analysis.