Numerical Analysis and Scientific Computing
Faculty: Professors Layton, Neilan, Trenchea, and Yotov
Adjunct Faculty: Professors Anitescu, Galdi, and Sussman
The diversity of this group is reflected in its research interests, which range over such areas as numerical analysis of partial differential equations, adaptive methods for scientific computing, computational methods of fluid dynamics and turbulence, numerical solution of nonlinear problems arising from porous media flow and transport, optimal control, and simulation of stochastic reaction diffusion systems. There are weekly seminars, as well as lectures and workshops at the Pittsburgh Supercomputing Center, on current trends in scientific supercomputing. More can be found at the group's Web site: Computational and Applied Mathematics Group.
Dr. Layton's research involves modeling the large eddies (such as storm fronts, hurricanes and tornadoes in the atmosphere) in turbulent flow, predicting their motion in computational experiments and validating mathematically the large eddy models and algorithms developed. Current approaches to LES seem to be presently confronting some barriers to resolution, accuracy and predictability. It seems likely that many of these barriers can be traced to the mathematical foundation of the models used, the boundary conditions imposed and the algorithms employed for the simulations. The goal of Dr. Layton's research is to develop these mathematical foundations as a guide for practical computation. This research promises to make it possible to extend the range of accuracy and reliability of predictions important to applications, such as those described above, where technological progress requires confronting turbulence! In all these aspects of analysis, modeling, algorithm development, numerical analysis and experimentation, common themes include: Mathematical analysis as a guide to practical computation and Numerical analysis as a guide to understanding phenomena rather than "solving equations".
Dr. Neilan's research interests include finite element methods and their convergence analysis for fully nonlinear partial differential equations. His current focus is on the construction and analysis of reliable and efficient numerical methods for the Monge-Ampere equation though the use of simple and practical finite elements. Other research interests of his include the numerical approximation of fluid flow (Stokes/Navier Stokes/Brinkman), the design and implementation of fourth and sixth order elliptic PDEs that arise in, e.g., plate bending and phase-field problems, the theory and construction of nonconforming finite element methods, and singular perturbation problems.
Dr. Trenchea’s expertise lies in the numerical analysis of semidiscrete and fully discrete space-time discretizations of control problems, convergence and error estimates, and the development of numerical algorithms for finding the optimal solutions. He is an expert on control theory in the abstract infinite dimensional space framework, the use of analysis and control theory for proving existence of optimal solutions (i.e., solutions that minimize the cost functional and satisfy the state equation), and getting necessary conditions of optimality for the continuous control problem.
Dr. Yotov’s research interests are in the numerical analysis and solution of partial differential equations and large scale scientific computing with applications to fluid flow and transport. His current research focus is on the design and analysis of accurate multiscale adaptive discretization techniques (mixed finite elements, finite volumes, finite differences) and efficient linear and nonlinear iterative solvers (domain decomposition, multigrid, Newton-Krylov methods) for massively parallel simulations of coupled multiphase porous media and surface flows. Other areas of research interest include estimation of uncertainty in stochastic systems and mathematical and computational modeling for biomedical applications. Dr. Yotov is also an adjunct faculty at the McGowan Institute for Regenerative Medicine.