## Professor Michele Benzi

Department of Mathematics and Computer Science

Emory University

400 Dowman Drive

Atlanta,GA 30322

Phone: 404-727-3638

E-mail: benzi@mathcs.emory.edu

Michele Benzi received a Laurea degree from the University of Bologna (Italy) and a PhD in Mathematics from North Carolina State University. After holding positions at the University of Bologna, at CERFACS (in Toulouse, France) and at Los Alamos National Laboratory, in 2000 he moved to Emory University where he is a professor in the Department of Mathematics and Computer Science. His research interests include numerical linear algebra (especially preconditioners and iterative methods for large sparse matrices), computational methods for Markov chains, and the history of numerical analysis and scientific computing.

**The Perron-Frobenius Theorem and Some of its Applications**

The Perron-Frobenius Theorem is one of the most beautiful and useful results in matrix theory. According to this theorem, the spectral radius of an irreducible matrix A with nonnegative elements is a simple eigenvalue of A; furthermore, the corresponding eigenvector can be chosen to have (strictly) positive entries. This theorem and its generalizations have found numerous applications in fields as diverse as probability theory (Markov chains), nuclear reactor studies, economics (Leontief models), biomathematics (population dynamics), numerical analysis, and so forth. More recently, Perron-Frobenius theory has been shown to provide the basis for the formulation and solution of ranking problems, including the ranking of NFL teams and Google's celebrated PageRank algorithm.

The primary goal of this expository talk is to highlight the main results of nonnegative matrix theory, using Google's PageRank algorithm to illustrate basic concepts. Another goal of the talk is to raise the students' awareness of the crucial role mathematics play in everyday activities like web surfing. The only prerequisite for this talk is a first course in Linear Algebra.

Level: Elementary

**Key Moments in the History of Numerical Analysis**

The talk will highlight some of the key moments in the evolution of numerical analysis into an independent mathematical discipline. The necessary context and background behind technical developments will be carefully exposed, as well as biographical information about the major figures in the field. Topics include numerical solution of PDEs, numerical linear algebra, solution of nonlinear equations, and mathematical programming.

Level: Intermediate

**Fast Iterative Solvers for the Incompressible Navier-Stokes Equations**

I will present recent work on iterative solvers for various discretizations of the incompressible Navier-Stokes equations, for both steady and unsteady flow cases. Picard-type linearizations for the standard form of the convection term will be compared with alternative linearizations using the rotation form, and efficient solvers for both forms will be presented.

The main focus of the talk is on preconditioning methods. I will focus on block triangular preconditioners based on an augmented Lagrangian formulation of the discrete saddle point problem and on preconditioners derived from the Hermitian/Skew-Hermitian splitting.

I will examine the performance of various solvers as the mesh size, Reynolds number, time step, and other problem parameters vary. Numerical tests show that fast convergence is achieved in many cases, independent of problem parameters.

This is joint work with Maxim Olshanskii (Moscow State University) and with Jia Liu (University of Western Florida).

Level: Advanced

**Localization Phenomena in Matrix Functions: Theory and Applications**

Many physical phenomena are characterized by strong localization, that is, rapid decay outside of a small spatial or temporal region. Frequently, such localization can be expressed as decay in the entries of a function f(A) of an appropriate sparse or banded matrix A that encodes a description of the system under study. Important examples include the decay in the entries of the density matrix for non-metallic systems in quantum chemistry (a function of the Hamiltonian), the localization of the eigenvectors in the Anderson model, and the decay behavior of the inverse of a banded symmetric positive definite matrix. Localization phenomena are of fundamental importance both at the level of the physical theory and at the computational level, because they open up the possibility of approximating relevant matrix functions in O(N) time, where N is a measure of the size of the system. In this talk I will give an overview of theoretical results, algorithms, and applications in various parts of computational physics and numerical analysis.

This is joint work with Nader Razouk.

Level: Advanced