Fri, Sep 18

Topology and Geometry Seminar

Vasudha Bharathram (Princeton University)

4:00 PM, 122 Mathematics Building

Tue, Sep 22

What is...? Seminar

Bill Menasco

TBA

4:00 PM, 250 Mathematics Building

TBA

Wed, Sep 23

Special Event

Myhill Lectures, Thanasis Fokas Cambridge University.

The Beauty and Usefulness of the Complex Plane

4:00 PM, Mathematics Building

An attempt will be made to demystify complex analysis. This naturally leads to the introduction of the Riemann-Hilbert and d-bar formalisms. For many years, the employment of the Wiener-Hopf technique to acoustics and other physical problems, was the only manifestation in applications of the Riemann-Hilbert formalism. However, in the last 50 years this formalism has appeared in many problems in mathematics and mathematical physics. In addition, in the early 1980s the d-bar formalism also began to emerge in applications. Several of these applications will be mentioned, including the solution of integrable nonlinear evolution PDEs in one, two, and three dimensions, as well as medical imaging

Thu, Sep 24

Special Event

Myhill Lectures, Thanasis Fokas Cambridge University.

The Unified Transform Method for Linear and Integrable Nonlinear PDEs

4:00 PM, Mathematics Building

The Unified Transform, also known as the  Fokas method,  provides a new and powerful method for analysing boundary value problems for linear and integrable nonlinear PDEs. For linear PDEs, it provides an unexpected extension of classical works of D’Alembert, Fourier, Laplace, Green and Kelvin. For nonlinear PDEs, perhaps the most striking application of the new method is the solution of  the x-periodic problem  for integrable PDEs, which was first considered in the early 1970s, and which attracted the involvement of many distinguished mathematicians. This lecture will review the above novel method with emphasis on linear evolution PDEs.

Fri, Sep 25

Applied Mathematics Seminar

Leo Rebholz (Clemson U.)

NGMRES convergence analysis and proof of acceleration for contractive and noncontractive iterations

4:00 PM, MATH250

We give the first convergence analysis and proof of acceleration for nonlinear GMRES (NGMRES) applied to contractive and noncontractive fixed point iterations (FPIs) for solving general nonlinear systems. Our main results are that in both the contractive and noncontractive cases, the ratio gain of the optimization problem is the mechanism responsible for accelerating (or enabling) convergence. Our analysis also reveals a second important quantity related to the optimization problem, which directly predicts the linear convergence rate at each iteration and proves it is at most 1; hence only higher order terms are responsible for NGMRES non-convergence. Numerical results for several challenging nonlinear test problems are given that illustrate the theory, show how the acceleration improves convergence, show that the quantity predicting the linear convergence rate is remarkably accurate and moreover can be useful for adaptively choosing NGMRES depth, show how restarts can improve convergence in noncontractive iterations, show how NGMRES is naturally suited for finding distinct solutions of a multi-solution PDE, and that NGMRES can perform better than Anderson acceleration when applied to superlinear FPIs.