Tuesday, September 9, 2014 at 3:00 PM in SAS 4201
Tim Kelley, NC State
Anderson acceleration is an algorithm for accelerating the convergence
of fixed point or Picard iteration. The method was invented in 1965
to accelerate the SCF iteration in electronic structure computations
and is now widely used in that field.
Anderson acceleration does not require the computation
or approximation of Jacobians or Jacobian-vector products, and this
can be an advantage over Newton-like methods.
There is little theory for the method. One can show that the method
is related to GMRES for linear problems and to multi-secant methods
for nonlinear problems. However, results of this type do not lead
to convergence theory for the way that the method is used in practice.
In this talk we will discuss the first convergence results for the
method, illustrate the results with an application to radiative transport, discuss issues with multi-physics coupling, and list a few open questions.
This is joint work with Alex Toth. This talk is the followup to my talk for the students in the First Year Seminar, where I promised to show them more math. Here is that more math.
Monday, September 15, 2014 at 3:00 PM in SAS 2102
Ivan Sudakov, U of Utah
The Modified RayleighBenard Convection Problem and its Application to Greenhouse Gas Emissions Simulation
The original RayleighBenard convection is a standard example of the system where bifurcations occur with changing of a control parameter. I will discuss the modified RayleighBenard convection problem which includes the radiative effects as well as the specific gas sources on a surface. Such formulation of this problem leads to identification a new kind of bifurcation phenomenon, besides the well-known Benard cells.
Modeling of greenhouse gas emissions into the atmosphere drives to difficult problems, involving the Navier-Stokes equations. There exist climate models with different levels of realism. Usually, one investigates these models by computer simulations. However, it is difficult to the estimate reliability of these computations, since it is connected with a complex mathematical problem on the structural stability of attractors. Taking into account the modified RayleighBenard convection problem, I will discuss a new approach which makes the problem of a climate catastrophe in the result of a greenhouse effect more mathematically tractable and allows us to describe catastrophic bifurcations in the atmosphere induced by soil greenhouse gas sources.
Tuesday, October 7, 2014 at 3:00 PM in SAS 4201
Hongkai Zhao, University of California, Irvine
Approximate Separability of Greens Function for Helmholtz Equation in the High Frequency Limit
Approximate separable representations of Greens functions for differential operators is a basic and important question in the analysis of differential equations, the development of efficient numerical algorithms and imaging. Being able to approximate a Greens function as a sum with few separable terms is equivalent to low rank properties of corresponding numerical solution operators. This will allow for matrix compression and fast solution techniques. Greens functions for coercive elliptic differential operators have been shown to be highly separable and the resulting low rank property for discretized system was explored to develop efficient numerical algorithms. However, the case of Helmholtz equation in the high frequency limit is more challenging both mathematically and numerically. We introduce new tools based on the study of relations between two Greens functions with different source points and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors to prove new lower bounds for the number of terms in the representation for the Greens function for Helmholtz operator in the high frequency limit. Upper bounds are also derived. We give explicit sharp estimates for cases that are common in practice and present numerical examples. This is a joint work with Bjorn Engquist.
Tuesday, November 4, 2014 at 3:00 PM in SAS 4201
Stuart Slattery, Oak Ridge National Laboratory
You can add or remove yourself from a seminar mailing list by visiting this link.