Seminars and Colloquia by Series

Monday, March 2, 2015 - 14:00 , Location: Skiles 005 , Professor Scott McCalla , Montana State University , Organizer: Martin Short
The existence, stability, and bifurcation structure of localized radially symmetric solutions to the Swift--Hohenberg equation is explored both numerically through continuation and analytically through the use of geometric blow-up techniques.  The bifurcation structure for these solutions is elucidated by formally treating the dimension as a continuous parameter in the equations.  This reveals a family of solutions with an anomalous amplitude scaling that is far larger than expected from a formal scaling in the far field.  One key advantage of the geometric blow-up techniques is that a priori knowledge of this scaling is unnecessary as it naturally emerges from the construction.  The stability of these patterned states will also be discussed.
Monday, February 16, 2015 - 14:00 , Location: Skiles 005 , Prof. Matthew Lin , National Chung Cheng University, Georgia Tech , mhlin@ccu.edu.tw , Organizer: Chi-Jen Wang

Reference[1] Moody T. Chu

, Nonnegative Inverse Eigenvalue and Singular Value Problems, SIAM J. Numer. Anal (1992).[2] Wei Ma and Zheng-J. Bai, A regularized directional derivative-based Newton method for inverse singular value problems, Inverse Problems (2012).

Nonnegative inverse eigenvalue and singular value problems have been a research focus for decades. It is true that an inverse problem is trivial if the desired matrix is not restricted to any structure. This talk is to present two numerical procedures, based on a  conquering procedure and an alternating projection process, to solve inverse eigenvalue and singular value problems for nonnegative matrices, respectively.  In theory, we also discuss the existence of nonnegative matrices subject to prescribed eigenvalues and singular values.  Though the focus of this talk is on inverse eigenvalue and singular value problems with nonnegative entries, the entire procedure can be straightforwardly applied to other types of structure with no difficulty.
Monday, February 9, 2015 - 14:00 , Location: Skiles 005 , Timo Eirola , Aalto University, Helsinki, Finland , Organizer: Martin Short
We consider three different approaches to solve the equations for electron density around nuclei particles. First we study a nonlinear eigenvalue problem and apply Quasi-Newton methods to this. In many cases they turn to behave better than the Pulay mixer, which widely used in physics community. Second we reformulate the problem as a minimization problem on a Stiefel manifold. One that formed from mxn matrices with orthonormal columns. Then for Quasi-Newton techniques one needs to transfer the secant conditions to the new tangent space, when moving on the manifold. We also consider nonlinear conjugate gradients in this setting. This minimization approach seems to work well especially for metals, which are known to be hard. Third (if time permits) we add temperature (the first two are for ground state). This means that we need to include entropy in the energy and optimize also with respect to occupation numbers. Joint work with Kurt Baarman and Ville Havu.
Monday, January 26, 2015 - 14:00 , Location: Skiles 005 , Raffaele D'Ambrosio , GA Tech , Organizer: Martin Short
The talk is the continuation of the previous one entitled "Structure-preserving numerical integration of ordinary and partial differential equations [8]" and is aimed to present both classical and more recent results regarding the numerical treatment of nonlinear differential equations, both for deterministic and stochastic problems. The perspective is that of introducing numerical methods which act as structure-preserving integrators, with special emphasys to numerically retaining dissipativity properties possessed by the problem.
Monday, December 1, 2014 - 14:00 , Location: Skiles 005 , Raffaele D'Ambrosio , GA Tech , Organizer: Martin Short
It is the purpose of this talk to analyze the behaviour of multi-value numerical methods acting as structure-preserving integrators for the numerical solution of ordinary and partial differential equations (PDEs), with special emphasys to Hamiltonian problems and reaction-diffusion PDEs. As regards Hamiltonian problems, we provide a rigorous long-term error analyis obtained by means of backward error analysis arguments, leading to sharp estimates for the parasitic solution components and for the error in the Hamiltonian. As regards PDEs, we consider structure-preservation properties in the numerical solution of oscillatory problems based on reaction-diffusion equations, typically modelling oscillatory biological systems, whose solutions oscillate both in space and in time. Special purpose numerical methods able to accurately retain the oscillatory behaviour are presented.
Monday, November 17, 2014 - 14:00 , Location: Skiles 005 , Dr. Mohammad Farazmand , GA Tech Physics , Organizer: Martin Short
Recent experimental and numerical observations have shown the significance of the Basset--Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. I discuss the governing equations of motion for the inertial particles, i.e. the Maxey-Riley equation, including a fractional order derivative in time. Then I show that the observed algebraic decay is a universal property of the Maxey--Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is O(\epsilon)-close to the fluid velocity, where 0<\epsilon<<1 is proportional to the square of the ratio of the particle radius to the fluid characteristic length-scale. These results follows from a sharp analytic upper bound that we derive for the particle velocity.
Friday, November 14, 2014 - 11:00 , Location: Skiles 005 , Professor Andre Martinez-Finkelshtein , Universidad de Almería , Organizer: Martin Short
The medical imaging benefits from the advances in constructiveapproximation, orthogonal polynomials, Fourier and numerical analysis,statistics and other branches of mathematics. At the same time, the needs of the medical diagnostic technology pose new mathematical challenges. This talk surveys a few problems, some of them related to approximation theory, that have appeared in my collaboration with specialists studying some pathologies of the human eye, in particular, of the cornea, such as:- reconstruction of the shape of the cornea from the data collected bykeratoscopes- implementation of simple indices of corneal irregularity- fast and reliable computation of the through-focus characteristics of a human eye.
Monday, November 3, 2014 - 14:00 , Location: Skiles 005 , Dr. Matthew Calef , Los Alamos National Lab , Organizer: Martin Short
Observations of high energy density environments, from supernovae implosions/explosions to inertial confinement fusion, are determined by many different physical effects acting concurrently. For example, one set of equations will describe material motion, while another set will describe the spatial flow of energy. The relevant spatial and temporal scales can vary substantially. Since direct measurement is difficult if not impossible, and the relevant physics happen concurrently, computer simulation becomes an important tool to understand how emergent behavior depends on the constituent laws governing the evolution of the system. Further, computer simulation can provide a means to use observation to constrain underlying physical models. This talk shall examine the challenges associated with developing computational multiphysics simulation. In particular this talk will outline some of the physics, the relevant mathematical models, the associated algorithmic challenges, some of which are driven by emerging compute architectures. The problem as a whole can be formidable and an effective solution couples many disciplines together.
Monday, October 20, 2014 - 14:00 , Location: Skiles 005 , Dr. Maria D'Orsogna , Cal State University Northridge , Organizer: Martin Short
Given their ubiquity in physics, chemistry and materialsciences, cluster nucleation and growth have been extensively studied,often assuming infinitely large numbers of buildingblocks and unbounded cluster sizes. These assumptions lead to theuse of mass-action, mean field descriptions such as the well knownBecker Doering equations.  In cellular biology, however, nucleationevents often take place in confined spaces, with a finite number ofcomponents, so that discrete and stochastic effects must be takeninto account.  In this talk we examine finite sized homogeneousnucleation by considering a fully stochastic master equation, solvedvia Monte-Carlo simulations and via analytical insight. We findstriking differences between the mean cluster sizes obtained from ourdiscrete, stochastic treatment and those predicted by mean fieldones. We also study first assembly times and compare results obtained from processes where only monomer attachment anddetachment are allowed to those obtained from general coagulation-fragmentationevents between clusters of any size. 
Monday, October 6, 2014 - 14:00 , Location: Skiles 005 , Dr. Maryam Yashtini , Georgia Tech Mathematics , Organizer: Martin Short
An alternating direction approximate Newton method (ADAN) is developedfor solving inverse problems of the form$\min \{\phi(Bu) +1/2\norm{Au-f}_2^2\}$,where $\phi$ is a convex function, possibly nonsmooth,and $A$ and $B$ are matrices.Problems of this form arise in image reconstruction where$A$ is the matrix describing the imaging device, $f$ is themeasured data, $\phi$ is a regularization term, and $B$ is aderivative operator. The proposed algorithm is designed tohandle applications where $A$ is a large, dense ill conditionmatrix. The algorithm is based on the alternating directionmethod of multipliers (ADMM) and an approximation to Newton's method in which Newton's Hessian is replaced by a Barzilai-Borwein approximation. It is shown that ADAN converges to a solutionof the inverse problem; neither a line search nor an estimateof problem parameters, such as a Lipschitz constant, are required.Numerical results are provided using test problems fromparallel magnetic resonance imaging (PMRI).ADAN performed better than the other schemes that were tested.

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