Seminars and Colloquia by Series

A Non-convex Approach for Signal and Image Processing

Series
Applied and Computational Mathematics Seminar
Time
Tuesday, March 24, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Yifei Lou – UT Dallas
A fundamental problem in compressed sensing (CS) is to reconstruct a sparsesignal under a few linear measurements far less than the physical dimensionof the signal. Currently, CS favors incoherent systems, in which any twomeasurements are as little correlated as possible. In reality, however, manyproblems are coherent, in which case conventional methods, such as L1minimization, do not work well. In this talk, I will present a novelnon-convex approach, which is to minimize the difference of L1 and L2 norms(L1-L2) in order to promote sparsity. Efficient minimization algorithms areconstructed and analyzed based on the difference of convex functionmethodology. The resulting DC algorithms (DCA) can be viewed as convergentand stable iterations on top of L1 minimization, hence improving L1 consistently. Through experiments, we discover that both L1 and L1-L2 obtain betterrecovery results from more coherent matrices, which appears unknown intheoretical analysis of exact sparse recovery. In addition, numericalstudies motivate us to consider a weighted difference model L1-aL2 (a>1) todeal with ill-conditioned matrices when L1-L2 fails to obtain a goodsolution. An extension of this model to image processing will be alsodiscussed, which turns out to be a weighted difference of anisotropic andisotropic total variation (TV), based on the well-known TV model and naturalimage statistics. Numerical experiments on image denoising, imagedeblurring, and magnetic resonance imaging (MRI) reconstruction demonstratethat our method improves on the classical TV model consistently, and is onpar with representative start-of-the-art methods.

Numerical schemes for stochastic backscatter in the inverse cascade of quasi-geostrophic turbulence

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yoonsang Lee – Courant Institute of Mathematical Sciences –
Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of two-dimensional and quasi-geostrophic turbulence, where the net transfer of energy is from small to large scales. A numerical scheme for stochastic backscatter in the two-dimensional and quasi-geostrophic inverse kinetic energy cascades is developed and analyzed. Its essential properties include a local formulation amenable to implementation in finite difference codes and non-periodic domains, smooth behavior at the coarse grid scale, and realistic temporal correlations, which allows detailed numerical analysis, focusing on the spatial and temporal correlation structure of the modeled backscatter. The method is demonstrated in an idealized setting of quasi-geostrophic turbulence using a low-order finite difference code, where it produces a good approximation to the results of a spectral code with more than 5 times higher nominal resolution. This is joint work with I. Grooms and A. J. Majda

Implicit interface boundary integral methods

Series
Applied and Computational Mathematics Seminar
Time
Friday, March 13, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Richard Tsai – University of Texas at Austin
I will present a new approach for computing boundary integrals that are defined on implicit interfaces, without the need of explicit parameterization. A key component of this approach is a volume integral which is identical to the integral over the interface. I will show results applying this approach to simulate interfaces that evolve according to Mullins-Sekerka dynamics used in certain phase transition problems. I will also discuss our latest results in generalization of this approach to summation of unstructured point clouds and regularization of hyper-singular integrals.

Transition path processes and coarse-graining of stochastic system

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Jianfeng Lu – Duke University –
Understanding rare events like transitions of chemical system from reactant to product states is a challenging problem due to the time scale separation. In this talk, we will discuss some recent progress in mathematical theory of transition paths. In particular, we identify and characterize the stochastic process corresponds to transition paths. The study of transition path process helps to understand the transition mechanism and provides a framework to design and analyze numerical approaches for rare event sampling and simulation.

Existence and Stability of Radially Symmetric Solutions to the Swift--Hohenberg Equation

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 2, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Scott McCalla – Montana State University
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.

Nonnegative Inverse Eigenvalue and Singular Value Problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 16, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Matthew Lin – National Chung Cheng University, Georgia Tech –

Please Note: 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.

Methods to solve Kohn-Sham equations for electron density

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Timo Eirola – Aalto University, Helsinki, Finland
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.

Nonlinear stability issues for the numerical solution of evolutionary problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 26, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Raffaele D'Ambrosio – GA Tech
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.

Structure-preserving numerical integration or ordinary and partial differential equations

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 1, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Raffaele D'Ambrosio – GA Tech
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.

Dynamics of inertial particles with memory: an application of fractional calculus

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 17, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Mohammad Farazmand – GA Tech Physics
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.

Pages