### Methods for multiscale inverse problems

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, September 28, 2015 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Dr. Christina Frederick – GA Tech

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- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, September 28, 2015 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Dr. Christina Frederick – GA Tech

I will discuss inverse problems involving elliptic partial differential
equations with highly oscillating coefficients. The multiscale nature of
such problems poses a challenge in both the mathematical formulation
and the numerical modeling, which is hard even for forward computations.
I will discuss uniqueness of the inverse in certain problem classes and
give numerical methods for inversion that can be applied to problems in
medical imaging and exploration seismology.

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, September 14, 2015 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Associate Professor Hongchao Zhang – Department of Mathematics and Center for Computational &amp; Technology (CCT) at Louisiana State University – hozhang@math.lsu.edu

In this talk, we discuss a very efficient algorithm for projecting a point onto a polyhedron. This algorithm solves the projeciton problem through its dual and fully exploits the sparsity. The SpaRSA (Sparse Reconstruction by Separable Approximation) is used to approximately identify active constraints in the polyhedron, and the Dual
Active Set Algorithm (DASA) is used to compute a high precision solution. Some interesting convergence properties and very promising numerical results compared with the state-of-the-art software IPOPT and CPLEX will be discussed in this talk.

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, April 20, 2015 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Dr. Antonio Cicone – L&#039;Aquila, Italy

Given a finite set of matrices F, the Markovian Joint Spectral
Radius represents the maximal rate of growth of products of matrices in
F when the matrices are multiplied each other following some Markovian law.
This quantity is important, for instance, in the study of the so called
zero stability of variable stepsize BDF methods for the numerical
integration of ordinary differential equations.
Recently Kozyakin, based on a work by Dai, showed that, given a set F of
N matrices of dimension d and a graph G, which represents the admissible
products, it is possibile to compute the Markovian Joint Spectral Radius
of the couple (F,G) as the classical Joint Spectral Radius of a new set
of N matrices of dimension N*d, which are produced as a particular
lifting of the matrices in F. Clearly by this approach the exact
evaluation or the simple approximation of the Markovian Joint Spectral
Radius becomes a challenge even for reasonably small values of N and d.
In this talk we briefly review the theory of the Joint Spectral Radius,
and we introduce the Markovian Joint Spectral Radius. Furthermore we
address the question whether it is possible to reduce the exact
calculation computational complexity of the Markovian Joint Spectral
Radius. We show that the problem can be recast as the computation of N
polytope norms in dimension d. We conclude the presentation with some
numerical examples.
This talk is based on a joint work with Nicola Guglielmi from the
University of L'Aquila, Italy, and Vladimir Yu. Protasov from the Moscow
State University, Russia.

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, April 20, 2015 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Professor Michael Malisoff – Louisiana State University

**Please Note:** Speaker’s Biography:Michael Malisoff received his PhD in 2000 from
the Department of Mathematics at Rutgers University in New Brunswick,
NJ. In 2001, he joined the faculty of the Department of Mathematics at
Louisiana State University in Baton Rouge (LSU), where he is now the Roy
Paul Daniels Professor #3 in theLSU College of Science. His main
research has been on controller design and analysis for nonlinear
control systems with time delays and uncertainty and their applications
in engineering. One of his projects is joint with the Georgia Tech
Savannah Robotics team, and helped develop marine robotic methods to
help understand the environmental impacts of oil spills. His more than
100 publications include a Springer monograph on constructive Lyapunov
methods. His awards include the First Place Student Best Paper Award at
the 1999 IEEE Conference on Decision and Control, two three-year
NationalScience Foundation Mathematical Sciences Priority Area
grants, and 9 Best Presentation awards in American Control Conference
sessions. He is an associate editor for IEEE Transactions on Automatic
Control and for SIAM Journal on Control and Optimization.

We present a new tracking controller for neuromuscular electrical stimulation, which is an emerging technology that can artificially stimulateskeletal muscles to help restore functionality to human limbs. We use a musculoskeletal model for a human using a leg extension machine. The novelty of our work is that we prove that the tracking error globally asymptotically and locally exponentially converges to zero for any positive input delay andfor a general class of possible reference trajectories that must be tracked, coupled with our ability to satisfy a state constraint. The state constraint is that for a seated subject, the human knee cannot be bent more than plus or minus 90 degrees from the straight down position. Also, our controller only requires sampled measurements of the states instead of continuousmeasurements and allows perturbed sampling schedules, which can be important for practical applications where continuous measurement of the states is not possible. Our work is based on a new method for constructing predictor maps for a large class of nonlinear time-varying systems, which is of independent interest. Prediction is a key method for delay compensation that uses dynamic control to compensate for arbitrarily long input delays. Reference: Karafyllis, I., M. Malisoff, M. de Queiroz, M. Krstic, and R.
Yang, "Predictor-based tracking for neuromuscular electrical
stimulation," International Journal of Robust and Nonlinear Control, to
appear. doi: 10.1002/rnc.3211

- Series
- Applied and Computational Mathematics Seminar
- Time
- Friday, April 17, 2015 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Stephen Sprigle – Schools of Industrial Design and Applied Physiology, Georgia Tech

The Rehabilitation Engineering and Applied Research Lab (REARLab) performs
both experimental research and product development activities focused on
persons with disabilities. The REARLab seeks collaboration from the School
of Mathematics on 2 current projects. This session will introduce
wheelchair seating with respect to pressure ulcer formation and present two
projects whose data analysis would benefit from applied mathematics.
3D Tissue Deformation- Sitting induces deformation of the
buttocks tissues. Tissue deformation has been identified as the underlying
cause of tissue damage resulting from external loading. The REARLab has
been collecting multi-planar images of the seated buttocks using MRI. This
data clearly shows marked differences between persons, as expected. We are
interested in characterizing tissue deformation as a combination of
displacement and distortion. Some tissues- such as muscle- displace
(translate within the sagittal, coronal and transverse planes) and distort
(change shape). Other tissue such as skin and subcutaneous fat, simple
distorts. We seek a mathematical means to characterize tissue deformation
that reflects its multi-planar nature.
Categorizing Weight-shifting behaviors - many wheelchair users have
limitations to their motor and/or sensory systems resulting in a risk of
pressure ulcers. Pressure ulcers occur when localized loading on the skin
causes ischemia and necrosis. In an attempt to reduce risk of pressure
ulcer occurrence, wheelchair users are taught to perform weight-shifts.
Weight shifts are movements that re-distribute loads off the buttocks for
short periods of time. The REARLab is measuring weight shifting behaviors
of wheelchair users during their everyday lives. We seek a means to
classify patterns of behavior and relate certain patterns to healthy
outcomes versus other patterns that result in unhealthy outcomes.

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, April 13, 2015 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Seong Jun Kim – Georgia Tech – skim396@math.gatech.edu

We introduce a new parallel in time (parareal) algorithm which couples multiscale integrators with fully resolved fine scale integration and computes highly oscillatory solutions for a class of ordinary differential equations in parallel.
The algorithm computes a low-cost approximation of all slow variables in the system. Then, fast phase-like variables are obtained using the parareal iterative methodology and an alignment algorithm. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances.

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, April 6, 2015 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Prof. Molei Tao – Georgia Tech School of Math. – mtao@gatech.edu

We show how to control an oscillator by periodically perturbing its stiffness, such that its amplitude follows an arbitrary positive smooth function. This also motivates the design of circuits that harvest energies contained in infinitesimal oscillations of ambient electromagnetic fields. To overcome a key obstacle, which is to compensate the dissipative effects due to finite resistances, we propose a theory that quantifies how small/fast periodic perturbations affect multidimensional systems. This results in the discovery of a mechanism that reduces the resistance threshold needed for energy extraction, based on coupling a large number of RLC circuits.

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, March 30, 2015 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Professor Andrei Martinez-Finkelshtein – University of Almería

The importance of the 2D Fourier transform in mathematical
imaging and vision is difficult to overestimate. For instance, the impulse
response of an optical system can be defined in terms of diffraction
integrals, that are in turn Fourier transforms of a function on a disk.
There are several popular competing approaches used to calculate
diffraction integrals, such as the extended Nijboer-Zernike (ENZ) theory.
In this talk, an alternative efficient method of computation of two
dimensional Fourier-type integrals based on approximation of the integrand
by Gaussian radial basis functions is discussed. Its outcome is a rapidly
converging series expansion for the integrals, allowing for their accurate
calculation. The proposed method yields a reliable and fast scheme for
simultaneous evaluation of such kind of integrals for several values of the
defocus parameter, as required in the characterization of the through-focus
optics.

- 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.

- 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 – ylee@cims.nyu.edu

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