Monday, February 20, 2012 - 14:00 , Location: Skiles 006 , Benjamin Berkels , South Carolina University , Organizer: Sung Ha Kang
Image registration is the task of transforming different images, or more general data sets, into a common coordinate system. In this talk, we employ a widely used general variational formulation for the registration of image pairs. We then discuss a general gradient flow based minimization framework suitable to numerically solve the arising minimization problems. The registration framework is next extended to handle the registration of hundreds of consecutive images to a single image. This registration approach allows us to average numerous noisy scanning transmission electron microscopy (STEM) images producing an improved image that surpasses the quality attainable by single shot STEM images.We extend these general ideas to develop a joint registration and denoising approach that allows to match the thorax surface extracted from 3D CT data and intra-fractionally recorded, noisy time-of-flight (ToF) range data. This model helps track intra-fractional respiratory motion with the aim of improving radiotherapy for patients with thoracic, abdominal and pelvic tumors.
Monday, January 30, 2012 - 14:00 , Location: Skiles 006 , David Mao , Institute for Mathematics and Its Applications (IMA) at University of Minnesota , Organizer: Sung Ha Kang
Binary function is a class of important function that appears in many applications e.g. image segmentation, bar code recognition, shape detection and so on. Most studies on reconstruction of binary function are based on the nonconvex double-well potential or total variation. In this research we proved that under certain conditions the binary function can be reconstructed from incomplete frequency information by using only simple linear programming, which is far more efficient.
Monday, January 23, 2012 - 14:05 , Location: Skiles 006 , Alper Erturk , Georgia Tech, School of Mechanical Engineering , Organizer:
The transformation of vibrations into low-power electricity has received growing attention over the last decade. The goal in this research field is to enable self-powered electronic components by harvesting the vibrational energy available in their environment. This talk will be focused on linear and nonlinear vibration-based energy harvesting using piezoelectric materials, including the modeling and experimental validation efforts. Electromechanical modeling discussions will involve both distributed-parameter and lumped-parameter approaches for quantitative prediction and qualitative representation. An important issue in energy harvesters employing linear resonance is that the best performance of the device is limited to a narrow bandwidth around the fundamental resonance frequency. If the excitation frequency slightly deviates from the resonance condition, the power output is drastically reduced. Energy harvesters based on nonlinear configurations (e.g., monostable and bistable Duffing oscillators with electromechanical coupling) offer rich nonlinear dynamic phenomena and outperform resonant energy harvesters under harmonic excitation over a range of frequencies. High-energy limit-cycle oscillations and chaotic vibrations in strongly nonlinear bistable beam and plate configurations are of particular interest. Inherent material nonlinearities and dissipative nonlinearities will also be discussed. Broadband random excitation of energy harvesters will be summarized with an emphasis on stochastic resonance in bistable configurations. Recent efforts on aeroelastic energy harvesting as well as underwater thrust and electricity generation using fiber-based flexible piezoelectric composites will be addressed briefly.
Monday, November 28, 2011 - 14:00 , Location: Skiles 006 , Christel Hohenegger , Mathematics, Univ. of Utah , Organizer:
One of the challenges in modeling the transport properties of complex fluids (e.g. many biofluids, polymer solutions, particle suspensions) is describing the interaction between the suspended micro-structure with the fluid itself. Here I will focus on understanding the dynamics of semi-dilute active suspensions, like swimming bacteria or artificial micro-swimmers modeled via a simple kinetic model neglecting chemical gradients and particle collisions. I will then present recent results on the linearized structure of such an active system near a state of uniformity and isotropy and on the onset of the instability as a function of the volume concentration of swimmers, both for a periodic domain. Finally, I will discuss the role of the domain geometry in driving the flow and the large-scale flow instabilities, as well as the appropriate boundary conditions.
Monday, November 14, 2011 - 14:00 , Location: Skiles 006 , Olof Widlund , Courant Institute,New York University, Mathematics and Computer Science , Organizer: Haomin Zhou
The domain decomposition methods considered are preconditioned conjugate gradient methods designed for the very large algebraic systems of equations which often arise in finite element practice. They are designed for massively parallel computer systems and the preconditioners are built from solvers on the substructures into whichthe domain of the given problem is partitioned. In addition, to obtain scalability, there must be a coarse problem, with a small number of degrees of freedom for each substructure. The design of this coarse problem is crucial for obtaining rapidly convergent iterations and poses the most interesting challenge in the analysis.Our work will be illustrated by overlapping Schwarz methods for almost incompressible elasticity approximated by mixed finite element and mixed spectral element methods. These algorithms is now used extensively at the SANDIA, Albuquerque laboratories and were developed in close collaboration with Dr. Clark R. Dohrmann. These results illustrate two roles of the coarse component of the preconditioner.Currently, these algorithms are being actively developed for problems posed in H(curl) and H(div). This work requires the development of new coarse spaces. We will also comment on recent work on extending domain decomposition theory to subdomains with quite irregular boundaries. This work is relevant because of the use of mesh partitioners in the decomposition of large finite element matrices.
Monday, November 7, 2011 - 14:00 , Location: Skiles 006 , Jingfang Liu , GT Math , Organizer: Sung Ha Kang
The empirical mode decomposition (EMD) was a method developed by Huang et al as an alternative approach to the traditional Fourier and wavelet techniques for studying signals. It decomposes signals into finite numbers of components which have well behaved intataneous frequency via Hilbert transform. These components are called intrinstic mode function (IMF). Recently, alternative algorithms for EMD have been developed, such as iterative filtering method or sparse time-frequency representation by optimization. In this talk we present our recent progress on iterative filtering method. We develop a new local filter based on a partial differential equation (PDE) model as well as a new approach to compute the instantaneous frequency, which generate similar or better results than the traditional EMD algorithm.
Fast Spectral-Galerkin Methods for High-Dimensional PDEs and Applications to the electronic Schrodinger equationMonday, October 31, 2011 - 14:00 , Location: Skiles 006 , Jie Shen , Purdue University, Department of Mathematics , Organizer: Haomin Zhou
Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations in both bounded and unbounded domains.As an application, we shall use the proposed sparse spectral method to solve the N-particle electronic Schrodinger equation.
Monday, October 24, 2011 - 14:00 , Location: Skiles 006 , Jun Lu , GT Math , Organizer: Sung Ha Kang
We propose a new fast algorithm for finding the global shortest path connecting two points while avoiding obstacles in a region by solving an initial value problem of ordinary differential equations (ODE's). The idea is based on the factthat the global shortest path possesses a simple geometric structure. This enables us to restrict the search in a set of feasible paths that share the same structure. The resulting search space is reduced to a finite dimensional set. We use a gradient descent strategy based on the intermittent diffusion (ID) in conjunction with the level set framework to obtain the global shortest path by solving a randomly perturbed ODE's with initial conditions.Compared to the existing methods, such as the combinatorial methods or partial differential equation(PDE) methods, our algorithm is faster and easier to implement. We can also handle cases in which obstacles shape are arbitrary and/or the dimension of the base space is three or higher.
Monday, October 10, 2011 - 14:00 , Location: Skiles 006 , Bradley Lucier , Purdue University, Department of Mathematics , Organizer: Sung Ha Kang
We consider a variant of Rudin--Osher--Fatemi variational image smoothing that replaces the BV semi-norm in the penalty term with the B^1_\infty(L_1) Besov space semi-norm. The space B^1_\infty(L_1$ differs from BV in a number of ways: It is somewhat larger than BV, so functions inB^1_\infty(L_1) can exhibit more general singularities than exhibited by functions in BV, and, in contrast to BV, affine functions are assigned no penalty in B^1_\infty(L_1). We provide a discrete model that uses a result of Ditzian and Ivanov to compute reliably with moduli of smoothness; we also incorporate some ``geometrical'' considerations into this model. We then present a convergent iterative method for solving the discrete variational problem. The resulting algorithms are multiscale, in that as the amount of smoothing increases, the results are computed using differences over increasingly large pixel distances. Some computational results will be presented. This is joint work with Greg Buzzard, Antonin Chambolle, and Stacey Levine.
Monday, October 3, 2011 - 14:00 , Location: Skiles 006 , Zhimin Zhang , Wayne State University , Organizer: Yingjie Liu
Finite element approximations for the eigenvalue problem of the Laplace operator are discussed. A gradient recovery scheme is proposed to enhance the ﬁnite element solutions of the eigenvalues. By reconstructing the numerical solution and its gradient, it is possible to produce more accurate numerical eigenvalues. Furthermore, the recovered gradient can be used to form an a posteriori error estimator to guide an adaptive mesh reﬁnement. Therefore, this method works not only for structured meshes, but also for unstructured and adaptive meshes. Additional computational cost for this post-processing technique is only O(N) (N is the total degrees of freedom), comparing with O(N^2) cost for the original problem.