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Date
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Speaker/Institution
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Title and
Abstract
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Aug. 24
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Organizational
meeting
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Aug. 31
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Jingyue Wang/KU
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Title:Error Bound for Numerical Methods for the Rudin-Osher-Fatemi Image Smoothing Model
Abstract: The Rudin-Osher-Fatemi variational model has been extensively studied and used in image analysis.
There have been several very successful numerical algorithms developed to compute the numerical solutions.
We study the convergence of the numerical solutions to various finite-difference approximation to this model.
We bound the difference between the solution to the continuous ROF model and the numerical solutions. These
bounds apply to ``typical'' images, i.e., images with edges or with fractal structure.
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Sept. 07
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Aijun Zhang/KU
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Title: Spatial Spread and Front Propagation Dynamics of Nonlocal Monostable Equations in Periodic Habitats
Abstract:
This talk is concerned with the spatial spread and front propagation dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. Such equations arise in modeling the population dynamics of many species which exhibit nonlocal internal interactions and live in spatially periodic habitats. Firstly, we establish a general principal eigenvalue theory for spatially periodic nonlocal dispersal operators. Secondly, applying such theory and comparison principle for sub- and super-solutions, we obtain the existence, uniqueness, and global stability of spatially periodic positive stationary solutions and the existence of a spatial spreading speed in any given direction of a general spatially periodic nonlocal equation. Such features are generic for nonlocal monostable equations in the sense that they are independent of the assumption of the existence of the principal eigenvalue of the linearized nonlocal dispersal operator at 0. Finally, under the above assumption we also investigate the front propagation feature for monostable equations with non-local dispersal in spatially periodic habitats. It remains open whether this feature is generic or not for spatially periodic nonlocal monostable equations.
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Sept. 14
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Shuguan Ji/KU & Jilin University
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Title: Poisson-Nernst-Planck Systems for Ion Flow with
Density Functional Theory for Hard-Sphere Potential
Abstract. In this talk, we will study the Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential. First, by combining the geometric singular perturbation theory with the functional analysis, we prove the existence of solution under the electro-neutrality. Then, by asymptotic expansion method, we derive an approximation of the I-V relation. Finally, the careful analysis for the I-V relation discovers two important critical voltage values, and obtain some nontrival consequences on the size effect on I-V relation.
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Sept. 21
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Mat Johnson/KU
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Title: Index Theorems for the Stability of Periodic Traveling Waves of KdV Type
Abstract: In this talk, I will discuss a variety of "index theorems" recently derived in the case of the generalized KdV equation which can be used to determine the spectral and nonlinear (orbital) stability of the associated periodic traveling wave solutions to specific classes of perturbations. These stability indices are geometric in nature, and are typically given in terms of Jacobians of various maps from the "natural" parameter space for the manifold of periodic traveling wave solutions to the conserved quantities of the gKdV flow. For power-law nonlinearities these Jacobians can be computed directly in terms of a finite number of moments of the underlying wave (something which, in turn, can be computed numerically rather easily). This leads to some new and rather surprising stability results which were previously unknown even in the well studied completely integrable cases of KdV and modified KdV.
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Sept. 28
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Changna
Lu/KU & NanJing University of Information Science and Technology
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Title: Simulations of Shallow Water Equations by Weighted Essential Non-oscillatory Schemes on Unstructured Meshes
Abstract: My talk is mainly about scientific computing. In this research, the third-order weighted essential non-oscillatory (WENO) schemes are used to simulate the two-dimensional shallow water equations with the source terms on unstructured triangle meshes. The balance of the flux and the source terms makes the shallow water equations fit to non-flat bottom problems. According to the tests of some typical examples and the simulation of a tidal bore on an estuary with trumpet shape and Qiantang river in China; the results show that the schemes can be used to simulate the current flow accurately and catch the stronger discontinuous in water wave, such as dam break and tidal bore effectively.
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Oct. 05
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Milena Stanislavova/KU
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Title:
Linear stability analysis for traveling waves of second order in time PDE's
Abstract:
Linear stability analysis of traveling waves of second order in time PDE's $u_{tt}+L u+N(u)=0$ will be the topic of my talk. This question will be reduced to the question for stability of quadratic pencils in the form $\lambda^2Id+2c \lambda \partial_x+H_c$.
If $H_c$ is a self-adjoint operator, with a simple negative eigenvalue and a simple eigenvalue at zero, then we completely characterize the linear stability of the wave. More precisely, we introduce an explicitly computable index $\omega^* \in (0, \infty]$, so that the wave is stable if and only if $|c| \geq \omega^*$.
The results are applicable both in the periodic case and in the whole line case.
The method of proof involves a delicate analysis of a function $G $, associated with $H_c$, whose positive zeros are exactly the positive (unstable)
eigenvalues of the pencil $\lambda^2Id+2c\lambda \partial_x+H_c$. The function $G$ is not the Evans function for the problem, but rather a new object that we define, which fits the situation well.
As an application, I will consider three classical
models - the ``good'' Boussinesq equation,
the Klein-Gordon-Zakharov (KGZ) system and the fourth order beam equation. In the whole line case, for the Boussinesq and KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable traveling waves.
For the beam equation, we provide an explicit formula, which works for all $p$ and for both the periodic and the whole line cases.
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Oct. 12
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Ronald Haynes/Memorial University of Newfoundland
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Title:Applications of Domain Decomposition Methods to Mesh Generation and Space-time Parallelism for PDEs
Abstract:
Adaptively choosing an underlying spatial grid for computation has proven to be a useful, if not essential, tool for the solution of boundary value problems and partial differential equations. One way of generating adaptive meshes is through the so-called equidistribution principle (EP). Recently, the RIDC approach has been shown to be a relatively easy way to add small scale parallelism (in time) to the solution of time dependent PDEs. In this talk I will show how large scale spatial parallelism can be added to both of these methodologies using relatively simple domain decomposition strategies. In the later case, this results in a truly parallel space-time method for PDEs.
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Oct. 19
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Jun Huan/EECS(KU)
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Title: Structured Sparse Learning in Boosting
Abstract:
Boosting is a very successful classification algorithm that produces a linear combination of "weak" classifiers (a.k.a. base learner) to obtain high quality classification models. In this talk we present a new boosting algorithm where base learners have structure relationships in the functional space. Towards an efficient incorporation of the structure information, we have designed a general model where we use an undirected graph to capture the relationship of base learners. In our method, we combine both L1 norm and Laplacian based L2 norm penalty with Logit loss function of Logit Boost. In this approach, we enforce model sparsity and smoothness in the functional space spanned by the basis functions. We have derived efficient optimization algorithms based on coordinate decent for the new boosting formulation and theoretically prove that it exhibits a natural grouping effect for nearby spatial overlapping features. Using comprehensive experimental study, we have demonstrated the effectiveness of the proposed learning methods.
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Oct. 26
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Brian Laird/Chemistry(KU)
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Title: Calculation of solid-liquid interfacial free energy for planar and curved interfaces
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Oct. 28
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Vu Hoang Linh/Vietnam National University
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Title: Approximation of spectral intervals and associated leading directions
for linear differential-algebraic equations via smooth SVD
Abstract: This talk is devoted to the numerical approximation of Lyapunov
and Sacker-Sell spectral intervals for linear differential-algebraic equations
(DAEs). The spectral analysis for DAEs is improved and the concepts of
leading directions and solution subspaces associated with spectral intervals
are extended to DAEs. Numerical methods based on smooth singular value
decomposition are introduced for computing all or only some spectral
intervals and their associated leading directions. The numerical algorithms
as well as implementation issues are discussed in detail and numerical
examples are presented to illustrate the theoretical results.
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Oct. 31
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Jue Yan/Iowa
State
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Title:
Direct discontinuous Galerkin method and its variations for convection diffusion problems
Abstract:
Discontinuous Galerkin method is a special class of finite element method that
use completely discontinuous piecewise polynomials as the approximate solution space.
The imposed discontinuity across the cell interface gives the method flexibility to easily
handle h-p adaptivity and the advantage to solve problems with discontinuities,
for example, the shocks for hyperbolic problems. Due to the lack of up-winding
mechanism (characteristics), discontinuous Galerkin method for diffusion problems
are not as well studied as for the hyperbolic problems.
In this talk, we will discuss the recent four discontinuous Galerkin methods for diffusion problems;
1) the Direct discontinuous Galerkin(DDG) method; 2) the DDG method with interface corrections;
3) the DDG method with symmetric structure; and 4) a new DG method with none symmetric structure.
Major feature of the DDG method is the introduction of the jumps of second or higher order
solution derivatives in the solution derivative numerical flux. The symmetric version of the DDG method helps us
obtain the optimal L2(L2) error analysis for the DG solution. For the non-symmetric version, we show
that the scheme performs better than the Baumann-Oden scheme or the NIPG method in the sense
that optimal convergence is recovered with even-th order polynomial approximations.
A series of numerical examples are presented to show the high order accuracy and the capacity of the methods.
At the end, we will discuss the recent studies of the maximum-principle-satisfying
or the positivity preserving properties of the DDG related methods.
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Nov. 02
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Lennard Kamenski /KU
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Title:
New condition number estimates for finite element equations with
general meshes
Abstract:
In d > 1 dimensions, it has been proven by Bank and Scott that
the condition number of finite element equations does not degrade
significantly on adaptive meshes if the mesh remains locally
quasi-uniform and an appropriate scaling of the resulting system
is used.
In this talk, I will present a generalization of the result by
Bank and Scott. The developed bound on the condition number is
valid for general meshes, without any assumptions on the shape of
mesh elements. As in isotropic case, an appropriately chosen,
mesh-dependent diagonal scaling can be used to improve the
conditioning of the resulting linear system.
An interesting result is that the bound on the condition number
of the scaled system is mostly the same as for the uniform case
even if the mesh contains highly anisotropic elements, provided
the number of anisotropic elements is relatively small. A similar
result is also achieved for d = 1 dimension.
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Nov. 09
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Mohamed Badawy/KU
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Oral Exam
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Nov. 16
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Tuoc Van Phan/University of Tennessee
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Tittle: Stationary Navier-Stokes equations with critically singular external forces: existence and stability results
Abstract:
We show the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces belonging to a critical space.
To the best of our knowledge, this is the largest critical space that is available up to now for this kind of existence.
This result can be viewed as the stationary counterpart of the existence result obtained by H. Koch and D. Tataru for the free non-stationary
Navier-Stokes equations with initial data in $\textup{BMO}^{-1}$. The stability of the stationary solutions in such spaces is also
obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the
corresponding semigroup.
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Nov. 23
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No talk
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Thanksgiving
Break
Begins
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Nov. 30
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Mingji Zhang/KU
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Title: A pitch-fork bifurcation of a steady-state
Poisson-Nernst-Planck system with nonzero permanent charge
function
Abstract: In this work, we study the Poisson-Nernst-Planck(PNP) system for two types of ions with five regions of piecewise constant permanent charge, assuming the Debye number is large, because the electric field is so strong compared to diffusion. Using the method of asymptotic expansion, first we established the existence of a singular orbit(the solution to the zeroth-order system in \epsilon),and by analyzing the resulting algebraic equations, we concluded that the system experiences a pitch-fork bifurcation at $Q=0,$ where $Q$ is the permanent charge function, finally, we perturbed the parameters involved in the system and numerically, we found three solutions for the zeroth-order system in $\epsilon,$ which is consistent with our second conclusion.
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Dec. 07
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No Talk
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********
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Spring 2012
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*********
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Jan. 25
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Organizational
meeting
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Feb 1
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Xuemin Tu
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Title: FETI-DP Domain Decomposition Methods for Incompressible Stokes Equation
Abstract: A unified framework of FETI-DP algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. Several previously developed FETI-DP algorithms can be represented under this framework. Their condition number estimates are also simplified using this framework. A distinctive feature of this framework is that both continuous and discontinuous pressures can be used in the finite element space, while previous FETI-DP algorithms are valid only for the case of using discontinuous pressures. Both lumped and Dirichlet type preconditioners are analyzed and scalable convergence rates are proved.
Numerical experiments of solving a two-dimensional incompressible
Stokes problem also demonstrate the performances of the
discussed FETI-DP algorithms represented by the same
framework.
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Feb 8
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Atanas Stefanov
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Title:
Linear Stability Analysis for Periodic Traveling Waves of the
Boussinesq Equation and the KGZ System
Abstract:We study the linear stability of spatially periodic waves for the Boussinesq equation (for the quadratic and cubic models) and the Klein-Gordon-Zakharov system. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability respectively), when the perturbations are taken with the same period. In particular, our results allow
us to completely recover the linear stability results for the whole line case. Joint work with Hakkaev and Stanislavova.
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Feb 15
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Perry Alexander/EECS/ITTC(KU)
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Title: Verifying the Trusted Processor Module
Abstract:
The Trusted Processor Module (TPM) found in virtually all enterprise PCs is a security coprocessor used for establishing trust and protecting secrets. The TPM is defined in the same fashion as most hardware using structured text to define operations over state. However, the critical nature of the TPM warrants a more formal approach. This talk will outline the structure of a correctness proof for the TPM focusing on the use of formal verification tools, monadic models of state, and the use of bisimulation to define correctness conditions.
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Feb 22
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Ming-Jun Lai
(University of Georgia)
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Titile: On Unconstrained Nonconvex Minimization for Sparse Vector and Low-rank Matrix Recovery
Abstract:
I will start with a short survey how to recover a low-rank matrix from a small number
of its linear measurements, e.g., a subset of its entries. A typical application is image
reconstruction from its partial pixel values.
As such a problem share many common features with the recent
study of recovering sparse vectors in compressed sensing. Thus, I
shall give a quick review including some most updated research results.
Then I will introduce unconstrained \ell_q minimization approach and
an iteratively reweighted algorithm for recovering sparse vectors as well as
the counter-part for recovering low-rank matrices. A convergence analysis
of iterative algorithms to compute the sparse solution will be given.
Finally, I shall present some numerical results for recovering images
from their random sampling entries without and with noises.
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Feb 29
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Hongguo Xu
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Title: Convergence of QR algorithm with Rayleigh-quotient shift for normal matrices - updated results
Abstract: The QR algorithm is ranked as one of the top 10 algorithms in the 20th century.
Yet, its convergence behavior has not been fully understood even for Hermitian matrices.
We consider the QR algorithm with Rayleigh-quotient shift for normal matrices
and give detailed convergence analysis results. The results show how the convergence
depend on the eigenvalue location and the choice of initial vector for the first Hessenberg
reduction.
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Mar 7
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Jingyue Wang
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Title: Nonconvex Minimization for Sparse Solution with Iterative Hard Thresholdng
Abstract: We study an unconstrained \ell_q minimization model for recovering sparse vectors in compressed sensing. Two algorithm will be given to solve the minimization problem. The first one is applying the standard iteratively reweighted algorithm. The second is the improvement of the first one with iterative hard thresholding. We also give convergence analysis for these algorithms.
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Mar 14
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Zhongquan
Zheng/Aerospace Engineering(KU)
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Title: Development and Applications of a Direct-Forcing Immersed-Boundary Method
Abstract: Immersed-boundary (IB) methods for fluid-structure interaction problems typically discretize the equations of motion for fluid flow on a Cartesian grid, and such methods generally do not require that the geometry of the structure conform in any way to this Cartesian grid. Therefore, IB methods suit the computational needs for studying moving and/or morphing objects in fluid flow. Instead, the equations of motion for the fluid are augmented by an appropriately defined forcing term that typically is nonzero only in the vicinity of the objects.
According to the ways how the virtual boundary forces are determined, there are primarily two types of IB methods: feedback force methods and direct-forcing methods. Direct-forcing methods use a forcing term determined by the difference between the interpolated velocities on the boundary points and the desired (physical) boundary velocities. The forcing term generated in this manner thus directly compensates the errors between the calculated velocities and the desired velocities on the boundary. Direct-forcing methods are usually able to relax the stability limit in comparison to feedback force methods.
In this presentation, an improved direct-forcing method is introduced by discussing velocity interpolations on the immersed boundary, direct-forcing extrapolations to the grid points, resolution requirement of the immersed boundary points, internal layer treatment, and accuracy of the solution. Examples of applying this method to simulate fluid dynamics problems with moving and morphing objects are then presented.
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Mar 21
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Spring break
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Mar. 28
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Ognyan Christov/Sofia University
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Title:
Non-integrability of first order resonances in Hamiltonian systems in
three degrees of freedom
Abstract:
The normal forms of the Hamiltonian 1 : 2 : $\omega$ resonances to
degree three for $\omega = 1, 3, 4$ are studied for
integrability. We prove that these systems are non-integrable
except for the discrete values of the parameters which are well
known. We use the Ziglin-Morales-Ramis method based on the
differential Galois theory.
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Apr. 4
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Angel Zhivkov/Sofia University
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Title:
Short-time stability of the solar system
Abstract:
We consider the Sun, Mercury, Venus, Earth+Moon, Mars, Jupiter,
Saturn, Uranus and Neptune as point masses, moving according
Newton's law of universal gravitation.
With the help of two canonical transformations, all mean anomalies
have been eliminated up to the third order of the masses.
Our calculations are based on the observation that each mean motion
is very nearly to a square root of some not very large integer.
For example, the ratio between Jupiter's and Saturnian years
- the number of Laplace 2.483..., is nearly $\sqrt{37}:\sqrt{6}$.
Such a fact cancels any small divisor.
Finally we prove a theorem that the present configuration
will be stable at least during the next 10 million years in sence
that any semi-major axis would not change significantly and the
eccentricities and inclinations will remain bounded.
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Apr 11
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Myunghyun Oh
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Title:
Study of stability of solutions of a viscous isentropic gas flow
through a nozzle with viscosity
Abstract:
In this talk, we examine the stability of stationary non-transonic waves for viscous isentropic
compressible flows through a nozzle with varying cross-section areas. The main result in this
paper is, for small viscous strength, stationary non-transonic waves with certain density are
spectrally unstable; more precisely, we will establish the existence of positive eigenvalues for
the linearization along such waves. The result is achieved via a center manifold reduction of
the eigenvalue problem. The reduced eigenvalue problem is then studied in the framework
of the Sturm-Liouville Theory.
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Apr 18
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Anil Misra/Civil Engineering (KU)
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Title: Second gradient continuum mechanics theory and its micromechanical derivation for cohesive granular materials
Abstract: Cohesive granular materials are known for their pressure-dependent strain softening behavior that is intimately linked to their microstructure and grain interactions. We have been developing continuum theories for these discrete materials utilizing higher-order displacement gradients and micromechanics. In this presentation, we will first describe a second gradient continuum mechanics theory using the principal of virtual work (variational principle) to establish the governing equations and the boundary conditions. We will then describe the derivation of constitutive equations for this theory using a micromechanical approach. The applicability of the derived model will be demonstrated through examples with different imperfections and simulation of shear band failure. For numerical calculations, the weak form of the second gradient theory is derived and discretized using an element-free Galerkin (EFG) formulation. The model predictions will be shown to have both quantitative and qualitative consistency with the observed behavior of cohesive granular material.
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Apr 24 (4--5PM)
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Sarah
Kieweg/Mechanical Engineering(KU)
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Title
Mathematical models of the fluid mechanics of non-Newtonian
polymer solutions used to prevent HIV transmission
Abstract:
HIV/AIDS is a devastating global pandemic. Scientists are working towards both vaccines and antiretroviral treatments that attenuate existing infections, but these approaches also have major obstacles which must be overcome. Another approach is to prevent the initial infection with a microbicide. A microbicide is a product which a woman can use to protect herself from HIV. A microbicide can be a delivery vehicle (such as a gel, cream, or vaginal ring) containing anti-HIV drugs; the vehicle itself may also prevent HIV infection. Dr. Kieweg’s research program uses engineering approaches to design and optimize drug delivery vehicles – specifically, cellulose-based polymer solutions – to maximize their function. This seminar will describe mathematical models of fluid mechanics to describe the spreading and retention of these non-Newtonian polymer solutions. The gravity-driven flow of a finite bolus of non-Newtonian fluid with a free surface was described and used to consider the effects of shear-thinning, consistency, inclination, and surface tension. The non-linear PDE representing the evolution of the fluid’s free surface was modeled with numerical methods, similarity solutions, and travelling waves. An elastic sheet squeezing and spreading the non-Newtonian fluid was also considered in a model to represent the application of a microbicide to biological tissue. Finally, future work considering the viscoelastic nature of both the polymer solution and the biological tissue will be described.
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Apr. 25
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Stephen Bond/Sandia Lab
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Title: Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
Abstract:
The computation of "free energy" has been described as one of the most important and challenging problems in computational chemistry. Free energy is a thermodynamic state function which plays an integral role in multiscale modeling and coarse-grained simulation. The "solvation free energy" is a measure of the thermodynamic work of moving a macromolecule from a vacuum to a solvent environment. A popular coarse-graining method is to approximate the solvent (e.g., water) interactions by a dielectric continuum as described by the Poisson-Boltzmann equation (a nonlinear elliptic PDE). To calculate the solvation free energy for the coarse grained system, one must evaluate a linear functional of the solution to this PDE. In this talk, we show how one can calculate the solvation free energy using an adaptive finite element approximation to the solution of the Poisson-Boltzmann equation. In our scheme, the mesh refinement is driven by goal-oriented error estimation based on the free energy functional. Hierarchical basis methods are used to precondition the resulting algebraic systems arising in the multilevel finite element discretization.
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May 2
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Aijun Zhang
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Title:Principal Eigenvalues of Dispersal Operators and Their Applications
Abstract
In this talk, first we will introduce the principal eigenvalue theory for random, discrete and nonlocal dispersal operators. Then we will show their applications in studying stationary solutions and spreading speeds for the monostable equations in periodic environments. Finally we will show the effects of spatial variations on the principal eigenvalues of these dispersal operators and periodic boundary condition, and the effects of spatial variations and dispersal strategies on the spreading speeds of monostable equations.
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