|
| |
| Name: | Gerard Awanou |
| E-mail: | gawanou@math.uga.edu |
| Institute: | University of Georgia |
| Talk Title: | Spline approximations of the 3D Navier-Stokes Equations |
| Talk Abstract: | We consider numerical approximations of the 3D Navier-Stokes equations in velocity-pressure formulation.
The pressure is eliminated from the equations by using a space of velocity fields which are divergence free.
The later is discretized by means of splines of arbitrary degree and arbitrary smoothness. Energy arguments are used to derive the discrete equations satisfied by the equations. The pressure term is computed by solving a Poisson problem with Neumann boundary. |
| | |
| Name: | Yuliya Babenko |
| E-mail: | yuliya.v.babenko@vanderbilt.edu |
| Institute: | Vanderbilt University |
| Talk Title: | About Kolmogorov
problem for (r-1)-monotone functions on the half-line |
| Talk Abstract: | The
talk is mainly about Kolmogorov-type inequalities that estimate
the $L_q$-norm of the intermediate derivative $x^{(k)}$ of
an $(r-1)$-monotone
function $x$ defined on a half-line in terms of
the $L_p$-norm of $x$
and the $L_\infty$-norm of its derivative $x^{(r)}$.
In particular, we will give a solution of the following
Kolmogorov problem:
find necessary and sufficient conditions such that
for any given $M_{0,\infty},M_{k,q},M_{r,\infty}$,
there exists an $(r-1)$-monotone function $x$ with
$M_{0,\infty}=\left\| x\right\| _{\infty},M_{k,q}=\left\|
x^{\left( k\right) }\right\| _{q},M_{r,\infty}=\left\|
x^{\left( r\right) }\right\|_{\infty}$.
|
| | |
| Name: | Katherine Balazs |
| E-mail: | kilgokb@auburn.edu |
| Institute: | Department of Mathematics Auburn University |
| Talk Title: | Simultaneous Approximation by Interpolation |
| Talk Abstract: | TBA |
| | |
| Name: | Victoria Baramidze |
| E-mail: | vbaramid@math.uga.edu |
| Institute: | University of Georgia |
| Talk Title: | Spherical Cubic Interpolating Splines |
| Talk Abstract: | We study several numerical schemes solving a scattered data interpolation problem on a sphere using spherical cubic splines. |
| | |
| Name: | Sudeshna Basu |
| E-mail: | sbasu@howard.edu |
| Institute: | Howard University |
| | |
| Name: | Peter Binev |
| E-mail: | binev@math.sc.edu |
| Institute: | University of South Carolina |
| Talk Title: | Local Subdivision on Adaptive Triangulations |
| Talk Abstract: | Representation of surfaces in computer graphics often
requires consecutive approximations with different levels
of accuracy. A multiresolution representation proves to be
very effective, but it usually involves a global (regular)
subdivision procedure. On the other hand adaptive local
refinement can provide an effective algorithm. Here we
propose to use a local refinement of irregular triangulations
known as {\it newest vertex bisection}, and an appropriate
subdivision scheme based on a quasi-interpolation operator.
|
| | |
| Name: | Kai Bittner |
| E-mail: | bittner@math.umsl.edu |
| Institute: | Department of Mathematics & CS, University of Missouri - St. Louis |
| Talk Title: | Spline Modulation of Sinusoids for Signal Representation |
| Talk Abstract: | We investigate biorthogonal Wilson bases generated by cardinal B-splines.
Such bases provide very efficient representations of sinusoids modulated
by spline functions.
Exact Riesz bound for such bases are given. Furthermore, we derive an
explicit formula for the computation of the dual window functions and
determine their decay rate.
|
| | |
| Name: | Derek Bruff |
| E-mail: | bruff@math.vanderbilt.edu |
| Institute: | Vanderbilt University |
| | |
| Name: | Carlos Cabrelli |
| E-mail: | cabrelli@math.gatech.edu |
| Institute: | University of Buenos Aires and Georgia Institute of Technology |
| | |
| Name: | Steven Damelin |
| E-mail: | damelin@gasou.edu |
| Institute: | Georgia Southern University |
| Talk Title: | Distribution of general interpolation
arrays for exponential weights |
| Talk Abstract: | We discuss the distribution of
general interpolation arrays for a large class of exponentially
decaying weights on $(-1,1)$ and the line. |
| | |
| Name: | Arthur Danielyan |
| E-mail: | adaniely@pegasus.cc.ucf.edu |
| Institute: | University of Central Florida |
| Talk Title: | On a Localization Theorem for Approximation in Complex Domain |
| Talk Abstract: | We consider the problem of representation of Baire first
class functions on the compact sets of the complex plane by
convergent sequences of rational functions and prove a
corresponding localization theorem. We also present several
open problems. |
| | |
| Name: | Frank Deutsch |
| E-mail: | deutsch@math.psu.edu |
| Institute: | Penn State University
|
| | |
| Name: | Narendra Kumar Govil |
| E-mail: | govilnk@auburn.edu |
| Institute: | Auburn University, Auburn, AL 36849 |
| Talk Title: | On Growth of Polynomials |
| Talk Abstract: | Let $p(z) = \sum_{\nu=0}^n a_\nu
z^\nu$ be a polynomial of degree $n$, $M(p,R)
= \max\limits_{|z| = R \ge 1} |p(z)|$ and let
$\|p\| = \max\limits_{|z| = 1} |p(z)|$.
If $p(z) \neq 0$ in $|z| < 1$, then according to a well
known result of Ankeny and Rivlin, $M(p,R) \le
\left( \frac{R^n + 1}{2} \right) \|p\| $ for $R \ge 1$. In this
talk, we would be presenting some results that generalize
and sharpen this and some other related inequalities. |
| | |
| Name: | Doug Hardin |
| E-mail: | hardin@math.vanderbilt.edu |
| Institute: | Vanderbilt University |
| Talk Title: | The matrix Fej\'er-Riesz Theorem and the construction of local orthonormal shift-invariant bases. |
| Talk Abstract: | We present a constructive proof of the matrix-valued Fej\'er-Riesz Theorem with real coefficients and characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis. |
| | |
| Name: | Wenjie He |
| E-mail: | he@neptune.cs.umsl.edu |
| Institute: | Dept. of Math. and Computer Science, Univ. of Missouri, St. Louis |
| Talk Title: | Compactly supported symmetric tight wavelet frames |
| Talk Abstract: | We consider compactly supported tight frames
$\Psi=\{ \psi^1, \ldots, \psi^N \}$ for $L^2(\RR^2)$ that associate
with some compactly supported refinable functions. We introduce an
algorithmic approach for constructing tight frame generators.
This approach features a recipe for selecting the columns of the
Kronecker product of the orthonormal matrix extensions of the univariate
two-scale symbols in order to formulate the multivariate orthonormal matrix
extension of the multivariate two-scale symbol. We pay particular attention
to the case that each tight frame generator is either symmetric or
antisymmetric.
|
| | |
| Name: | Christopher Heil |
| E-mail: | heil@math.gatech.edu |
| Institute: | Georgia Tech |
| | |
| Name: | Don Hong |
| E-mail: | hong@math.vanderbilt.edu, hong@etsu.edu |
| Institute: | E T S U / Vanderbilt University |
| | |
| Name: | Bruce Kessler |
| E-mail: | bruce.kessler@wku.edu |
| Institute: | Western Kentucky University
|
| Talk Title: | Optimal Prefilters for a DGH 2-D
Orthogonal
Scaling Vector |
| Talk Abstract: | In the case where a single
scaling func
tion is used to represent a signal, the discrete data can be used as
scaling fun
ction coefficients (the identity prefilter) and polynomial degree of
data (up to
the approximation order of the $V_0$ space) is preserved. This is
not generall
y true, however, with scaling vectors of multiplicity $r>1$. This
talk will pre
sent a generalization of work by Hardin and Roach (1998) to find
prefilters for
a dilation-3 orthogonal scaling vector (created by Donovan, Geronimo,
and Hardin
(1995) and defined
over a uniform triangular lattice of ${\bf R}^2$) that will preserve
up to linea
r approximation order and also preserve the norm of the data.
Examples of the effectiveness of the prefilters will be provided. |
| | |
| Name: | Theodore Kilgore |
| E-mail: | kilgota@banach.math.auburn.edu |
| Institute: | Auburn University |
| Talk Title: | Simultaneous approximation on the line and on the half-line
for certain rational functions |
| Talk Abstract: | Some approximation properties on
$(-\infty,\infty)$ by
rational functions with denominator $(1+x^2)^{-n}$ will be addressed.
Analogous results will be presented, too, for
approximation on $[0,\infty)$ by
rational functions with denominator $(1+t)^{-n}$.
|
| | |
| Name: | Tom Kunkle |
| E-mail: | kunklet@cofc.edu |
| Institute: | College of Charleston |
| Talk Title: | Favard's Interpolation Problem in Several Variables |
| Talk Abstract: | \def\RR{{{\rm I}\kern-.2em {\rm R}}}
Let $n$ and $d$ be natural numbers
and consider the following problem. Construct
a smooth interpolant $F$ to function values $f$
given at points $m$ in $\RR^d$, where
$F$ depends locally and linearly on $f$, and $F$'s derivatives of total
degree $n$ are bounded by a constant $C$ times the corresponding
divided differences of $f$. Here $C$ may depend on $d$ and $n$, but
must be independent of $f$ and $m$.
\par
Favard gave an optimal solution to the problem in case
$d=1$ by bounding each of two consecutive derivatives by the corresponding
divided differences and allowing $m$ to be any discrete set in
$\RR$.
Such freedom would not be possible in the multivariate setting.
In fact, if $m$
is a tensor product grid in $\RR^d$,
i.e., the Cartesian product of $d$ increasing
sequences of real numbers, then such an interpolation scheme exists
if and only if $m$ has uniform spacing in each of the coordinate
directions. When $m$ is a subset of a tensor product grid with
uniform spacing, the interpolation problem may or may not have a solution,
depending on the geometry of the subset.
|
| | |
| Name: | John Lavery |
| E-mail: | lavery@aro.arl.army.mil |
| Institute: | Army Research Office |
| Talk Title: | Shape-preserving approximation of multiscale univariate data by cubic $L_1$ splines |
| Talk Abstract: | Spline fits are calculated by minimizing a data fitting
functional over a manifold of splines. Smoothing splines
are calculated by minimizing a linear combination of the
data fitting functional and the interpolating spline functional.
For multiscale data, that is, data with abrupt changes in
magnitude and/or spacing, currently available spline fits and
smoothing splines and spline fits typically have extraneous
oscillation. In this paper, a class of cubic $L_1$ spline fits
that do not have extraneous oscillation is investigated. Cubic
$L_1$ spline fits are compared with the conventional $L_2$
cubic spline fits and with cubic $L_1$ and $L_2$ smoothing splines.
Cubic $L_1$ spline fits are calculated by a Lagrange-multiplier-based
primal affine (interior point) algorithm. Computational results for
three data sets are given. $L_1$ spline fits preserve shape well.
$L_1$ spline fits are computationally more expensive than $L_1$
smoothing splines but require only half the storage and do not
depend on a parameter that the user must choose.
|
| | |
| Name: | Jian-Guo Liu |
| E-mail: | jliu@math.umd.edu |
| Institute: | University of Maryland |
| Talk Title: | Efficient Numerical Methods for
Incompressible Flow
|
| Talk Abstract: | In this talk, I will review a class of
numerical methods for 2D and 3D
unsteady incompressible Navier-Stokes
equations for large Reynolds
number based on local vorticity boundary
conditions or local pressure
boundary conditions. In each case a
convectionally stable explicit time
stepping is used for the momentum equation
and the incompressibility
constraint is realized by a derived Poisson
equation. The local vorticity
boundary conditions and local pressure
boundary conditions are used so
that the dynamic and kinematic equations are
decoupled. The result
in a class of very efficient Naver-Stokes
solver. At each time step,
only one Poisson equation needs to be solved
to achieve 2nd order accuracy.
With a modest increase in the computational
cost, higher order
approximations are possible using compact
finite difference approximations,
finite element approximations, or spectral
approximations. A patch mesh
refinement can be easily incorporated into
the schemes to resolve highly
unstable boundary layers. Various numerical
experiments will
be presented to demonstrate the accuracy and
efficiency of this class
of methods, including the flow past cylinder
at the Reynolds numeber up to
200,000 and an air bubble rising water.
|
| | |
| Name: | Doron Lubinsky |
| E-mail: | lubinsky@math.gatech.edu |
| Institute: | Georgia Institute of Technology |
| Talk Title: | Rogers-Ramanujan and the Baker-Gammel-Wills (Pade) Conjecture |
| Talk Abstract: | In 1961, Baker, Gammel and Wills conjectured that a subsequence of the diagonal Pade sequence to a function f meromorphic in the unit ball, converges away from the poles. We showed recently that the Rogers-Ramanujan continued fraction provides a counterexample. We discuss this and subsequent developments. No Pade background is assumed. |
| | |
| Name: | Hrushikesh Mhaskar |
| E-mail: | hmhaska@calstatela.edu |
| Institute: | California State University, Los Angeles |
| Talk Title: | Bit representation of band-dominant functions on the sphere |
| Talk Abstract: | A band-dominant function on the Euclidean sphere embedded in ${\bf R}^{q+1}$ is the restriction to this sphere of an entire function of $q+1$ complex variables having a finite exponential type in each of its variables. We develop a method to represent such a function using finitely many bits, using the values of the function at scattered sites on the sphere. The number of bits required in our representation is asymptotically the same as the metric entropy of the class of such functions with respect to any of the $L^p$ norms on the sphere.
|
| | |
| Name: | Erwin Mina |
| E-mail: | emina@math.vanderbilt.edu |
| Institute: | Vanderbilt University |
| | |
| Name: | Ram Mohapatra |
| E-mail: | ramm@pegasus.cc.ucf.edu |
| Institute: | University of Central Florida |
| Talk Title: | On Fractional Order Derivatives of Trigonometric Polynomials |
| Talk Abstract: | In this paper we consider Weyl derivatives of fractional order and obtain necessary and sufficvient conditions for a Szego type inequality to hold. We also extend to fractional order an estimate for the derivative of a self inversive polynomial. |
| | |
| Name: | Ursula Molter |
| E-mail: | umolter@math.gatech.edu |
| Institute: | University of Buenos Aires and Georgia Institute of Technology |
| | |
| Name: | Mike Neamtu |
| E-mail: | neamtu@math.vanderbilt.edu |
| Institute: | Vanderbilt University |
| Talk Title: | Multivariate Splines - Recent Progress |
| Talk Abstract: | In this survey talk I will discuss recent developments
in the theory of multivariate splines constructed via simplex splines. |
| | |
| Name: | David Roach |
| E-mail: | david.roach@murraystate.edu |
| Institute: | Murray State University |
| Talk Title: | Explicit Parameterizations of
Wavelets |
| Talk Abstract: | In this talk, we give a
complete and simple parameterization for the length six
one-dimensional filters which satisfy the necessary conditions
for orthogonality. This parameterization includes
all compactly supported univariate scaling functions contained
within the interval $[0,5]$ using dilation factor 2.
These formulae are a convenient way to generate a continuum of
wavelets with varying approximation properties ranging from
scaling functions which only reproduce constants to ones which
reproduce linears and quadratics. In addition, solutions for
the length eight and ten filters are given where the parameters
have transcendental constraints. We conclude the talk with some
interesting numerical experiments presenting a parameterized
wavelet of length six which performs better than the standard
wavelets of Haar, D4, and D6 in an image compression scheme
on a variety of images. |
| | |
| Name: | Larry Schumaker |
| E-mail: | s@mars.cas.vanderbilt.edu |
| Institute: | Vanderbilt university |
| Talk Title: | Surface Compression Using $C^1$
Cubic Splines on Triangulated Quadrangulations |
| Talk Abstract: | The space of $C^1$ cubic
splines defined
on triangulated quadrangulations is used to construct
a Faber-type multiresolution interpolation method for
approximating and compressing surfaces defined
by bivariate functions. |
| | |
| Name | Boris Shekhtman |
| E-mail | boris@math.usf.edu |
| Institution | University of South Florida |
| Talk Title | Interpolation by Polynomials in
Several Variables. |
| Abstract | We discuss the dimension of
uniformly interpolating subspaces in several variables. |
| | |
| Name: | Tatyana Sorokina |
| E-mail: | sorokina@math.vanderbilt.edu |
| Institute: | Vanderbilt University |
| Talk Title: | Quintic Spline Interpolation on
Type-4 Tetrahedral Partitions |
| Talk Abstract: | Given a rectangular box $B$,
we define the associated
type-4 tetrahedral partition $\tri$ by first splitting $B$
into suboxes, then splitting each subbox into 5 tetrahedra.
The partition can be regarded as a cross-cut partition which
is created by cutting $B$ using planes with four different
orientations. We construct a method for
interpolating Hermite data
at the vertices of $\tri$ using a space of
$C^1$ quintic supersplines. We prove that the method
has full approximation power,
and also present several numerical results. |
| | |
| Name: | Joachim Stoeckler |
| E-mail: | joachim.stoeckler@math.uni-dortmund.de |
| Institute: | University of Dortmund |
| Talk Title: | Spline frames on the real line and bounded intervals |
| Talk Abstract: | We present new concepts for the construction of tight frames
on the real line with maximum vanishing moments.
Similar concepts
can be introduced for the construction of tight frames on bounded intervals.
As an example we study frames that are linear combinations of
of cubic B-splines with arbitrary knots. |
| | |
| Name: | Paul Wenston |
| E-mail: | Paul@math.uga.edu |
| Institute: | University of Georgia |
| Talk Title: | Bivarate C^1 Cubic Splines for Exterior Biharmonic Problems |
| Talk Abstract: | C^1 cubic splines and the infinite element method are used
to solve the both the homogeneous and nonhomogenous biharmonic
equation exterior to a bounded domain. |
| | |
| Name: | Frank Zeilfelder |
| E-mail: | zeilfeld@euklid.math.uni-mannheim.de |
| Institute: | University of Mannheim, Germany |
| Talk Title: | Local Lagrange Interpolation by Bivariate Splines |
| Talk Abstract: | There exists a vast literature on local Hermite interpolation
by smooth splines on triangulations. On the other hand, since recently
no results of this type were known for Lagrange interpolation.
We develop the first local Lagrange interpolation methods for
$C^1$ splines on triangulations and triangulated quadrangulations.
The methods are based on a suitable coloring of the triangles
and quadrilaterals, respectively. The Lagrange interpolating
splines yield optimal approximation order and can be computed
with linear complexity. We give numerical tests which demonstrate
the efficiency of the methods. The results were obtained by
joint work with G’nther N’rnberger and Larry L. Schumaker.
|
| | |