http://interval.louisiana.edu/conferences/2007_Stenger/Optimal_algorithms_abstracts.html
Optimal Algorithms and Computational
Complexity for Numerical Problems
Schedule,
Abstracts, and Slides of Some of the Talks
Note: All talks, as well as the official conference meals, will
be in the Officer's Club, located next to the Guesthouse. Signs
with directions will be posted.
Registration for the conference will begin Monday at 8:00 a.m, at the
Officer's club. The general registration fee will be $50, and $20 for
students. This fee will include all meals and coffee breaks,
as follows:
Monday
Continental breakfast
Lunch
Reception dinner
Tuesday
Continental Breakfast
Lunch
+ 4 coffee breaks
Schedule
Click on a particular talk title to be directed to the abstract.
Click on an author's name to be directed to his home page.
Monday, May 7
9:00--9:15: Opening Remarks
9:15--9:40: Professor
Richard A. Askey, Some
Things for Frank Stenger from School Mathematics
9:45--10:10: Dr.
David H. Bailey, High-Precision
Numerical Integration and Experimental Mathematics
10:15--10:40: Professor
József Szabados , Weighted
Approximation and Interpolation on Infinite Intervals (a survey)
10:45--11:00: Refreshment Break
11:00--11:25: Jared Tanner, The Surprising Structure of Gaussian Point
Clouds and Implications for Signal Processing
11:30--11:55: Dr.
Ronnie Ramlau , Regularization
of Inverse Problems Using Sparsity Constraints -- Analysis and
Applications
12:00--13:30: Lunch
13:30--13:55: Professor
Hidesada Kanda , Laminar-Turbulent
Transition: Calculation of Minimum Critical Reynolds Number in Pipe Flow
14:00--14:25: Professor
Avraham Sidi, Recent Developments in Variable
Transformations for Numerical Integration
14:30--14:55: Professor
Ahmed Zayed , A
Comparison between the Sinc-Galerkin, Adomian Decomposition , and
Wavelets
Methods in Solving Some Non-Linear Problems
15:00--15:15: Refreshment break
15:15--15:40: Professor
Sven-Åke
Gustafson , Estimating
and Obtaining Ultimate Accuracy with Sinc Expansions
15:45--16:10: Professor
Jean-Paul Berrut , A
Formula for the Error of Finite Sinc-Interpolation with an Even Number
of Nodes
16:15--16:40: Brian Bockelman
, Sinclib:
An High-Level Optimized Library for Sinc Computation
18:00: Dinner reception
Tuesday, May 8
9:00--9:25: Professor Marek A.
Kowalski , Children
of the Sincs: The Prolate Spheroidals
9:30--9:55: Professor
Henryk Woźniakowski , Smoothness
and Tractability of Multivariate Approximation
10:00--10:25: Professor
Grzegorz W. Wasilkowski , Adaption
Makes it Easy to Approximate Piecewise Smooth Functions with
Unknown
Singularities
10:30--10:45: Refreshment Break
10:45--11:10: Christopher
Sikorski , From
Poland to Utah in Communist Times:
Nonliner Problems /
Algorithms in Collaboration with Frank
11:15--11:40: Professor
R. Baker Kearfott , Degree Computation and Global
Optimization: A Personal Perspective
11:45--13:30: Lunch
13:30--13:55: Open
14:00--14:25: Professor
Fritz Keinert , A
Linear Algebra Approach to Boundary Multiwavelets
14:30--14:55: Professor
Lothar Reichel, Iterative methods for
ill-posed problems
15:00--15:15: Refreshment break
15:15--15:40: Professor
Gerhard Opfer, Experimental approximations with shifts of
exp(az)/z
15:45--16:10: Professor
Frank Stenger, Sinc Solution
of PDE's by Separation of Variables
16:15--16:40: Professor
Richard S. Varga , Title
to
be announced
People whose circumstances prevented them from attending:
Toshihiro
Yamamoto , Sinc
Methods for a Boundary Integral Equation
Professor
Ronald A. DeVore , Analog
to Digital Conversion: A Mathematician's View
Professor
Emilio Spedicato, ABS
Methods for Continuous and Integer Linear Systems: Some Rrecent
Results
Alphabetical
List of Speakers and Abstracts
Some Things for Frank
Stenger from School Mathematics
Professor
Richard A. Askey
Two recent results coming from school
mathematics
will be discussed. One is how sines and cosines help explain some
results on Fibonacci numbers. The other is how to sum an infinite
series, which was posed as a problem in a comic strip a bit over 10
years
ago, and the need for some numerical analysis to get an approximate
answer
if you are unable sum the series.
High-Precision
Numerical
Integration and Experimental Mathematics
Dr.
David H. Bailey
Frank Stenger was a pioneer in advanced
methods for highly accurate numerical integration. As it turns
out,
these methods, extended and refined by others, are now a centerpiece in
the emerging discipline of "experimental
mathematics," wherein numerical computations are used to discover new
facts of mathematics. As a single example, recent research using
numerical methods has uncovered numerous heretofore unknown relations
in
the Ising theory of mathematical physics. This talk will give an
overview of these methods together with several examples of results
obtained
using this methodology.
A
Formula for the Error of Finite Sinc-Interpolation with an Even Number
of Nodes
Professor
Jean-Paul Berrut
Recently we gave a formula for the error
committed when truncating the sinc-series of a function that does not
decrease
sufficiently rapidly for the discarded tails of the series to be
negligible.
The main part of the formula is a polynomial in the distance between
the
nodes whose coefficients contain derivatives of the function at the
extremities.
The middle term of a sinc-series usually corresponds to the origin,
so that its symmetric truncation contains an odd number of terms,
one for every node. In our talk we give the formula for the case of an
even number of nodes.
Sinclib:
An High-Level Optimized Library for Sinc Computation
Sinclib is a freely-available library for computing PDEs and ODEs using
Sinc methods. The methods for solution draws primarily on the
works
of Stenger, Lund, and Bower. One of the primary goals for Sinclib was
for
it to be optimized to the point of being usable in applications, but
have
the source code still be accessible and readable. To this end, a
popular
interpreted language, "Python," was chosen, combined with a
numerical library, "`NumPy," written in Python and C.
Sinclib demonstrates several numerical speedups to common
algorithms.
Some of the implemented optimizations include iterative methods, taking
advantage of special Toeplitz structures, and symbolic manipulation of
kronecker products.
Analog
to Digital Conversion: A Mathematician's View
Professor
Ronald A. DeVore
The digital world is preferred for
signal
processing since one can take advantage of the fact that the signal
only
takes values in a finite set of possibilities when doing computation.
On
the other hand, most signals are inherently analog. This make the
operations
of Analog to Digital conversion (A/D) and the reverse Digital to Analog
conversion (D/A) cornerstones of signal processing. The story of how
one
should perform this conversion is an interesting one from the
viewpoints
of both mathematicians and engineers. The mathematical solution does
not
match the preferred engineering solution (Sigma-Delta modulation). We
shall
try to explain some possible mathematical reasons why Engineers choose
Sigma-Delta Modulation over more obvious and seemingly better choices.
Estimating
and Obtaining Ultimate Accuracy with Sinc Expansions
Professor
Sven-Åke Gustafson
I have been thinking of the question
that
the fact that numbers in a computer are represented by a finite
accuracy
only and you have to balance truncation and round-off errors. I [will
talk
about some] interesting results [along this line].
Laminar-Turbulent
Transition: Calculation of Minimum Critical Reynolds Number in Pipe Flow
Professor
Hidesada Kanda
For laminar-turbulent transition in pipe
flows, there is the unsolved minimum critical value Rc(min) of
approximately
2030. From many previous experimental investigations and ours, it
became
clear that under natural calm disturbances, (i) the transition occurs
in
the entrance region, (ii) the entrance shape, or bellmouth diameter,
affects
Rc, and (iii) Rc(min) is obtained when the contraction ratio of
bellmouth
diameter to pipe diameter is minimum, i.e., in the case of a straight
pipe.
Computationally, we found that there exists a normal force (NWS) at the
wall near the inlet. NWS is derived from the Navier-Stokes equation in
a vector form as the radial component of the curl of vorticity at the
wall.
Let the dimensionless power done by NWS be PW. In the entrance region,
the velocity profile develops from uniform to parabolic profile and the
kinetic energy of fluid increases; its dimensionless magnitude is named
KE (physical unit is power). Here, note that NWS and PW decrease as the
Reynolds number (Re) increases, but KE is a constant regardless of Re,
unlike NWS and PW. Thus, we assume that the judgement condition for the
occurrence of the transition depends on whether the power PW is higher
or lower than the required acceleration power KE for establishing the
fully
developed flow: (i) when PW < KE, transition takes place and (ii)
when
PW > KE, flow is stable. Consequently, Rc(min) of 2040 was obtained
for
a straight pipe flow.
Degree Computation and Global
Optimization: A Personal Perspective
Professor
R. Baker Kearfott
In automatically verified global
optimization,
floating point arithmetic is used in conjunction with directed rounding
and exhaustive search to produce mathematically rigorous bounds on
answers.
In other words, if the program produces a list of bounds on the domain
variables, completion of the program execution constitutes a
mathematical
proof that any global optimizers must lie in that list of bounds. The
speaker
will describe how his interest in verified global optimization grew out
of his dissertation work under Professor Stenger on numerical
computation
of the topological degree. Time permitting, he will also explain
successes and challenges in verified global optimization.
A
Linear Algebra Approach to Boundary Multiwavelets
Wavelets are naturally defined on the whole real line. Applying them on
a finite interval requires the introduction of additional endpoint
functions
and corresponding changes in the decomposition and reconstruction
algorithms.
This talk will present a linear algebra approach to this problem.
Professor
Marek A. Kowalski
(Click
on the title for the abstract.)
Professor
Gerhard Opfer
(Click on the title for the abstract.)
(Click
on the title for the abstract.)
Iterative methods for ill-posed problems
Ill-posed problems often arise when one is interested in
determining
the cause of an observed effect, such as when one is interested in
restoring an available image that has been contaminated by blur and
noise. Image restoration gives rise to large-scale problems, because of
the typically large number of pixels that make up an image. We review
available iterative methods and focus, in particular, on recently
proposed multilevel methods. The talk presents joint work with Serena
Morigi, Fiorella Sgallari, and Andrei Shyshkov.
(Click on the title for the abstract.)
From
Poland to Utah in Communist Times:
Nonliner Problems
/ Algorithms in Collaboration with Frank
Christopher
Sikorski
We review algorithms for the solution
of nonlinear equations, fixed point problems and the computation of
topological degree that resulted from the collaboration with Frank.
Optimal complexity properties, numerical implementations and tests will
be discussed. The focus will be on multivariate bisection and ellipsoid
type methods.
<>
ABS
Methods for Continuous and Integer Linear Systems: Some Rrecent
Results
Professor
Emilio Spedicato
We review the main properties of the ABS
class for linear systems and linearly constrained nonlinear
optimization.
We discuss the recent application to Diophantine linear systems and LP
problems. We show how ABS methods can reduce the conditioning and
better
exploit the structure of the linear system arising in the Newton method
used in the primal dualinterior point method (a joint result with
Florian
Potra).
Professor
Frank Stenger
(Click
on the title for the abstract.)
Weighted
Approximation and Interpolation on Infinite Intervals (a survey)
Professor
József Szabados
Weighted Lagrange and Hermite-Fejér interpolation on the real
line
means new challenges compared to the unweighted case on a finite
interval.
We give a survey of the problems encountered in this topic. Another
difficult
problem is to establish Jackson type approximation theorems in this
setting.
We list the possible ways of defining suitable moduli of continuity
which
can measure the rate of polynomial approximation.
The Surprising
Structure of Gaussian Point Clouds and Implications for Signal
Processing
Jared Tanner
<>We will explore connections
between the structure of high-dimensional
convex polytopes and information acquisition for compressible signals.
A classical result in the field of convex polytopes is that if N points
are distributed Gaussian iid at random in dimension n<<N, then
only order (log N)^n of the points are vertices of their convex
hull. Recent results show that provided n grows slowly with N,
then with high probability all of the points are vertices of its convex
hull. More surprisingly, a rich "neighborliness" structure
emerges in the faces of the convex hull. One implication of this
phenomenon is that an N-vector with k non-zeros can be recovered
computationally efficiently from only n random projections with n=2e k
log(N/n). Alternatively, the best k-term approximation of a signal in
any basis can be recovered from 2e k log(N/n) non-adaptive
measurements, which is within a log factor of the optimal rate
achievable for adaptive sampling. Additional implications for
randomized error correcting codes will be presented.
This work was joint with David L. Donoho.
Title
to be Announced
(Click
on the title for the abstract.)
Sinc Methods for a
Boundary
Integral Equation
The aim of this work is to propose a novel numerical method to solve
a differential equation by the use of Sinc methods. To solve a
differential
equation, this method exploits a boundary integral equation that is
usually
used in the boundary element method. Because boundary integral
equations
of this kind have singularities, in the proposed method, an integral
involved
in the integral equation is evaluated by the trapezoidal rule if the
integral
does not have any singular point, and by Sinc convolution
otherwise.
We introduce how this method solves differential equations, showing
some
examples of numerical computations, and discuss some aspects of this
method.
A Comparison between
the Sinc-Galerkin, Adomian Decomposition , and Wavelets Methods in
Solving
Some Non-Linear Problems
In this talk we introduce a modified Adomian decomposition method
and the wavelet-Galerkin method for solving nonlinear ordinary
differential
equations with boundary conditions and then compare the results with
those
obtained by using the Sinc-Galerkin methods.
Slides of Some of the
Presentations
(Click on the title to
obtain PDF copies of the presentation.)