https://www.reliable-computing.org/archive/conferences/2007_Stenger/Optimal_algorithms_abstracts.html

Optimal Algorithms and Computational Complexity for Numerical Problems

Schedule, Abstracts, and Slides of Some of the Talks

Monday, May 7 to Tuesday, May 8, 2007

/ Schedule /

/ Alphabetical List of Speakers and Abstracts /

/ Slides of Some of the Presentations /



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 SzabadosWeighted 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 KandaLaminar-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 ZayedA 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 GustafsonEstimating and Obtaining Ultimate Accuracy with Sinc Expansions

15:45--16:10: Professor Jean-Paul BerrutA Formula for the Error of Finite Sinc-Interpolation with an Even Number of Nodes

16:15--16:40: Brian BockelmanSinclib: 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 WoźniakowskiSmoothness 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 SikorskiFrom 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 KeinertA 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 YamamotoSinc Methods for a Boundary Integral Equation
Professor Ronald A. DeVoreAnalog 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 

Brian Bockelman
 
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

Professor Fritz Keinert
 
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.
 

Children of the Sincs: The Prolate Spheroidals

Professor Marek A. Kowalski
(Click on the title for the abstract.)
 

Experimental approximations with shifts of exp(az)/z 

Professor Gerhard Opfer
 
(Click on the title for the abstract.)

Regularization of Inverse Problems Using Sparsity Constraints -- Analysis and Applicationss

Dr. Ronnie Ramlau
(Click on the title for the abstract.)
 

Iterative methods for ill-posed problems

Professor Lothar Reichel

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.

Recent Developments in Variable Transformations for Numerical Integration

Professor Avraham  Sidi
(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).

Sinc Solution of PDE's by Separation of Variables

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

Professor Richard S. Varga
 

Adaption Makes it Easy to Approximate Piecewise Smooth Functions  with Unknown Singularities

Professor Grzegorz W. Wasilkowski
 
(Click on the title for the abstract.)

Smoothness and Tractability of Multivariate Approximation

(Click on the title for the abstract.)

Sinc Methods for a Boundary Integral Equation

Toshihiro Yamamoto


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

Professor Ahmed Zayed


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.)


High-Precision Numerical Integration and Experimental Mathematics

 Dr. David H. Bailey




Degree Computation and Global Optimization: A Personal Perspective

Professor R. Baker Kearfott


A Linear Algebra Approach to Boundary Multiwavelets

Professor Fritz Keinert
 

Experimental approximations with shifts of exp(az)/z 

Professor Gerhard Opfer


Iterative methods for ill-posed problems

Professor Lothar Reichel



Homer and Orosius:  A Key to Explain the Deucalion Flood, Exodus, and Other Tales

Professor Emilio Spedicato



A Super-Tunguska Event Circa 1447 BC: A Scenario for the Phaethon Explosion, the Indo-Aryan Migration and the Exodus Events

Professor Emilio Spedicato


SINC-PACK, and Separation of Variables

Professor Frank Stenger

Smoothness and Tractability of Multivariate Approximation

Professor Henryk Woźniakowski