Nonlinear Methods in Computational Mathematics
Org: Kirill Kopotun (Manitoba) [PDF]
 A. BASS BAGAYOGO, University College of Saint Boniface
Hybrid Octree Grid Based Method and Application to an
Aircraft Geometry
[PDF] 
Geometry modelling and grid generation over complex objects is one of
the important and essential aspect in Computational Fluid Dynamics.
As the complexity grows with size, it becomes difficult to visualize
and modify the grids. In this talk I will present and hybrid grid
generate by using and Octree/Quadtree based method, the emphasis will
be on the rapid acquisition of the geometry, the special design data
structures, and some aspects related to the intersection algorithms.
 FENG DAI, University of Alberta, Edmonton, Alberta
An inequality on mterm approximation by ultraspherical
polynomials and its applications
[PDF] 
In this talk, I shall show a useful inequality on mterm
approximation by ultraspherical polynomials on [1,1]. As an
application, I shall show how to use this inequality to construct a
sequence of polynomials y_{j}, j = 1,2,... with the following
properties:
(i) y_{j} Î span {P_{2j1+1}^{l}, P_{2j1+2}^{l}, ..., P_{2j}^{l}}, where
P_{k}^{l} denotes the usual ultraspherical polynomial of degree
k and index l on [1,1].
(ii) y_{j}_{2,l} » y_{j}_{¥} with
the constant of equivalence independent of j, here
·_{2,l} denotes the L^{2} norm on [1,1] computed
with respect to the weight (1t^{2})^{l}.
 ZEEV DITZIAN, University of Alberta, Edmonton, Alberta
Sharp Jackson inequalities
[PDF] 
Sharp Jackson inequalities are given for some systems of orthogonal
functions on various domains.
 GERMAN DZYUBENKO, International Mathematical Center of NAS of Ukraine, 01601,
Tereschenkivska Str., 3, Kyiv4, Ukraine
Shape preserving approximation of periodic functions
[PDF] 
Let 2s, s Î N, fixed points y_{i} p £ y_{2s} < y_{2s1} < ¼ < y_{1} < p are given and for the other indexes i Î Z, the points y_{i} are defined periodically, i.e., by
the equality y_{i} = y_{i+2s} + 2p, Y : = {y_{i}}_{i Î Z}.
>From the space C of continuous 2pperiodic functions f :R ® R with the norm f : = max_{x Î R} f(x), we extract three sets
D^{(q)}(Y), q = 0,1,2, of all functions f which are,
respectively, nonnegative/nondecreasing/convex on [y_{1},y_{0}],
nonpositive/nonincreasing/concave on [y_{2},y_{1}] and so on. Let
E_{n}^{(q)}(f) : = 
inf
T_{n} Î T_{n} ÇD^{(q)}(Y)

fT_{n}, n Î N, 

where T_{n} is the space of trigonometric polynomials of
order £ n1.
Theorem 1
If f Î D^{(q)}(Y) then
E_{n}^{(q)}(f) £ c(s) w_{k}(f,p/n), n ³ N(Y), k = 
ì í
î




where w_{k} (f,t) is the kth modulus of continuity of f,
c(s) and N(Y) are the constants depending only on s and on
min_{i=1,...,2s} {y_{i}y_{i+1}}, respectively.
Remark 1
Each of these three estimates is wrong with a greater k. It follows
from the Whitney inequality that the constants c(s) and N(Y) can
be both replaced simultaneously by c(Y) and 1, respectively. The
respective estimates with c(s) and 1 are wrong.
The case q=0 was proved by the author and J. Gilewicz, q=1 by the
author and M. G. Pleshakov, q=2 by the pupil of the author
V. D. Zalizko.
 QIANG GUO, York University
Adaptive wavelet method for aerosol dynamic equation
[PDF] 
A new and robust waveletbased splitting method has been developed to
solve the general aerosol equations. The considered models are the
nonlinear integropartial differential equations on time, size and
space, which describe different processes of atmospheric aerosols
including condensation, ucleation, coagulation, deposition, and
sources as well as turbulent mixing.
The proposed method reduces the complex general aerosol dynamic
equation to two directional splitting equations. Because there are
steeply varying number densities across a size range, an adaptive
wavelet strategy is developed to solve the size splitting equation
effectively. And further the wavelet method and the finite difference
method are alternately used for two directional splitting equations at
each time interval.
 TOM HOGAN, The Boeing Company, Seattle, WA, USA
Implications of design optimization on geometry generation
in aerospace
[PDF] 
Airplane design, and vehicle design in general, is evolving. The
traditional technique was for an experienced designer/engineer with a
real talent for design and a large personal knowledge base to draft a
single vehicle in a CAD system; analyze it for pertinent properties
(like the lift provided by the wings, the drag of the vehicle, its
structural integrity, predicted fuel consumption, etc.); and decide if
it meets the market's needs. If it doesn't, which is typical, the
next step was essentially to go back to the drawing board to see if it
can be tweaked to do so. More recently, the designer may provide a
baseline design to which small perturbations can be made. Then an
optimization package can try to hone in on an acceptable design... as long as there is one that is nothing more than a minor modification
of the baseline.
The next step in this evolution is for the designer to design an
entire family of vehicles that depend on a set of parameters, i.e., a
bunch of virtual knobs that can be turned to morph the vehicle,
allowing for more significant changes so a larger set of vehicles can
be studied. In this presentation we show why existing CAD packages
are illequipped for this new approach, present some of the tools we
have developed to address the issues and give a taste of the kinds of
mathematics behind these new tools.
 YINGKANG HU, Georgia Southern University, Dept. of Math. Sci.,
Statesboro, GA 304608093, USA
Global Optimization using hyperbolic cross points with
application in clustering
[PDF] 
Erich Novak and Klaus Ritter developed in 1996 a global optimization
algorithm that uses hyperbolic cross points (HCPs). We modify this
algorithm in many ways to improve its efficiency and developed a local
search strategy that results in much better chance to find the global
minimizer. The ideas are implemented on the computer for optimization
in clustering. The program has been tested extensively with very
promising results.
 FRANCISCOJAVIER MUÑOZDELGADO, Universidad de Jaen, 23071 Jaen, Spain
Optimal shape perserving linear operators with different
types of data
[PDF] 
In 1980, H. Berens and R. DeVore (A characterization of
Bernstein polynomials, in Approximation Theory III, Proc. Conf.,
Austin, Texas, 1980, 213219) showed that classical Bernstein
operators are the best in certain sense. They proved that if L is a
linear operator mapping real functions defined on [0,1] onto
polynomial functions of degree less or equal to n, preserving the
positivity and the sign of all the derivatives and fixing the linear
polynomial, then the eigenvalue corresponding to the polynomial
functions of degree two, l_{2}, verifies l_{2} £ [(n1)/(n)], and the identity is satisfied only by Bernstein operators.
Now, we consider linear polynomial operators that use certain type of
data (values of functions in some points, derivatives, moments, etc.)
and we consider the preservation of the sign of only one derivative.
For each case, we look for a optimal operator. We show that Bernstein,
BernsteinKantorovich and BernsteinDurrmeyer operators are optimals
in certain cases. In others, we show new Bernsteintype operators.
 BOJAN POPOV, Texas A&M University
L_{1} approximations of HamiltonJacobi equations
[PDF] 
L_{1}based minimization method for stationary HamiltonJacobi
equations
H(x,u,Du) = 0, x Î W with u_{¶ W} = 0 

is developed. The case considered is of a 2D bounded domain with a
Lipschitz boundary. The general assumption is that the viscosity
solution u of the problem is unique, u Î W^{1,¥}(W),
and the gradient Du is of bounded variation. We approximate the
solution to this problem using continuous finite elements and by
minimizing the residual in L_{1}. In the case of a convex (with
respect to Du) and uniformly continuous hamiltonian, it is shown
that, upon introducing an appropriate entropy, the sequence of
approximate solutions based on quasiuniform shape regular finite
element triangulations converges to the unique viscosity solution u.
The main features of the method are that it is an arbitrary polynomial
order and it does not have any artificial viscosity. The fact that
the residual in minimized in L_{1} is a key. Numerical examples and
possible application of this method to other hyperbolic equations will
be discussed.
 ANDRIY PRYMAK, CAB 632, Department of Mathematical and Statistical Sciences,
University of Alberta, Edmonton, AB, T6G 2G1
Ul'yanovtype inequality for bounded convex sets in R^{d}
[PDF] 
For W Î R^{d}, a convex bounded set with nonempty interior,
the moduli of smoothness w^{r}(f,t)_{Lq(W)} and the norm
f_{Lq(W)} are estimated by an Ul'yanovtype expression
involving w^{r}(f,t)_{Lp(W)} where 0 < p < q £ ¥.
The main result for q < ¥ is given by
w^{r} (f,t)_{q} £ C 
ì í
î


ó õ

t
0

u^{qq} w^{r}(f,u)^{q}_{p} 
du
u


ü ý
þ

1/q

, 0 < t £ diamW, q = 
d
p

 
d
q

. 

A corresponding estimate of f_{Lq(W)} is, in fact, an
embedding theorem involving Besov spaces with a range of q more
general than known today. The power q achieved is optimal.
 IGOR SHEVCHUK, National Taras Shevchenko University of Kyiv, Ukraine
Convex and coconvex polynomial approximation in the uniform
norm
[PDF] 
A survey on the results by K. Kopotun, D. Leviatan, the author, and
others.
Let s Î N, 1 < y_{s} < ¼ < y_{1} < 1, Y_{s} = {y_{i}}_{i=1}^{s}, D^{(2)}(Y_{s}) be the set of continuous on
[1,1] functions, which are convex on [y_{1},1], concave on
[y_{2},y_{1}], etc., D^{(2)}(Y_{0}) be the set of convex continuous
on [1,1] functions, · be a uniform norm on [1,1],
P_{n} be the space of algebraic polynomials of degree less
that n, and
E_{n}^{(2)} (f,Y_{s}) : = 
inf
P_{n} Î P_{n}ÇD^{(2)}(Y_{s})

fP_{n} 

be the error of the best uniform coconvex approximation of f.
For k Î N, r Î N_{0} and function f Î C^{(r)} ÇD^{(2)} (Y_{s}) we will discuss the validity of the
inequality
E_{n}^{(2)} (f,Y_{s}) £ 
C
n^{r}

w_{k} 
æ è


1
n

, f^{(r)} 
ö ø

, n ³ N, 

where w_{k} are moduli of smoothness of different types.
 PING ZHOU, St. Francis Xavier University
Newton's interpolation formula in study of multivariate
functions
[PDF] 
We discuss the use of Newton's interpolation formula, i.e., divided
differences, in the following studies of multivariate functions:
1. Finite sum representations of some multivariate functions,
e.g. the Lauricella function F_{D}.
2. Explicit constructions of multivariate Padé approximants
for pseudomultivariate functions.
3. Arithmetical results on certain multivariate power series,
i.e., the proof of the irrationality and transcendence of some
multivariate power series.
