CMS-MITACS Joint Conference 2007

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.

In this talk, I shall show a useful inequality on m-term 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) yj Î span {P2j-1+1l, P2j-1+2l, ..., P2jl}, where Pkl denotes the usual ultraspherical polynomial of degree k and index l on [-1,1].
(ii) ||yj||2,l » ||yj||¥ with the constant of equivalence independent of j, here ||·||2,l denotes the L2 norm on [-1,1] computed with respect to the weight (1-t2)l.

Sharp Jackson inequalities are given for some systems of orthogonal functions on various domains.

Let 2s, s Î N, fixed points y_i -p £ y_2s < y_2s-1 < ¼ < y₁ < 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 2p-periodic 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 [y₁,y₀], nonpositive/nonincreasing/concave on [y₂,y₁] and so on. Let

Theorem 1 If f Î D^(q)(Y) then

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.

A new and robust wavelet-based splitting method has been developed to solve the general aerosol equations. The considered models are the nonlinear integro-partial 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.

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 ill-equipped 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.

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.

In 1980, H. Berens and R. DeVore (A characterization of Bernstein polynomials, in Approximation Theory III, Proc. Conf., Austin, Texas, 1980, 213-219) 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, l₂, verifies l₂ £ [(n-1)/(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, Bernstein-Kantorovich and Bernstein-Durrmeyer operators are optimals in certain cases. In others, we show new Bernstein-type operators.

L₁-based minimization method for stationary Hamilton-Jacobi equations

For W Î R^d, a convex bounded set with non-empty interior, the moduli of smoothness w^r(f,t)_Lq(W) and the norm ||f||_Lq(W) are estimated by an Ul'yanov-type expression involving w^r(f,t)_Lp(W) where 0 < p < q £ ¥. The main result for q < ¥ is given by

A survey on the results by K. Kopotun, D. Leviatan, the author, and others.

Let s Î N, -1 < y_s < ¼ < y₁ < 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 [y₁,1], concave on [y₂,y₁], etc., D⁽²⁾(Y₀) 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

For k Î N, r Î N₀ and function f Î C^(r) ÇD⁽²⁾ (Y_s) we will discuss the validity of the inequality