Preprint Series 2002

2002:22
S. Konyagin,
B. Popov, and
O. Trifonov
A class of nonoscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered. This class of methods contains the classical LaxFriedrichs and the secondorder NessyahuTadmor schemes. In the case of linear flux, new l_{2} stability results and error estimates for the methods are proved ...
[Full Abstract] 
2002:21
P. Petrushev
This article is a survey of some recent developments which concern two multilevel approximation schemes: (a) Nonlinear nterm approximation from piecewise polynomials generated by anisotropic dyadic partitions in R^{d}, and (b) Nonlinear nterm approximation from sequences of hierarchical spline bases generated by multilevel triangulations in R^{2}. A ...
[Full Abstract] 
2002:20
A. Kamont and
V. Temlyakov
We study nonlinear mterm approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis H in $L _ p([0,1]),1<p<\infty$) a greedy type algorithm realizes nearly best mterm approximation ...
[Full Abstract] 
2002:19
P. Binev,
W. Dahmen,
R. DeVore, and
P. Petrushev
Adaptive Finite Element Methods (AFEM) are numerical procedures that approximate the solution to a partial differential equation (PDE) by piecewise polynomials on adaptively generated triangulations. Only recently has any analysis of the convergence of these methods [10, 13] or their rates of convergence [2] become available. In the latter paper ...
[Full Abstract] 
2002:18
V. Temlyakov
We continue to study efficiency of approximation and convergence of greedy type algorithms in uniformly smooth Banach spaces. Two greedy type approximations methods, the Weak Chebyshev Greedy Algorith (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA), have been introduced and studied in [T1]. These methods (WCGA and WRGA) are very ...
[Full Abstract] 
2002:17
K. Oskolkov
Let P be a bivariate algebraic polynomial of degree n with the real senior part, and $Y=\{y _ j\} _ 1^n$ an nelement collection of pairwise noncolinear unit vectors on the real plane. It is proved that there exists a rigid rotation $Y^\varphi$ of Y by ...
[Full Abstract] 
2002:16
S. Konyagin and
V. Temlyakov
This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasiBanach space. The last decade was very successful in studying nonlinear approximation ...
[Full Abstract] 
2002:15
S. Brenner and
L. SungMultigrid methods for the computation of singular solutions and stress intensity factors III: Interface singularities (file not available)
(Computer Methods in Applied Mechanics and Engineering 192 (2003), 46874702)
It is shown in this paper that twodimensional interface problems with large jumps in the coefficients can be solved effectively by the piecewise linear finite element method on quasiuniform grids. This is achieved by combining the full multigrid methodology, the ...
[Full Abstract] 
2002:14
É. Czabarka,
O. Sykora,
L. Székely, and
I. Vrto
We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs. We find a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We find ...
[Full Abstract] 
2002:13
B. Karaivanov,
P. Petrushev, and
R. Sharpley
In this article algorithms are developed for nonlinear nterm Courant element approximation of functions in $L _ p$ ($0 < p \le \infty$) on bounded polygonal domains in $R^2$. Redundant collections of Courant elements, which are generated by multilevel nested triangulations allowing arbitrarily sharp angles, are investigated. Scalable algorithms ...
[Full Abstract] 
2002:12
O. Davydov and
P. Petrushev
We study nonlinear nterm approximation in $L _ p(R^2)$ ($0<p\leq\infty$) from hierarchical sequences of stable local bases consisting of differentiable (i.e., $C^r$ with $r\geq1$) piecewise polynomials (splines). We construct such sequences of bases over multilevel nested triangulations of $R^2$, which ...
[Full Abstract] 
2002:11
P. Binev and
R. DeVore
Adaptive methods of approximation arise in many settings including numerical methods for PDEs and image processing. They can usually be described by a tree which records the adaptive decisions. This paper is concerned with the fast computation of near optimal trees based on n adaptive decisions. The best tree based ...
[Full Abstract] 
2002:10
A. Lutoborski and
V. Temlyakov
Our objective is to study nonlinear approximation with regard to redundant systems. Redundancy on the one hand offers much promise for great efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. Greedy type approximations proved to be convenient and ...
[Full Abstract] 
2002:09
S. Konyagin and
V. Temlyakov
We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trignonometric polynomial of the form $G _ m(f):=\sum _ {k\in\Lambda}\hat{f}(k)e^{i(k,x)}$, where $\Lambda\subset{Z ...
[Full Abstract] 
2002:08
S. Konyagin and
V. Temlyakov
We consider convergence of thresholding type approximations with regard to general complete minimal systems $\{e _ n\}$ in a quasiBanach space X. Thresholding approximations are defined as follows. Let $\{e^ * _ n\}\subset{X}^ * $ be the conjugate (dual) system to the $\{e _ n\}$; then define for $\epsilon>0$ and ...
[Full Abstract] 
2002:07
L. Székely
This paper surveys how the concept of crossing number, which used to be familiar only to a limited group of specialists, emerges as a significant graph parameter. This paper has dual purposes: first, it reviews foundational, historical, and philosophical issues of crossing numbers; second, it shows a new lower bound ...
[Full Abstract] 
2002:06
V. Temlyakov
The main goal of this paper is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical intergation), the discrepancy theory, and nonlinear approximation. In Section 1 we discuss...
[Full Abstract] 
2002:05
S. BrennerKorn's inequalities for piecewise H^{1} vector fields (file not available)
(Mathematics of Computation 73 (2004), 10671087)
We develop a family of lockingfree elements for the ReissnerMindlin plate using Discontinuous Galerkin (DG) techniques, one for each odd degree, and prove optimal error estimates. A second family uses conforming elements for the rotations and nonconforming elements for the transverse displacement, generalizing the ...
[Full Abstract] 
2002:04
D. Dix
The group $G _ a$ of rigid motions, i.e. the semidirect product of $R^n$ (translations) and $SO(n)$ (proper rotations), acts on $R^n$. Let N be a set with N elements. Let $(R^n)^N$ denote the set of all mappings $i\mapsto{R _ i}$ from ...
[Full Abstract] 
2002:03
R. Gribonval and
M. Nielsen
We study various approximation classes associated with mterm approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best mterm approximation is compared to new classes obtained by considering mterm approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied ...
[Full Abstract] 
2002:02
R. Gribonval and
M. Nielsen
We characterize the approximation spaces associated with the best nterm approximation in L_{p}(R) by elements from a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between L_{p} and classical Besov spaces, and the result coincide with the ...
[Full Abstract] 
2002:01
S. BrennerPoincaréFriedrichs inequalities for piecewise H^{1} functions (file not available)
(SIAM Journal on Numerical Analysis 41 (2003), 306324)
PoincaréFriedrichs inequalities for piecewise H^{1} functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods.
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