Preprint Series 1997

1997:18
P. Petrushev
We investigate the efficiency of approximation by linear combinations of ridge functions in the metric of $L _ 2(B^d)$ with $B^d$ the unit ball in $R^d$. If $X _ n$ is an ndimensional linear space of univariate functions in $L _ 2(I),I=[1 ...
[Full Abstract] 
1997:17
M. AlLawatia,
R. Sharpley, and
H. Wang
We develop a characteristicbased domain decomposition and spacetime local refinement method for firstorder linear hyperbolic equations. The method naturally incorporates various physical and numerical interfaces and generates accurate numerical solutions even if large time steps are used. The method fully utilizes the transient and strongly local behavior of the solutions ...
[Full Abstract] 
1997:16
B. Ersland and
H. Wang
The mathematical model that describes the process of an immiscible displacement of oil by water in reservoir production or other twophase fluid flows in porous media leads to a strongly coupled system of a degenerated nonlinear advectiondiffusion equation for saturation and an elliptic equation for pressure and velocity. The hyperbolic ...
[Full Abstract] 
1997:15
M. AlLawatia,
R. Sharpley, and
H. Wang
We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order RungeKutta approximation of the characteristics within the framework of the EulerianLagrangian localized adjoint method. These methods naturally incorporate all three types of boundary conditions in their formulations are fully mass conservative ...
[Full Abstract] 
1997:14
H. Dahle,
M. Espedal,
R. Ewing,
S. Man,
R. Sharpley, and
H. Wang
We develop an ELLAM (EulerianLagrangian localized adjoint method) scheme to solve twodimensional advectiondispersion equations with all combinations of inflow and outflow Dirichlet, Neumann, and flux boundary conditions. The ELLAM formalism provides a systematic framework for implementation of general boundary conditions, leading to massconservative numerical schemes. The computational advantages of the ...
[Full Abstract] 
1997:13
V. Temlyakov
We study efficiency of $L _ p$greedy algorithm with regard to a multivariate system which is equivalent to the multivariate Haar system. In a place of multivariate Haar system we take the corresponding tensor product of univariate Haar systems. We prove that for $1<p<\infty$ the $L _ ...
[Full Abstract] 
1997:12
V. Temlyakov
We investigate a generalization of Kolmogorov’s width which is suitable for estimating best mterm approximation. We generalize the Carl’s inequality which gives lower estimate of Kolmogorov widths in terms of the entropy numbers. Application of these new inequalities gives some progress in the problem of estimating best mterm ...
[Full Abstract] 
1997:11
V. Temlyakov
We study the efficiency of greedy type algorithms with regard to redundant dictionaries in Hilbert space. In Section 2 we prove a general result which gives a sufficient condition on a dictionary to guarantee that Pure Greedy Algorithm is near best in the sense of power decay of error of ...
[Full Abstract] 
1997:10
V. Temlyakov
Two theorems on nonlinear mterm approximation in $L _ p,1<p<\infty$, are proved in this paper. The first one (Theorem 2.1) says that if a basis $\Psi:=\{\psi _ I\} _ I$ is $L _ p$equivalent to the Haar basis then near best mterm ...
[Full Abstract] 
1997:09
K. Oskolkov
Free (non linear) ridge $L^2$ approximation $NRA _ n(f),n=1,2,\ldots,$ of a function $f(x)=f(x _ 1,x _ 2)$ in the unit disc $I\!\!B^2$ is considered:
$\left\f\sum^n _ 1F _ j(x\cdot\xi _ j);L ...
[Full Abstract] 
1997:08
K. Oskolkov
Let
$h(t,x):=p.v.\ \sum _ {n\in{Z}\backslash\{0\}}\frac{e^{\pi{i}(tn^2+2xn)}}{2\pi{in}}=\lim _ {N\to\infty}\sum _ {0<n\leq{N}}\frac{e^{\pi{i}(tn^2+2xn)}}{2\pi{in}}$
($i=\sqrt{1};t,x=$ real ...
[Full Abstract] 
1997:07
A. Cohen,
R. DeVore,
P. Petrushev, and
H. Xu
Given a function $f\in{L _ 2}(Q),Q:=[0,1)^2$ and a real number $t>0$, let $U(f,t):=\inf _ {g\in{BV(Q)}}\fg\^2 _ {L _ 2(1)}%2BtV _ Q(g)$, where the infimum is taken over all functions $g\in ...
[Full Abstract] 
1997:06
A. Cohen,
R. DeVore, and
R. Hochmut
Approximation by a linear combination of n wavelets is a form of nonlinear approximation that occurs in several applications including image processing, statistical estimation, and the numerical solution of differential equations. In this paper, we shall consider variants of n term approximation which we call restricted approximation. As explained further ...
[Full Abstract] 
1997:05
S. BrennerThe condition number of the Schur complement in domain decomposition (file not available)
(Numer. Math. 83 (1999), 187203)
It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order $d^{1}h^{2m+1}$, where d measures the diameters of the subdomains and ...
[Full Abstract] 
1997:04
S. Brenner and
L. SungBalancing domain decomposition for nonconforming plate elements (file not available)
(Numer. Math. 83 (1999), 25  52)
In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number of the preconditioned system is shown to be bounded by $C\left[1+\left(\ln\frac{H}{h}\right)\right]^2$, where H measures the diameters of ...
[Full Abstract] 
1997:03
A. Fokas,
L. Sung, and
D. TsoubelisThe inverse spectral method for colliding gravitational waves (file not available)
(Math. Phys. Anal. Geom. 1 (1999), 313330)
The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular region 1 ≤ x < y ≤ 1 for a certain 2 × 2 matrix valued nonlinear equation. This equation, which is a particular exact reduction of the vacuum Einstein equations, is ...
[Full Abstract] 
1997:02
A. Andreev,
Z. Gao, and
R. Sharpley
Two elementary algorithms are introduced for image compression each of which is based on efficient, lossless encoding of quantized biorthogonal wavelet coefficients. Application of this type of algorithm is applied to several standard test images using regular and hyperbolic wavelet bases, and comparisons are given to Shapiro’s EZW algorithm ...
[Full Abstract] 
1997:01
L. Johnson,
A. Kaulgud, and
R. Sharpley
G3D is a graphical user interface designed to facilitate model development and understanding of complex dynamics governed by partial differential equations. G3D assists in the preparation of data for simulators based on three dimensional logically rectangular grids. The grid may be created by modifying an existing grid or by constructing ...
[Full Abstract]