145 research outputs found
A real algebra perspective on multivariate tight wavelet frames
Recent results from real algebraic geometry and the theory of polynomial
optimization are related in a new framework to the existence question of
multivariate tight wavelet frames whose generators have at least one vanishing
moment. Namely, several equivalent formulations of the so-called Unitary
Extension Principle by Ron and Shen are interpreted in terms of hermitian sums
of squares of certain nonnegative trigonometric polynomials and in terms of
semi-definite programming. The latter together with the recent results in
algebraic geometry and semi-definite programming allow us to answer
affirmatively the long standing open question of the existence of such tight
wavelet frames in dimension ; we also provide numerically efficient
methods for checking their existence and actual construction in any dimension.
We exhibit a class of counterexamples in dimension showing that, in
general, the UEP property is not sufficient for the existence of tight wavelet
frames. On the other hand we provide stronger sufficient conditions for the
existence of tight wavelet frames in dimension and illustrate our
results by several examples
A real algebra perspective on multivariate tight wavelet frames
Recent results from real algebraic geometry and the theory of polynomial optimization
are related in a new framework to the existence question of multivariate tight wavelet
frames whose generators have at least one vanishing moment. Namely, several equivalent
formulations of the so-called Unitary Extension Principle (UEP) from [33] are interpreted
in terms of hermitian sums of squares of certain nongenative trigonometric polynomials
and in terms of semi-definite programming. The latter together with the results in
[31, 35] answer affirmatively the long stading open question of the existence of such tight
wavelet frames in dimesion d = 2; we also provide numerically efficient methods for
checking their existence and actual construction in any dimension. We exhibit a class
of counterexamples in dimension d = 3 showing that, in general, the UEP property is
not sufficient for the existence of tight wavelet frames. On the other hand we provide
stronger sufficient conditions for the existence of tight wavelet frames in dimension d ≥ 3
and illustrate our results by several examples
Symmetric interpolatory dual wavelet frames
For any symmetry group and any appropriate matrix dilation we give an
explicit method for the construction of -symmetric refinable interpolatory
refinable masks which satisfy sum rule of arbitrary order . For each such
mask we give an explicit technique for the construction of dual wavelet frames
such that the corresponding wavelet masks are mutually symmetric and have the
vanishing moments up to the order n. For an abelian symmetry group we
modify the technique such that each constructed wavelet mask is -symmetric.Comment: 22 page
Multivariate tile B-splines
Tile B-splines in are defined as autoconvolutions of the
indicators of tiles, which are special self-similar compact sets whose integer
translates tile the space . These functions are not
piecewise-polynomial, however, being direct generalizations of classical
B-splines, they enjoy many of their properties and have some advantages. In
particular, the precise values of the H\"older exponents of the tile B-splines
are computed in this work. They sometimes exceed the regularity of the
classical B-splines. The orthonormal systems of wavelets based on the tile
B-splines are constructed and the estimates of their exponentional decay are
obtained. Subdivision schemes constructed by the tile B-splines demonstrate
their efficiency in applications. It is achieved by means of the high
regularity, the fast convergence, and small number of the coefficients in the
corresponding refinement equation.Comment: 45 pages, 37 figure
Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
Multigrid methods and stationary subdivisions
Multigridmethods are fast iterative solvers for sparse large ill-conditioned linear systems of equations derived, for instance, via discretization of PDEs in fluid dynamics, electrostatics and continuummechanics problems. Subdivision schemes are simple iterative algorithms for generation of smooth curves and surfaces with applications in 3D computer graphics and animation industry.
This thesis presents the first definition and analysis of subdivision based multigrid methods.
The main goal is to improve the convergence rate and the computational cost of multigrid taking advantage of the reproduction and regularity properties of underlying subdivision.
The analysis focuses on the grid transfer operators appearing at the coarse grid correction step in the multigrid procedure. The convergence of multigrid is expressed in terms of algebraic properties of the trigonometric polynomial associated to the grid transfer operator. We interpreter the coarse-to-fine grid transfer operator as one step of subdivision. We reformulate the algebraic properties ensuring multigrid convergence in terms of regularity and generation properties of subdivision. The theoretical analysis is supported by numerical experiments for both algebraic and geometric multigrid. The numerical tests with the bivariate anisotropic Laplacian ask for subdivision schemes with anisotropic dilation. We construct a family of interpolatory subdivision schemes with such dilation which are optimal in terms of the size of the support versus their polynomial generation properties. The numerical tests confirmthe validity of our theoretical analysis
Reconstruction of sparse wavelet signals from partial Fourier measurements
In this paper, we show that high-dimensional sparse wavelet signals of finite
levels can be constructed from their partial Fourier measurements on a
deterministic sampling set with cardinality about a multiple of signal
sparsity
Approximation orders of shift-invariant subspaces of
We extend the existing theory of approximation orders provided by
shift-invariant subspaces of to the setting of Sobolev spaces, provide
treatment of cases that have not been covered before, and apply our
results to determine approximation order of solutions to a refinement equation
with a higher-dimensional solution space.Comment: 49 page
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