Lubomir T. Dechevsky

A new generation of wavelet shrinkage: adaptive strategies based on composition of Lorentz-type thresholding and Besov-type non-threshold shrinkage

This article is a systematic overview of compression, smoothing and denoising techniques based on shrinkage of wavelet coefficients, and proposes an advanced technique for generating enhanced composite wavelet shrinkage strategies.

Wavelet compression, data fitting and approximation based on adaptive composition of lorentz-type thresholding and besov-type non-threshold shrinkage

In this study we initiate the investigation of a new advanced technique, proposed in Section 6 of [3], for generating adaptive Besov–Lorentz composite wavelet shrinkage strategies We discuss some advantages of the Besov–Lorentz approach compared to firm thresholding.

Computing n -variate orthogonal discrete wavelet transforms on graphics processing units

In [4,5] an algorithm was proposed for isometric mapping between smooth n-variate m-dimensional vector fields and fractal curves and surfaces, by using orthonormal wavelet bases This algorithm matched only the orthonormal bases of scaling functions (the “V-spaces” of multiresolution analyses) In the present communication we shall consider a new algorithm which matches the orthonormal bases of wavelets (the “W-spaces” of multiresolution analyses) Being of Cantor diagonal type, it was applicable for both bounded and unbounded domains, but the complexity of its implementation was rather high In [3] we proposed a simpler algorithm for the case of boundary-corrected wavelet basis on a bounded hyper-rectangle In combination with the algorithm for the “V-spaces” from [4,5], the new algorithm provides the opportunity to compute multidimensional orthogonal discrete wavelet transform (DWT) in two ways – via the “classical” way for computing multidimensional wavelet transforms, and by using a commutative diagram of mappings of the bases, resulting in an equivalent computation on graphics processing units (GPUs) The orthonormality of the wavelet bases ensures that the direct and inverse transformations of the bases are mutually adjoint (transposed in the case of real entries) orthogonal matrices, which eases the computations of matrix inverses in the algorithm 1D and 2D orthogonal wavelet transforms have been first implemented for parallel computing on GPUs using C++ and OpenGL shading language around the year 2000; our new algorithm allows to extend general-purpose computing on GPUs (GPGPU) also to higher-dimensional wavelet transforms If used in combination with the Cantor diagonal type algorithm of [4,5] (the “V-space” basis matching) this algorithm can in principle be applied for computing DWT of n-variate vector fields defined on the whole ℝn However, if boundary-corrected wavelets are considered for vector-fields defined on a bounded hyper-rectangle in ℝn, then the present algorithm for GPU-based computation of n-variate orthogonal DWT can be enhanced with the new simple “V-space”-basis matching algorithm of [3] It is this version that we consider in detail in the present study.

First Instances of Generalized Expo‐Rational Finite Elements on Triangulations

In this communication we consider a construction of simplicial finite elements on triangulated two‐dimensional polygonal domains. This construction is, in some sense, dual to the construction of generalized expo‐rational B‐splines (GERBS). The main result is in the obtaining of new polynomial simplicial patches of the first several lowest possible total polynomial degrees which exhibit Hermite interpolatory properties. The derivation of these results is based on the theory of piecewise polynomial GERBS called Euler Beta‐function B‐splines. We also provide 3‐dimensional visualization of the graphs of the new polynomial simplicial patches and their control polygons.

Wavelet-based isometric conversion between dimension and resolution and some of its applications

In this paper we provide an overview about an orthonormal (multi) wavelet-based method for isometric immersion of smooth n-variate m-dimensional vector fields onto fractal curves and surfaces. This method was proposed in an earlier publication by two of the authors, with the purpose of extending the applicability of emerging GPU-programming to rich diversity of multidimensional problems. Here we propose (in Section 3) several directions for upgrading the method, with respective new applications.

Hermite interpolation using ERBS with trigonometric polynomial local functions

Given a sequence of knots t0 <t1 <⋯<tn+1 , an expo-rational B-spline (ERBS) function f(t) is defined by

On the numerical performance of FEM based on piecewise rational smooth resolutions of unity on triangulations

f(t)=\sum_{k=1}^n \ell_k(t)B_k(t), \quad t\in[t_1,t_n],

Blending functions for hermite interpolation by beta-function b-splines on triangulations

where Bk (t) are the ERBS and lk (t) are local functions defined on (tk−1 ,tk+1 ). Consider the Hermite interpolation problem at the knots 0≤t1 <t2 <⋯<tn <2π of arbitrary multiplicities. In [3] a formula was suggested for Hermite-interpolating ERBS function with lk (t) being algebraic polynomials. Here we construct Hermite interpolation by an ERBS function with trigonometric polynomial local functions. We provide also numerical results for the performance of the new trigonometric ERBS (TERBS) interpolant in graphical comparison with the interpolant from [3]. Potential applications and some topics for further research on TERBS are briefly outlined.