Adaptive Haar Type Wavelets on Manifolds
- PDF / 249,313 Bytes
- 17 Pages / 594 x 792 pts Page_size
- 48 Downloads / 188 Views
Journal of Mathematical Sciences, Vol. 251, No. 6, December, 2020
ADAPTIVE HAAR TYPE WAVELETS ON MANIFOLDS Yu. K. Dem’yanovich St. Petersburg State University 28, Universitetskii pr., Petrodvorets, St. Petersburg 198504, Russia [email protected]
UDC 517.9
We consider embedded Haar type spaces associated with cell subdivisions of a smooth manifold. We use an adaptivity criterion connected with a nonnegative set function possessing certain monotonicity properties. We propose an algorithm for constructing embedded spaces satisfying the adaptivity criterion. To construct the wavelet decomposition, we apply the nonclassical approach and obtain the adaptive wavelet decomposition of the Haar type space on the manifold. Some model examples are given. Bibliography: 6 titles.
1
Introduction
Adaptive approximations are used to significantly speed up computations and improve the accuracy of approximations of functions by the finite element method, grid method, and wavelet processing of numerical information flows (cf. [1]–[5]). In the case of spline and finite-element approximations, the adaptivity is achieved by modifying the grid and related approximation elements provided that the approximating spaces remain embedded. The embedded spaces are associated with embedded grids in the one-dimensional case and embedded subdivisions of the domain of functions under consideration in the multi-dimensional case. The adaptivity of methods and algorithms means that their structure is determined by the data. In the above-mentioned cases, the adaptivity property appears if the system of embedded spaces is restructured, depending on incoming data. Moreover, such a restructuring can be related to some class of data, as well as an individual datum (a function or a numerical flow). From this point of view, the development of adaptive methods and algorithms is far from complete in most of the areas. A certain progress has been achieved with the nonclassical approach to the wavelet (spline–wavelet) decomposition of numerical information flows in the one-dimensional case (cf. [6]). However, multi-dimensional adaptive approximations are required in most of such methods. In this paper, we construct multi-dimensional adaptive Haar type approximations and the corresponding wavelet decomposition. We deal with embedded Haar type spaces associated with cell subdivisions of smooth manifolds. The adaptivity criterion is connected with a nonnegative set function, called the pseudomeasure, possessing certain monotonicity properties. We propose an algorithm for constructing embedded spaces satisfying the adaptivity criterion. To construct Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 23-37. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2516-0797
797
the wavelet decomposition, we use the nonclassical approach. As a result, we obtain adaptive Haar type wavelets on the manifold under consideration. Some examples are also given to illustrate the results obtained in the paper.
2
Preliminaries
We consider a smooth n-dimen
Data Loading...
