TY - JOUR

T1 - The generalized haar spaces and their adaptive decomposition

AU - Demjanovich, A. Yuri K.

AU - Safonova, Tatjana A.

AU - Terekhov, Mikhail A.

AU - Belyakova, V.

AU - Le, Bich T.N.

N1 - Publisher Copyright: © 2020, North Atlantic University Union. All rights reserved.

PY - 2020

Y1 - 2020

N2 - This paper is devoted to the numerical information flows and adaptive decompositions of the general Haar functions connected with them. The aim of this paper is to propose an adaptive wavelet decomposition using an adaptive compression algorithm for a flow of numerical information of length M with complexity O(M) and with a given precision of Ɛ> 0. The numerical flows are associated with irregular spline grids. This paper discusses the calibration relations, the embedding of the general Haar spaces and their wavelet decompositions. The structure of the decomposition/ reconstruction algorithms are done. The cases of the finite and the infinite flows are considered. The paper discusses various methods of adaptive Haar approximations for the flow of function values. Assuming that the values of the first derivative of the approximated function is known (exactly or approximately), the complexity of using an adaptive grid is estimated for a priori specified approximation accuracy. The number of K knots in the adaptive grid determine the required amount of memory for storage of the compression results. The number of M knots of the initial grid characterizes the number of operations required to obtain the adaptive compression. In the case of access to the derivative values (or their approximations) the number of digital operations is proportional to the number M. If it does not have access to the last ones then the number of required operations has the order of M2 (in the general case). If additionally, the approximated flow is convex, then the number of required operations has the order of M log2M. In all cases the result requires the computer memory amount to be of the order of K.

AB - This paper is devoted to the numerical information flows and adaptive decompositions of the general Haar functions connected with them. The aim of this paper is to propose an adaptive wavelet decomposition using an adaptive compression algorithm for a flow of numerical information of length M with complexity O(M) and with a given precision of Ɛ> 0. The numerical flows are associated with irregular spline grids. This paper discusses the calibration relations, the embedding of the general Haar spaces and their wavelet decompositions. The structure of the decomposition/ reconstruction algorithms are done. The cases of the finite and the infinite flows are considered. The paper discusses various methods of adaptive Haar approximations for the flow of function values. Assuming that the values of the first derivative of the approximated function is known (exactly or approximately), the complexity of using an adaptive grid is estimated for a priori specified approximation accuracy. The number of K knots in the adaptive grid determine the required amount of memory for storage of the compression results. The number of M knots of the initial grid characterizes the number of operations required to obtain the adaptive compression. In the case of access to the derivative values (or their approximations) the number of digital operations is proportional to the number M. If it does not have access to the last ones then the number of required operations has the order of M2 (in the general case). If additionally, the approximated flow is convex, then the number of required operations has the order of M log2M. In all cases the result requires the computer memory amount to be of the order of K.

KW - Calibration relations

KW - Generalized Haar spaces

KW - Irregular grids

KW - Wavelet decomposition

UR - http://www.scopus.com/inward/record.url?scp=85092064234&partnerID=8YFLogxK

U2 - 10.46300/9106.2020.14.71

DO - 10.46300/9106.2020.14.71

M3 - Article

AN - SCOPUS:85092064234

VL - 14

SP - 548

EP - 560

JO - International Journal of Circuits, Systems and Signal Processing

JF - International Journal of Circuits, Systems and Signal Processing

SN - 1998-4464

ER -