Advertisement

Multidimensional Polynomial Splines

  • Dhananjay Singh
  • Madhusudan Singh
  • Zaynidinov Hakimjon
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

This paper showcases the study of multidimensional polynomial splines viewing them from a signal processing perspective. Combining the advantages of the tabular-algorithmic methods and using them for reproduction of functions and multidimensional B-splines lead to the implementation of parallel-pipeline multivariable computation structures that ensure the highest performance. Multiple multiprocessor high-performance computation structures were evaluated based on tabular-algorithmic method of processing known for top-speed and multidimensional spline approximation renowned for high performance. A parallel-pipeline computation structure was developed to implement two-dimensional basic spline approximation. This saves memory for storage of values of basic splines twice on a limited number of processors. A parallel-pipeline computation structure is proposed for recovering values of three-variable functions that have a limited number of processors and are noted for high performance. Some main characteristics of the implementation of tabular-algorithmic computation structures were obtained for processing of signals in piecewise-polynomial basics. It was proven that the tabular-algorithmic computation structures based on basic splines function faster classical polynomials of the same levels. Parabolic basic splines are 1.76 times faster and cubic basic splines 2.53 times faster than their respective classic counterparts. It has been proven that the hardware costs for implementation of the computation structures based on cubic splines are higher at the same time.

References

  1. 1.
    R.Z. Morawski, On teaching measurement applications of digital signal processing. Measurement 40(2), 213–223, ISSN 0263-2241, (2007).  https://doi.org/10.1016/j.measurement.2006.06.015CrossRefGoogle Scholar
  2. 2.
    A. Sotiras, C. Davatzikos, N. Paragios, Deformable Medical image registration: a survey. IEEE Trans. Med. Imaging 32(7), 1153–1190, (2013).  https://doi.org/10.1109/tmi.2013.2265603. (PMC. Web. 21 Sept. 2018)CrossRefGoogle Scholar
  3. 3.
    I. Garcia Marco, P. Koiran, T. Pecatte, Polynomial equivalence problems for sum of affine powers. PRoceddInt3rpp 303–310, (2018).  https://doi.org/10.1145/3208976.3208993
  4. 4.
    V. Agrawal, K. Bhattacharya, Shock wave propagation through a model one dimensional heterogeneous medium. Int. J. Solids Struct. 51(21–22), 3604–3618, ISSN 0020-7683, (2014).  https://doi.org/10.1016/j.ijsolstr.2014.06.021CrossRefGoogle Scholar
  5. 5.
    I.V. Anikin, K. Alnajjar, Primitive polynomial selection method for peseudo-random number generator. J. Phys.: Conf. Ser. 944(1), 012003, (2018). http://stacks.iop.org/1742-6596/944/i=1/a=012003Google Scholar
  6. 6.
    D. Botes, P.M. Bokov, Polynomial interpolation of few-group neutron cross sections on sparse grids. Ann. Nucl. Energy 64, 156–168, ISSN 0306-4549, (2014)CrossRefGoogle Scholar
  7. 7.
    G. Raspa, F. Salvi, G. Torri, Probability mapping of indoor radon-prone areas using disjunctive kriging. Radiat. Prot. Dosimetry 138(1), 3–19 (2010).  https://doi.org/10.1093/rpd/ncp180CrossRefGoogle Scholar
  8. 8.
    T. Hämäläinen, J. Saarinen, K. Kaski, TUTNC: a general purpose parallel computer for neural network computations. Microprocess. Microsyst. 19(8), 447–465, ISSN 0141-9331, (1995).  https://doi.org/10.1016/0141-9331(96)82010-2CrossRefGoogle Scholar
  9. 9.
    M. Viegers, E. Brunner, O. Soloviev, C.C. de Visser, M. Verhaegen, Nonlinear spline wavefront reconstruction through moment-based Shack-Hartmann sensor measurements. Opt. Express 25(10), 11514–11529 (2017).  https://doi.org/10.1364/OE.25.011514CrossRefGoogle Scholar
  10. 10.
    Y.-M. Wang, L. Ren, High-order compact difference methods for caputo-type variable coefficient fractional sub-diffusion equations in consecutive form. J. Sci. Comput. 76(2), 1007–1043 (2018).  https://doi.org/10.1007/s10915-018-0647-4MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205(2), 719–736 (2005). http://dx.doi.org/10.1016/j.jcp.2004.11.025MathSciNetCrossRefGoogle Scholar
  12. 12.
    Z.Q. Li, Z.Q. Liang, Y.B. Yan, High-order numerical methods for solving time fractional partial differential equations. J. Sci. Comput. 71, 785–803 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Dhananjay Singh
    • 1
  • Madhusudan Singh
    • 2
  • Zaynidinov Hakimjon
    • 3
  1. 1.Department of Electronics EngineeringHankuk University of Foreign Studies (Global Campus)YonginKorea (Republic of)
  2. 2.School of Technology Studies, Endicott College of International StudiesWoosong UniversityDaejeonKorea (Republic of)
  3. 3.Head of Department of Information TechnologiesTashkent University of Information TechnologiesTashkentUzbekistan

Personalised recommendations