Skip to main content

Other Useful Applications of Wavelet

  • Chapter
  • First Online:
Wavelets Theory and Its Applications

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

  • 1475 Accesses

Abstract

Many of the ideas behind wavelets have been in existence for a long time. However, wavelet analysis as we now know it really began in the mid-1980s, when it was developed to interrogate seismic signals. The wavelet has emerged as a powerful tool from time–frequency analysis (discussed in Sect. 2.3) to signal processing. The tools of signal processing can also be used for the investigation of biosignals (e.g., electrocardiogram (ECG), heart rate variability (HRV), etc.) which takes its applications to medical science. There are already good books on signal processing [1, 2]. Interest in wavelet analysis remained within a small, mainly mathematical community during the rest of the 1980s. The different applications of wavelets in science and engineering really began to take off at the beginning of the 1990s.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 39.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Mallat, A Wavelet Tour of Signal Processing (Elsevier, USA, 2009)

    Google Scholar 

  2. G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge press, USA, 1996)

    Google Scholar 

  3. M. Lounsbery, T.D. Derose, J. Warren, Multiresolution surfaces of arbitrary topological type. ACM Trans. Graphics 16, 34–73 (1997)

    Article  Google Scholar 

  4. M. Benzi, M. Tuma, A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19, 141–183 (1998)

    Article  MathSciNet  Google Scholar 

  5. T.F. Chan, W.P. Tang, W.L. Wan, Wavelet sparse approximate inverse preconditioners. BIT 37, 644–660 (1997)

    Article  MathSciNet  Google Scholar 

  6. M. Grote, T. Huckle, Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18, 838–853 (1997)

    Article  MathSciNet  Google Scholar 

  7. K. Chen, Discrete wavelet transforms accelerated sparse preconditioners for dense boundary element systems. Electron. Trans. Numer. Anal. 8, 138–153 (1999)

    MathSciNet  MATH  Google Scholar 

  8. J. Ford, Ke Chen, Wavelet-based preconditioners for dense matrices with non-smooth local features. BIT 41, 282–307 (2001)

    Article  MathSciNet  Google Scholar 

  9. B.V. Rathish Kumar, M. Mehra, Wavelet based preconditioners for sparse linear systems. Appl. Maths. Comput. 171, 203–224 (2005)

    Google Scholar 

  10. U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1999)

    Google Scholar 

  11. J. Liandrat, F. Moret-Bally, The wavelet transform:some applications to fluid dynamics and turbulece. Eur. J. Mech. B Fluids 9, 1–19 (1990)

    Google Scholar 

  12. M. Farge, K. Schneider, N.K.-R. Kevlahan, Non-gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis. Phys. Fluids 11, 2187–2201 (1999)

    Article  MathSciNet  Google Scholar 

  13. K. Schneider, N.K.-R. Kevlahan, M. Farge, Comparison of an adaptive wavelet method and nonlinearity filtered pseudospectral methods for two dimensional turbulence. Theor. Comput. Fluid Dyn. 9, 191–206 (1997)

    Article  Google Scholar 

  14. P. Charton, V. Perrier, A pseudo-wavelet scheme for the two-dimensional navier-stokes equation. Comput. Appl. Math. 15, 139–160 (1996)

    MathSciNet  MATH  Google Scholar 

  15. B.V. Rathish Kumar, M. Mehra, A time-accurate pseudo-wavelet scheme for two-dimensional turbulence. Int. J. Wavelets Multiresolut. Inf. Process. 3(4), 1–13 (2005)

    Google Scholar 

  16. R.N. van yen et al., Wavelets meet burgulence:cvs–filtered Burgers equation. Phys. D 237, 2151–2157 (2008)

    Google Scholar 

  17. V.E. Henson, W.L. Briggs, Wavelets and multigrid. SIAM J. Sci. Comput. 14, 506–510 (1993)

    Article  MathSciNet  Google Scholar 

  18. A. Brandt, A multilevel adaptive solutions of boundary value problems. Math. comput. 31, 333–390 (1977)

    Article  Google Scholar 

  19. W. Hackbusch, Iteratisolution of Large Sparse Systems of Equations (Springer, Berlin, 1993)

    Google Scholar 

  20. A. Limon, H. Morris, A multilevel adaptive solver based on second-generation wavelet thresholding techniques. Numer. Linear Algebra Appl. 13, 251–273 (2006)

    Article  MathSciNet  Google Scholar 

  21. S.G. Mikhlin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics, and Technology (Macmillan, New York, 1964)

    MATH  Google Scholar 

  22. S. Esmaeili, M. Shamsi, M. Dehghan, Numerical solution of fractional differential equations via a volterra integral equation approach. Central Eur. J. Phys. 11, 1470–1481 (2013)

    Google Scholar 

  23. N. Morita, Integral equation formulation for heat conduction in closed regions with arbitrarily shaped boundary surfaces. J. Appl. Phys. 56, 00 (1984)

    Google Scholar 

  24. M.A. Abdou, Fredholm volterra integral equations and generalised potential kernel. J. Appl. Math. Comput. 131, 81–94 (2002)

    Article  Google Scholar 

  25. M.A. Abdou, Fredholm volterra integral equations with singular kernels. J. Appl. Math. Comput. 137, 231–243 (2003)

    Article  MathSciNet  Google Scholar 

  26. A. Adawi, F. Awawdeh, A numerical method for solving integral equations. Int. J. Contemp. Math. Sci. 4, 485–496 (2009)

    MathSciNet  MATH  Google Scholar 

  27. K. Atkinson, Survey of Numerical methods for the solution of Fredholm integral equations of second kind (SIAM, 1976)

    Google Scholar 

  28. K. Atkinson, F. Potra, Galerkin’s method for nonlinear integral equations. SIAM J. Numer. Anal. 24, 1352–1373 (1987)

    Article  MathSciNet  Google Scholar 

  29. A.M. Frieze, R. Kannan, S. Vempala, Fast monte-carlo algorithms for finding low-rank approximations, in Proceedings of 39th IEEE Symposium on Foundation of Computer Science (1998), pp. 370–378

    Google Scholar 

  30. P. Drineas, R. Kannan, M.W. Mahoney, Fast monte carlo algorithms for matrices i: approximating matrix multiplication. SIAM J. Comput. 36, 132–157 (2006)

    Article  MathSciNet  Google Scholar 

  31. B. Alpert, V. Rokhlin, A fast algorithm for evaluation of legendre expansions. SIAM J. Sci. Stat. Comput. 12, 158–179 (1991)

    Article  MathSciNet  Google Scholar 

  32. H. Cai, A fast Galerkin method for solving integral equation of second kind with weakly singular kernels. J. Appl. Math. Comput. 32, 405–415 (2010)

    Article  MathSciNet  Google Scholar 

  33. Z. Chen, Y. Xu, H. Yang, Fast collocation methods for solving ill-posed integral equations of the first kind. Inverse Probl. 24 (2008)

    Google Scholar 

  34. G. Vainikko, A. Kivinukk, J. Lippus, Fast solvers of integral equations of second kind: wavelet methods. J. Complex. 21, 243–273 (2005)

    Article  MathSciNet  Google Scholar 

  35. G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms. Commun. Pure Appl. Math. XLIV, 141–183 (1991)

    Google Scholar 

  36. B. Alpert, A class of bases in \(l^2\) for the sparse representaion of integral operators. SIAM J. Math. Anal. 24, 246–262 (1993)

    Google Scholar 

  37. E. Hairer, C. Lubich, A. Schlichte, Fast numerical solution of nonlinear volterra convolution equations. SIAM J. Sci. Stat. Comput. 6, 532–541 (1985)

    Article  MathSciNet  Google Scholar 

  38. B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, Wavelets for the fast solution of second kind integral equations. SIAM J. Sci. Comput. 14, 159–184 (1993)

    Article  MathSciNet  Google Scholar 

  39. T. Von Petersdorff, C. Schwab, Wavelet discretization of parabolic integrodifferential equations. SIAM J. Numer. Anal. 41, 159–180 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mani Mehra .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mehra, M. (2018). Other Useful Applications of Wavelet. In: Wavelets Theory and Its Applications. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-2595-3_11

Download citation

Publish with us

Policies and ethics