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Wavelets & Applications to PDE

Wavelets & Applications to PDE. Ref: J. C. Goswami & A. K. Chan, Fundamentals of Wavelets – Theory, Algorithms, and Applications . chs.5-7,10, John Wiley & Sons, 1999. Continuous Wavelet Transform. vanishing moments of order m time-frequency window: b: location of wavelet window

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Wavelets & Applications to PDE

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  1. Wavelets & Applications to PDE Ref: J. C. Goswami & A. K. Chan, Fundamentals of Wavelets – Theory, Algorithms, and Applications. chs.5-7,10, John Wiley & Sons, 1999.

  2. Continuous Wavelet Transform vanishing moments of order m time-frequency window: b: location of wavelet window 1/a: a measure of frequency Inverse wavelet transform Fig.4.7 admissibility condition

  3. Example of CWT Morlet wavelet

  4. Discrete Wavelet Transform Wavelets: Orthogonality: Wavelet series:

  5. Orthogonal Basis for any integer l

  6. Multi-Resolution Analysis • Approximation Subspace, • Wavelet Subspace, (t): scaling function (t): wavelet t t t

  7. Splitting of MRA Subspaces As+1 As Ws As-1 Ws-1 As-2 Ws-2 Orthogonal decomposition for integer l

  8. Construction of Orthonormal Wavelets Two-scale relation Note: Decomposition relation

  9. Haar Wavelets for 0<t<1 elsewhere for 0<t<1/2 for ½<t<1 elsewhere Two-scale relation: Decomposition relation:

  10. Shannon Scaling Functions

  11. Meyer Scaling Functions

  12. Battle-Lemarie Scaling Functions

  13. Wavelet Decomposition Algorithm as+1 as h0 2 h1 2-point decimator x[n] y[n] 2 2 y[n]=x[2n] ws

  14. Reconstruction Algorithm as+1 as g0 2 2-point interpolator g1 y[n] x’[n] 2 x’[n] = y[n/2] for even n = 0 for odd n 2 ws

  15. Denoising Noisy Data • Wavelet shrinkage & thresholding • Transformed image to wavelet domain using Coiflets with three vanishing moments • Applied a threshold at two standard deviations • Inverse-transformed image to signal domain “Before” and “after” illustrations of a NMR signal • Ref. D. Donoho, “Nonlinear wavelet methods for recovery of signals, densities, & spectra from indirect & noisy data,” Different Perspectives on Wavelets, Proc. of Symposia in Applied Mathematics, Vol. 47, I. Dauhechies ed., 1993, pp.173-205

  16. Integral Equation e.g., TM scattering off PEC

  17. Method of Moments xm full matrix Point matching Galerkin’s method

  18. Use of Fast Wavelet Algorithm Ref: R. L. Wagner, P. Otto, & W. C. Chew, “Fast waveguide mode computation using wavelet-like basis functions,” IEEE MGWL, pp. 208-210, 1993 H. Kim & H. Ling, “On the application of fast wavelet transform to the integral-equation of electromagnetic scattering problems,” MOTL, pp. 168-173, 1993. Aw: nearly sparse

  19. Direct Application of Wavelets Ref. J. C. Goswami, A. K. Chan, & C. K. Chui, “On solving first-kind integral equations using wavelets on a bounded interval,” IEEE AP-43, pp.614-22, 1995 Vanishing moments property  [A,] : full, [A,]: almost diagonal, all others near zero

  20. Numerical Examples • Circular cylindrical surfaces : threshold parameter Relative error m=2: linear spline m=4: cubic spline Percent sparsity

  21. Numerical Example (cont’)

  22. S-MRTD I(z,t) Ref: M. Krumpholz & L. Katehi, “MRTD: new time-domain schemes based on multi-resolution analysis,” IEEE MTT-44, pp.555-571, 1996 P.D.E. V(z,t) h0(t) Fig.1 basis -1/2 1/2 Battle-Lemarie scaling function

  23. Numerical Result • For 3D FDTD; • Stability cond.: t  tmax=0.368z/c • Choose a smaller t, sayt = tmax/5, space division can be as coarse as /2 without causing large dispersion error

  24. Some Home Pages • http://www.c3.lanl.gov/~brislawn/main.html • http://www.mat.sbg.ac.at/~uhl/wav.html • http://playfair.stanford.edu/~wavelab/ (wavelab Matlab software) • http://www.cs.ubc.ca/nest/imager/contributions/bobl/wvlt/top.html (U. British Columbia CS Dept.) • http://www.mathsoft.com/wavelets.html (Matlab wavelet resources) • http://www.math.scarolina.edu:/~wavelet/ • http://www.best.com/~agraps/current/wavelet.html Ref.: A. Graps, “An introduction to wavelets,” Signal & Image Processing, IEEE Computational Science & Engineering, pp.50-59, 1995.

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