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Amartunga Pde

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    0 1 2 32

    1

    0

    1

    2

    0 1 2 32

    1

    0

    1

    2

    0 1 2 32

    1

    0

    1

    2

    0 1 2 32

    1

    0

    1

    2

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    8 6 4 2 0 2 4 6 80

    2

    4

    6h[k]

    8 6 4 2 0 2 4 6 80

    2

    4

    6h[k] = h[0k]

    8 6 4 2 0 2 4 6 80

    2

    4

    6h[3k] : in general h[nk]

    -

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    x

    Outermost

    Pole

    REAL

    IMAG

    Unit Circle

    ROC

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    x

    Outermost

    Pole

    REAL

    IMAG

    Unit Circle

    ROC

    REAL

    IMAG

    Unit Circle

    ROCa 1

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    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

    0.5

    1

    1.5

    2

    2.5

    Angular frequency (normalized by pi)

    Fouriertransformm

    agnitude

    Frequency response magnitude for Daubechies 6tap filters

    Lowpass Highpass

    c [k]m

    H(z) 2

    G(z) 2d [k]

    m-2

    H(z) 2c [k]

    m-1

    G(z) 2

    d [k]m-1

    c [k]m-2

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    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

    0.5

    1

    1.5

    2

    Angular frequency (normalized by pi)

    Fouriertransformm

    agnitude

    Haar frequency response

    Lowpass Highpass

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    1 0 1 2 32

    0

    2

    4

    6

    8

    10

    12

    k

    c_

    m1[k]

    .. Circular convolution

    . Symmetric extension (with duplication)

    Symmetric extension (without duplication)

    Wavelet extrapolation

    1 0 1 2 33

    2.5

    2

    1.5

    1

    0.5

    0

    0.5

    1

    k

    d_

    m1[k]

    .. Circular convolution

    . Symmetric extension (with duplication)

    Symmetric extension (without duplication)

    Wavelet extrapolation

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    101

    102

    103

    103

    104

    105

    106

    10

    7

    108

    L

    conditionnumber

    o with preconditioning

    + without preconditioning

    1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

    0.5

    1

    1.5

    2

    2.5x 10

    5

    P

    conditionnumber

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    101

    102

    103

    104

    103

    104

    105

    106

    107

    108

    L

    numberofoperations

    o hierarchical algorithm

    + nonhierarchical algorithm

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    28 2 1>.:. AlIlaral UU!-(ll't.-Galerkiu Solutiou of \\"aw El111atiou

    and adaptive solution strategies. Tlw discussion of tlw p l ' ( ~ c e d i n g sections suggl'sts sl'wralgeneral solution strategies f(Jr solving m u l t i s c a l ( ~ equations.

    1. Nou-h:ic'f'(J,1'(;hical (J,pp1'O(J,che.'i. In a non-hierarchical approach. tlw goal is t.o COlllputI' tlw lllllnerical solution. 1/,1/1(:1:). for a single vahw of 1T/.. wlwl' l l l ' l ~ c l e c lin order to determine '/lin (;r.). A non-hierarchicaJ approach can })(' i l l l p k n w n t . ( ~ d usingeither a single scale formulation or a lllultiscah f(H'lnulation. so consideration lH'edsto })(' given t o whe ther th e f()l'lnation of the nlllltiscak equations is . i u s t i f i ( ~ d . I f wehave (J, IJ7'i01"i knowledge of the behavior of the solution. as f()r e X l u n p h ~ in t.he caseof s t l ' ( ~ S S concentrations around a hole in a stressed Plastic plate. then tlw forlllationof the multiseale equations will al low us to eliminate sonH' of tlw degrpes of f r e ( ~ d ( J l l lwhkh (10 not lie within the region of high gradient.13eylkin. Coifman and Rokhlin [22] have devPloped fast algorithms f(n' tlw applicatiollof the multiHcale wavelet-Galerkill diffenmtial operator (awl othpl' oIH'rators) 1.0 al'-

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