Quadrature integration for orthogonalwavelet systems

Bruce R. Johnson,a Jason P. Modisette,b

Peter J. Nordlander,b and James L. Kinsey a

a Department of Chemistry and Rice Quantum InstituteRice University

Houston, TX 77251-1892

b Department of Physics and Rice Quantum InstituteRice University

Houston, TX 77251-1892

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Abstract

Wavelet systems can be used as bases in quantum mechanical applications where

localization and scale are both important. General quadrature formulae are developed

for accurate evaluation of integrals involving compact support wavelet families, and their

use is demonstrated in examples of spectral analysis and integrals over anharmonic

potentials. In contrast to usual expectations for these uniformly-spaced basis functions, it

is shown that nonuniform spacings of sample points are readily allowed. Adaptive

wavelet quadrature schemes are also presented for the purpose of meeting specific

accuracy criteria without excessive oversampling.

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I. Introduction

There is now an extensive wavelet literature within a variety of applications, e.g.,

digital signal processing, image processing, fingerprint classification, remote sensing,

target recognition, etc. 1-7 There has been some exploration of the use of wavelets in

chemical applications,8-20 but of only limited extent to date. Nevertheless, wavelet

methods promise a distinct advantage over Fourier methods for problems requiring

different levels of resolution in different locations, e.g., spectral analysis and

compression, molecular calculations of electronic or nuclear motion, etc. Regarded as

basis functions for calculation or analysis of quantum wave functions, wavelet systems

offer certain unusual properties. In fact, only with the introduction of the orthonormal

compact support wavelets by Daubechies1,3 did it become clear that there exist basis sets

that are capable of being simultaneously localized, multiscale, multicenter, and

orthonormal. The purpose of this paper is to provide efficient and accurate quadrature

methods for required integrals over compact support wavelet families and their

generalizations.

Such functions do not have simple closed forms, but are instead defined by

recursion between different scales. The original Daubechies bases are generated by two

related functions, the scaling function φ(x) and the wavelet ψ(x). Both are nonzero only

for 0 < x < L–1, where L is an even integer. Figure 1 shows these functions for L = 8.

From these functions one obtains others that are just copies widened by factors 2j and

translated by steps k 2j, where j and k are integers,

φkj x( ) = 2− j /2 φ 2− j x−k( ) , (1)

ψ kj x( ) = 2− j /2 ψ 2− j x− k( ) , (2)

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with φ00 = φ , ψ 0

0 = ψ . By construction, the set of scaling functions at a particular

resolution level j, φkj{ }, form a basis which is orthogonal though not complete. A more

complete set φkj −1{ } is also orthogonal but is twice as dense. The difference in detail

between these two bases is spanned by the set of wavelets at level j, so that

φkj −1{ } = φ k

j{ }⊕ ψ kj{ } . This is expressed in terms of the recursion relations,

φkj x( ) = h ′ k φ2 k+ ′ k

j−1 x( )′ k = 0

L −1

∑ , (3)

ψ kj x( ) = g ′ k φ2 k + ′ k

j −1 x( )′ k =0

L −1

∑ , (4)

where the constants are related by gk = − 1( )k +1hL− 1− k .2 Ultimately, every quantity that

must be calculated boils down to some function of the hk, so that the latter take on a

central role in calculations. An orthonormal multiresolution basis can be comprised of

scaling functions φkJ{ } on a coarsest level j = J and wavelet functions ψ k

j{ } from this

coarsest level down to a finest scale j = 0 which provide the various additional octaves

of detail. The approximate expansion of a function f in this basis then takes the form

f x( ) ≈ ψ kj f ψ k

j x( )k

∑j = 0

J

∑ + φkJ f φ k

J x( )k

∑ . (5)

Detail functions ψ kj{ } at each scale which are unimportant (have small expansion

coefficients ψ kj f ) may simply be dropped, leaving a finite basis with customized

resolution. This immediately generalizes to multidimensional Cartesian cases by using

products of such basis functions.

For quantum applications, the needs arise to calculate the projection integrals

φ kj f , ψ k

j f for the vector basis and the matrix elements φ ′ k ′ j f φ k

j , φ ′ k ′ j f ψ k

j ,

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ψ ′ k ′ j f ψ k

j , where f may be a general function of x . Use of the recursion relations in

Eqs. (3) and (4) allows all of these integrals to be expressed in terms of integrals with a

scaling function (or functions) on a single finest scale, but accurate methods are still

required for calculation of the remaining set of integrals. One method mentioned by

Daubechies3 (p. 166) for calculation of the vector projection integrals uses Fourier

convolution. While this method is not widely used, Wei and Chou17 have adapted it to

their density functional calculations. Alternatively, a quadrature method with uniformly-

spaced sample points was derived for the vector elements by Sweldens and Piessens.21,22

The latter method was used by Modisette, et al.,15 for wavelet expansion of a semi-

singular double-well potential; by expanding the wave functions in the same basis, matrix

elements could then be evaluated in terms of exactly-calculable "connection coefficients,"

or integrals of products of three functions from the wavelet basis.7,23,24 The portability of

this approach is complicated by the large number of connection coefficients generally

required, however. A recent adaptive wavelet collocation method developed for O(N)

solution of PDE's 25 directly addresses calculation of matrix elements, though a linear

system of equations depending on the sample grid must generally be solved.

In the present paper, parallel treatments are given of polynomial-based quadrature

formulae for direct calculation of φ kj f and φ ′ k

j f φ kj . An unusual aspect is that there

is no requirement that sample points lie on a uniformly spaced grid. For a given uniform

or irregular grid, the quadrature weights are explicitly expressed in terms of Lagrange

interpolating polynomials and polynomial moments of the scaling functions. The latter

are independent of the grid choice and need only be calculated once. Furthermore, use of

the scaling function recursion in Eq. (3) leads naturally to adaptive evaluation whereby a

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chosen accuracy goal can be reached without excessive over-sampling. Straightforward

generalizations of these developments also hold for biorthogonal compact support

wavelet systems.26

II. Projection Integral Quadrature

For a given j and k, a change of variables transforms the general vector integral

for scaling functions into a standardized form,

φ kj f = 2− j /2 dx φ 2− j x−k( )∫ f x( ) = 2 j / 2 dx φ x( )∫ f 2 j x + k( )[ ]. (6)

Thus it is only necessary to consider the j = k = 0 case, for which the integral is

approximated by quadrature,

dx φ x( )∫ f x( ) ≈ ω q f xq( )q =1

r

∑ . (7)

The only restriction imposed on the sample points is that the xq are assumed to be

ordered, x1 < x2 … < xr. One can specify theωq in terms of the xq by requiring that Eq.

(6) be exact for f equal to a polynomial of order r–1 or less:

ωq xqp

q= 1

r

∑ = mp ≡ dx φ x( )∫ x p , 0 ≤ p ≤ r −1. (8)

The general solution to these equations (see Appendix A) is

ωq =mp

p!p= 0

r− 1

∑ Lr,qp( ) 0( ) , (9)

using the Lagrange interpolating polynomials

Lr , q x( ) =x − x ′ q

xq − x ′ q ′ q ≠ q

r

∏ (10)

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and their derivatives Lr , qp( ) at x = 0 . The Lagrange polynomials are the coefficients for

interpolation of smooth functions,27

Lr ,q x( ) f xq( )q= 1

r

∑ ≈ f x( ), (11)

with equality holding for f(x) a polynomial in x of order less than r. Their connection

with numerical quadrature is well known,28,29 although the simple result Eq. (9) (which is

not really restricted to wavelet systems) appears to be new. Lagrange polynomials are

also familiar in wavelet contexts from their use in interpolatory refinement schemes, e.g.,

the symmetric iterative interpolation process of Deslauriers and Dubuq.30 The latter

scheme uses a "fundamental solution" that was subsequently shown31 to coincide with

the autocorrelation function of the compact support scaling functions (see Appendix B).

The Lagrange polynomials29,32 and their derivatives are easily evaluated once the

quadrature nodes are specified. As for the moments mp, their calculation may be

accomplished by the simple recursive equation derived by Gopinath and Burrus,33

mp = 1− 2− p( )−1 p

′ p

m ′ p µp− ′ p

′ p = 0

p −1

∑ , (12)

where the µp are discrete moments of the scaling coefficients of Eq. (3),

µp =1

2hk k p

k =0

L−1

∑ . (13)

The leading moment m0 = 1 as a matter of normalization, providing the starting point for

the recursion. (Not all of the resulting moments mp are independent of each other, as is

discussed in Appendix B.) The moments are thus determined completely by the scaling

coefficients hk, whereas the quadrature coefficients are determined in terms of both the hk

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and the nodes xq. In calculations for a given compact support basis, the moments need

only be calculated once.

Sweldens and Piessens21 considered the important case of uniformly-spaced xq.

Such an arrangement allows the sample points for scaling functions of adjacent k to

overlap, requires only a single set of quadrature coefficients for all k, minimizes the

number of function evaluations needed, and yields a higher order error of O ∆x r( ), where

∆x is the spacing. This is a Newton-Cotes type of quadrature28 for which is allowed a

single shift of the entire set of sample points with respect to the origin; this degree of

freedom can be used to make the quadrature exact for polynomials up to order r instead

of r – 1. One may achieve even greater accuracy by defining a Gauss quadrature for

which all r of the xq are chosen (non-uniformly) so as to exactly include polynomials up

to order 2r – 1; in this case there is no overlap between the sample points for adjacent

scaling functions. In the present work, the disposition of the xq is left open for purposes

of adaptability.

For compact support wavelets, it is known21,33 that m2 = m12, although higher

moments are not simple powers of m1 (Appendix B). Introducing the differences

∆mp = mp −m1p , (14)

Eq. (8) can be rearranged,

ωq =m1

p

p!p= 0

r − 1

∑ Lr, qp( ) 0( )+

∆mp

p!p= 3

r − 1

∑ L r,qp( ) 0( )

= Lr ,q m1( ) +∆mp

p!p= 3

r − 1

∑ Lr, qp( ) 0( ) , (15)

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where the first series has been recognized as the Taylor expansion of the Lagrange

interpolation polynomial evaluated at m1. To the extent that the second series can be

ignored, Eqs. (7) and (11) show that

dx φ x( )∫ f x( ) ≈ Lr ,q m1( ) f xq( )q= 1

r

∑ ≈ f m1( ) (16)

This directly reflects the fact determined by Gopinath and Burrus that a one-point

quadrature using m1 as the sample point has O ∆x3( ) error. Higher order error generally

requires a multi-point quadrature. An exception is found in the Coifman wavelet

family,34,35 where m1 = m2 = …= mM -1 = 0 for chosen integer M, leading to a one-point

quadrature error of O ∆x M( ) .

For many applications, it is worthwhile to use a more flexible framework than

orthogonal wavelet families. In biorthogonal families,26 the functions φ and ψ are

orthogonal to dual functions ˜ φ and ˜ ψ rather than to themselves. The scaled and

translated versions of these functions, ˜ φ kj and ˜ ψ k

j , satisfy their own scaling relations, cf.,

Eqs. (3) and (4),

˜ φ kj x( ) = ˜ h ′ k

˜ φ 2k+ ′ k j−1 x( )

′ k ∑ , (17)

˜ ψ kj x( ) = ˜ g ′ k

˜ φ 2k+ ′ k j−1 x( )

′ k ∑ , (18)

The extra freedom within biorthogonal families allows, for instance, φ and ψ to possess

definite symmetry, an impossibility for the orthogonal compact support families.1,3 It

also provides the framework for the Lifting scheme introduced by Sweldens36,37 for

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purposes of adaptable multiresolution analyses. The general expansion of a function in a

biorthogonal basis may be expressed as

f x( ) ≈ ˜ ψ kj f ψ k

j x( )k

∑j = 0

J

∑ + ˜ φ kJ f φ k

J x( )k

∑ (19)

rather than Eq. (5). Similarly the analogs of Eqs. (7) and (8) are

dx ˜ φ x( )∫ f x( ) ≈ ˜ ω q f xq( )q =1

r

∑ , (20)

˜ ω q xqp

q =1

r

∑ = ˜ m p ≡ dx ˜ φ x( )∫ x p , 0 ≤ p ≤ r −1. (21)

The structure of the equations is otherwise unchanged so that the quadrature coefficent

expressions immediately adapt to the biorthogonal wavelet case.

III. Wavelet Transform of UV Ozone Absorption Spectrum

One of the potential uses of wavelet analysis is to molecular spectra with complex

features corresponding to action on different scales. The ozone molecule provides a

particular example in its Hartley UV absorption band (see Johnson, et al.,38-40 and

references therein), as shown in Fig. 2. Photoexcitation to a steeply–repulsive

dissociative region of the potential energy surface for the ˜ D 1B2 state leads to a single

intense band which is continuous across several thousand cm–1. In addition, smaller

features appear across much of the band as minor irregular structure with average spacing

250 cm–1. Under Fourier transformation, the time domain autocorrelation function

exhibits completely–resolved features at shorter (< 10 fs) and longer (40–150 fs) times

corresponding, respectively, to direct and indirect dissociation contributions. Dynamical

analysis of the indirect contribution, corresponding to Franck-Condon-region exit and

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return, has progressed in recent years with the aid of both classical40 and quantum41-43

propagations. More definitive information is currently being obtained by Le, et al.,44, in

the form of detailed Raman excitation profiles for the different accessible final

vibrational states; these exhibit structure that reflects upon the same excited-state

dynamics, but in a vibration-specific manner.

One limitation of the Fourier transform (FT) is that it completely defocuses

information about the old variable ν. The tremendous breadth of the total band implies

that there are many different energetic regimes mixed together in the autocorrelation

function, and that a joint time-frequency analysis with partial resolution in both variables

is to be preferred. The first such application to ozone used a sliding window on the

spectral data prior to the FT operation,39 thereby allowing the following of time features

between partially-localized regions of frequency. Gabor45 and Vibrogram/Husimi46

transform techniques have also proven to be valuable alternatives for time-frequency

analysis of classical and quantum dynamics in other systems, though they have not been

applied to ozone. In the present work, an alternative separation of the distinct dynamical

regimes is obtained through wavelet multiresolution analysis of the absorption spectrum.

Using one of the orthonormal Daubechies wavelet families (L = 8), it is possible

to calculate projection integrals of translated and scaled scaling functions on a level j = 0,

dν1

∆νφ

ν − ν0

∆ν− k

∫ σ ν( )

= ∆ν dx φ x( )∫ σ ν0 + k∆ν + x∆ν( )

≈ ∆ν ωqσ ν0 + k∆ν + xq∆ν( )q=1

r

∑ . (22)

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An absolutely typical initial complication arises from the fact that the absorption cross-

section σ was measured47 in uniform wavelength steps of 0.1 Å, i.e., the frequency

variable ν has non-equidistant spacings. One may perform interpolation of different

types in order to obtain uniform steps in ν, but this is unnecessary if the quadrature

described above is used. A total of r irregularly–spaced experimental points (with

average spacing close to ∆ν) were selected from within the support of each φk0 and the

associated projections then calculated by the rules described above. With the average

spacing chosen to be 3 cm–1, comparison of integrals for r = 7 and r = 8 generally showed

convergence to 5 significant figures. This reflects, of course, a combination of both the

inherent quadrature errors and any point-to-point experimental uncertainties.

Using the recursions in Eqs. (3) and (4), the projections onto an orthonormal basis

of wavelets ψ kj{ }, j ≤ J, and scaling functions φk

J{ } on level J may then be calculated in

the normal fashion. In this way, the spectrum is decomposed simultaneously into

different scales and positions. The smoothed version of the absorption cross-section

shown in Fig. 2 was obtained by summing over all k the projections onto the φkJ for J = 8.

This partial reconstruction process requires an algorithm that accurately calculates the

values of the scaling functions at particular points (see Burrus, et al.6). Figure 3 shows

the corresponding summed ψ kj projections, representing the differences in detail between

different scales j. It is seen that the spectral structure is separated from the smooth

component and is concentrated primarily in three octaves of scale change spanned by the

wavelets for j = 5–7. One obtains localized frequency information even from these

summed quantities, e.g., it is clear from j = 5 that higher-frequency components of the

structure are more prominent on the low-frequency side of band center.

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The corresponding wavelet coefficients shown in Fig. 4 of course contain much

more detailed local frequency information. All of the information so obtained is

preserved under Fourier transformation of the wavelets and scaling functions to the time

domain, so that one can determine how the longer-time features in the autocorrelation

function change with both scale and general frequency region, subject only to the

constraints of the Uncertainty Principle. (Visual analysis, e.g., for comparison with the

windowed FT analysis of Johnson and Kinsey,39 would be simpler using the Continuous

Wavelet Transform3 although the latter is redundant and less efficient to implement.) A

more important advantage of this type of decomposition should be realized in the analysis

of the Raman experiments in ozone. Transformation of a Raman excitation profile to the

time domain is not a simple linear operation since the phase is not measured and must be

recovered using Hilbert transform techniques.48-52 The wavelet decomposition should

prove useful here for separation of the early and later time domain features; from these

results the shapes of the excited state potential and transition dipole in the Franck-Condon

geometry region and beyond can be determined53,54 (see also Shapiro55).

It merits emphasis that the irregular sampling is accommodated easily in this

application. This type of irregularity stands in contrast to that in the Lifting scheme,36,37

where the biorthogonal basis itself is tailored. In the present case, the underlying wavelet

basis is regularly-spaced as usual, but the sample points are not tied to that arrangement.

The numerical quadrature performed for the expansion coefficients on the finest scale

does not change the O N( ) nature of the wavelet recursion, but does increase the work

needed for initialization by a factor proportional to the quadrature order r. In the

language of filter banks used in engineering contexts, this corresponds to a "prefiltering"

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operation on the data,5,56,57 and the quadrature scheme used here represents a polynomial

prefilter for irregularly-spaced samples.

IV. Morse Potential Expansion and Adaptive Quadrature

In wavelet applications, easy compression is one of the usual goals. If a particular

wavelet coefficient is calculated to be small, that element of the basis may be eliminated.

On the other hand, efficiency and/or storage in numerical applications may require that

these coefficients should not even be calculated if they are known to be small, for

instance, if overlapping elements from coarser scales have already been excluded. This

argues for a refinement approach3,30 where finer scales are used only as needed by the

problem under investigation. In this section, the results of Section II are embedded in an

adaptive wavelet quadrature.

A prototypical anharmonic molecular potential function is given by the Morse

potential,

V x( ) = De exp − 2a x − x e( )[ ]−2De exp − a x − xe( )[ ], (23)

characterized by a steeply repulsive inner wall, a more slowly-varying neighborhood of

the minimum, and a flat outer region. Parameters appropriate to molecular hydrogen are

used, a = 1.9425 Å–1, xe = 0.7416 Å, De = 4.7446 eV. Projections are calculated of this

function on the scaling functions from the L = 10 Daubechies system,

φkJ x( ) = 2 J λ( )−1/2

φ x / 2J λ − k( ) , (24)

with spacing 2J λ = 0.4 Å (λ and J are not independent parameters here).

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For the sake of simplicity (but not necessity), the uniformly-spaced quadrature

associated with the basis functions on each level is used, xq = k + q −1( ) 2 j λ . The same

quadrature weights are therefore used for each functionφkj , and r – 1 of the points for

neighboring values of k overlap. In Fig. 5 the absolute values of the differences ∆

between integrals for orders r and r + 1 are displayed; it is very clear that significant lack

of convergence occurs for low r when only the coarsest quadrature is used. The highly

inaccurate case r = 1 is the usual basis of the Fast Wavelet Transform. Given the

O ∆x r( )error, proceeding to finer scales unfortunately provides only slow linear

convergence with respect to spacing (or cubic in the special case that the sampling is

moved to the position of the first moment). Errors are exponentially reduced for higher r

in the range covered, r ≤ 10. (Even higher orders r are allowed but, as is generally the

case with Newton-Cotes quadratures, can eventually lead to divergence.) The asymptotic

rates of convergence (slopes on the right-hand side of the semi-log plot) all conform. The

absolute errors, however, differ systematically for different k. As is evident from the top

of Fig. 6, the source of the differences is the strongly varying behavior of the anharmonic

Morse potential over the region spanned by the different φkj 's.

Greater accuracy for level J can be obtained by evaluation of the scaling function

integrals on levels J – 1, J – 2, …, until the desired level of accuracy is achieved,

followed by use of Eqs. (3) and (4) to recur back to level J . The best procedure is clearly

to use a value of r large enough to obtain satisfactory convergence rates and thus to

avoid needlessly high numbers of recursions, but coupled with selectivity over which

regions are subjected to refinement. Convergence may be measured either by (i)

comparison of the numerical results using quadrature at level j with those of quadrature at

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level j – 1 followed by recursion back to j, or (ii) direct quadrature at level j using orders

r and r + 1 as done for the highest level in Fig. 5. The latter (simpler) choice was found

to be adequate in the calculations here, though this need not always be the case.

In the bottom section of Fig. 6 are shown the results for a demanding coarsest-

level convergence tolerance of 10–8 eV. For each level of refinement, the tolerance is

correspondingly reduced by a factor of 2. The markers in each row correspond to the

left-hand sides of included scaling functions; these functions are shown explicitly for the

coarsest level j = 4. On the finest level required, j = 0, the adaptive procedure has

reduced the 199 potentially-required integrals to only 20. One may obtain greater

compression for a smaller choice of L, although this comes at the price of reduced

smoothness in the basis functions.

Efficiency in the calculations can be gained by utilizing the fact that only

approximately half of the sample points required on each finer level are new.

Furthermore, each φkj for which the integral is recalculated using refinement requires the

integrals for φ2kj −1 , φ2k +1

j −1 , …, φ2k +L − 1j −1 , but there is a multiplexing advantage sinceφk ±1

j

require L – 2 of the same integrals, etc. Finally, when performing calculations with a

basis consisting of functions from different levels of resolution, many of the intermediate

integrals calculated will be of use for basis function integrals at more than one scale j .

Thus, the intermediate results can be stored and re-used to speed execution time. If

storage becomes an issue (e.g., in multidimensional wavelet calculations), however, it

may be more desirable to allow for a certain amount of recalculation of integrals to trade

speed for storage savings.

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V. Matrix Element Quadrature

The considerations above have only considered integrals linear in the vector basis.

Bilinear integrals are standard in quantum mechanics, where one typically evaluates

matrix elements of kinetic and potential energy operators between two basis functions.

While kinetic energy operators in a compact support basis may be calculated by

straightforward methods,23,52 the same is not true for matrix elements of a potential

which is some general function of the coordinate (or, in multiple dimensions,

coordinates). Nevertheless, the quadrature methods described above translate

straightforwardly to the case of such bilinear integrals.

As before, the problem is first simplified by virtue of the recursions in Eqs. (3)

and (4). For a particular finite multiresolution basis of wavelets and scaling functions, all

necessary matrix elements can be reduced to integrals between scaling functions on the

same scale. It is therefore sufficient to consider the j = 0 case,

φ ′ k 0 f φ k

0 = dx∫ φ x − ′ k ( ) f x( )φ x − k( ). (25)

This matrix is automatically banded. Since the scaling functions vanish for arguments

outside (0, L –1), functions of index k and k' spatially overlap only for ′ k − k ≤ L − 2 .

Due to symmetry, there are L –1 distinct cases, exemplified by the specific choices k' = 0,

k = 0, 1, 2, …, L–2.

A quadrature strategy is adopted which allows for distinct sets of quadrature

points xq,k and weights ωq,k for each of these L – 1 values of k,

dx∫ φ x( ) f x( )φ x − k( ) ≈ ωq ,k f xq, k( )q= 1

rk

∑ . (26)

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The xq,k may be chosen to overlap for different values of k, reducing the number of

samples required. Insisting that the quadrature be exact for powers 0 through rk – 1 leads

to

ωq, k xq ,kp

q =1

rk

∑ = mp ,k ≡ dx φ x( )∫ xp φ x − k( ), 0 ≤ p ≤ rk − 1. (27)

While the moments are different from before, the structure of the equations is identical.

Therefore the weights are given by

ωq, k =mp,k

p!i =0

rk −1

∑ Lrk ,q ,kp( ) 0( ). (28)

The particular moments in Eq. (27) may be obtained in the manner used by Beylkin52 to

obtain matrix representations of operators. This method uses change of scales for both

scaling functions in Eq. (27) and change of integration variables to obtain linear

equations that the moments must satisfy,

mp,k = 2− pp

′ p

′ p = 0

p

∑ m ′ p ,2 k + ′ k ap− ′ p , ′ k ′ k =1− L

L−1

∑ . (29)

The ap,k are the easily–calculated discrete sums

ap, k = hn +k hn n p

n∑ , (30)

where the scaling function coefficients hn are zero unless 0 ≤ n ≤ L–1. Also, from Eq.

(27) the moments for k < 0 can be shown to be related to those for k > 0 by

mp,− k =p

′ p

′ p =0

p

∑ − k( )p − ′ p m ′ p ,k . (31)

Thus Eqs. (29) need only be solved for the moments with non-negative k (using standard

methods), and this is required only once.

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The polynomial-based quadrature above can be adapted straightforwardly to

matrix elements between the dual sets of functions in a biorthogonal basis. The latter

type of integral can arise, for instance, in solution of differential equations using the

biorthogonal systems in the Wavelet Galerkin method.58-61 In quantum contexts,

biorthogonal matrices may have application to problems in which non-Hermitian

Hamiltonian-type operators arise, e.g., in the use of complex rotation methods for L2 basis

calculations of resonance eigenvalues.62,63 One loses the advantages of symmetry of the

matrices in these cases, but gains in flexibility.

Following the Morse potential example of the last section, an adaptive quadrature

may be implemented to evaluate in tandem the potential matrix elements for the L = 10

scaling function basis. The same uniformly-spaced quadrature is used, but this time the

starting point of the grid for each matrix element must be chosen for two scaling

functions which partially overlap. A reasonable choice is to start at the left-hand side of

their region of intersection, i.e., at 2j λ max(k', k ). The multiplexing advantage stemming

from overlapping grids still applies despite the fact that a matrix is evaluated in this case.

The number of samples required at the first stage is only linearly proportional to the

distance spanned by the scaling function basis at level j.

Convergence may again be monitored by comparing results for adjacent orders rk

and rk + 1. All matrix elements with the same starting samples, i.e., the same max(k', k ),

are regarded jointly; if any of them fail to converge, then the entire subset is earmarked

for more accurate evaluation by recursion of the relevant integrals from the next finer

level. This starts a recursive refinement effectively focusing on the local behavior of the

potential sample points. The procedure is continued to finer scales, tightening the

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convergence threshold by a factor of 2 each time, until the desired convergence is

reached and the results may be backtracked to the original scale. The alternative

convergence criterion mentioned in the vector case applies to the matrix case as well: one

may monitor convergence by agreement of quadrature at level j and of quadrature at level

j – 1 with recursion to level j. This would require one extra scale as well as a greater

amount of computation at intermediate steps.

While the matrix integrals are thus capable of being calculated in close analogy to

the vector integrals, there is one chief difference. The recursion between scales must be

implemented for both the bra and ket scaling functions at the same time, thereby

increasing the computational effort. In this regard, it is in principle possible to streamline

things by combining the two separate matrix operations into a single compound

operation.

For the level–j basis, 0 ≤ k ≤ 4, the required matrix elements were sought with a

convergence threshold of 10–5 eV. Quadrature orders 6 and 7 were used for checking of

convergence for all matrix elements. Table 2 shows the rapid convergence as a function

of scale for all of the matrix elements needed to evaluate the top–level integrals. The

number of distinct function evaluations is kept minimal by the current procedure. The

required matrix elements φ ′ k j V φ k

j are indicated explicitly in Fig. 7, with the starting

points of the two scaling functions mapped onto separate axes. Thus one can see

immediately the spatial distribution of the matrix elements required and, in particular, the

adaptive increase in density for the region near the steep inner wall of the potential.

21

VI. Conclusion

Integration by polynomial-based quadrature for orthonormal (or bi-orthonormal)

compact support wavelet families has been investigated for the purpose of use in

chemical and physical applications. In the above, a quadrature formula has been

developed in terms of easily–calculated derivatives of Lagrange interpolation

polynomials for the chosen coordinate grid and particular moments which are

grid–independent. Easy application to irregular grids was demonstrated in the wavelet

decomposition of the Hartley absorption spectrum of ozone. Adaptive quadrature was

developed for integrals of products of reasonably smooth functions and one or two

scaling functions, and applied to integrals of the anharmonic Morse potential of the types

that will arise in numerical quantum mechanical calculations using wavelet bases.

Acknowledgments

This work was supported by the National Science Foundation and the Robert A.

Welch Foundation. The authors also wish to thank R. O. Wells, C. S. Burrus, J. Tian and

W. Sweldens for conversations.

22

Appendix A

Derivation of Eq. (9) for the quadrature weights proceeds directly from the

interpolation formula in Eq. (11). The equality sign holds for the latter if f x( ) is a power

of x less than r,

Lr ,q x( ) xqn

q=1

r

∑ = xn . (A1)

Differentiating this with respect to x p times and setting x equal to zero yields

Lr ,qp( ) 0( ) xq

n

q= 1

r

∑ = p!δp, n, (A2)

or, multiplying both sides by mp / p!,

mp

p!Lr ,q

p( ) 0( )

xq

n

q =1

r

∑ = mp δp ,n . (A3)

If this is summed over p up to r – 1, the Kronecker delta is unity for p = n and zero

otherwise. Thus,

mp

p!Lr ,q

p( ) 0( )p = 0

r −1

∑

xqn

q=1

r

∑ = mn . (A4)

Since this is true for all n < r, the quantity in brackets is the solution of Eq. (8).

Appendix B

Nonlinear relations exist between various moments of the scaling function in

orthogonal compact support wavelet systems. This is most easily seen by considering the

autocorrelation function (the fundamental function of the Deslauriers-Dubuq scheme30)

Φ y( ) = dx∫ φ x( )φ x + y( ) . (B1)

23

The moments of Φ y( ) may be determined after a little algebra in terms of the moments

mp of the scaling function defined in Eq. (8),64

Mn = dy∫ yn Φ y( ) = −1( )n − p n

p

mn − p mp

p =0

n

∑ (B2)

On the other hand, Beylkin52 and Tian and Wells35 have shown that Mn = 0 for n = 0, 1,

…, L–1. For odd n this yields no new information; for even n, however, one finds

m2 = m12 , L ≥ 4, (B3)

m4 = 4 m3 m1 − 3m14 , L ≥ 6, (B4)

m6 = 6m5 m1 +10m32 − 60m3 m1

3 + 45m16, L ≥ 8, (B5)

m8 = 8m7 m1 +56m5 m3 −168m5 m13 +2520m3 m1

5 − 840m32 m1

2 −1575m18,

L ≥ 10, (B6)

etc. The first of these relations was derived by Gopinath and Burrus33 for the Daubechies

scaling functions and later shown to be more general by Sweldens and Piessens.21 To the

knowledge of the authors, the higher-order relations do not appear to have been noted

before. They are valid for all of the compact support wavelet families since the latter all

share the same autocorrelation function.

24

Bibliography

1 I. Daubechies, Comm. Pure Appl. Math. 41, 909 (1988).

2 S. G. Mallat, IEEE Trans. Pattern Anal. Machine Intell. 11, 674 (1989).

3 I. Daubechies, Ten Lectures on Wavelets (SIAM Publications, Philadelphia,

1992).

4 Wavelets: Mathematics and Applications, Vol. , edited by J. J. Benedetto and M.

W. Frazier (CRC Press, Boca Raton, 1994).

5 G. Strang and T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge

Press, Wellesley, 1996).

6 C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet

Transforms (Prentice-Hall, 1996).

7 H. L. Resnikoff and R. O. Wells, Jr., Wavelet Analysis: The Scalable Structure of

Information (Springer Verlag, 1998).

8 T. Paul, in Wavelets: Time-Frequency Methods and Phase Space, edited by J. M.

Combes, A. Grossman, and P. Tchamitchian (Springer Verlag, Berlin, 1989), pp. 315.

9 D. Permann, Phys. Rev. Lett. 69, 2607 (1992).

10 Z. Li, A. Borrmann, and C. C. Martens, Chem. Phys. Lett. 214, 362 (1993).

11 K. Cho, T. A. Arias, J. D. Joannopolous, and P. K. Lam, Phys. Rev. Lett. 71, 1808

(1993).

12 P. Fischer and M. Defranceschi, Int. J. Quant. Chem. 45, 619 (1993).

13 D. Permann and I. Hamilton, J. Chem. Phys. 100, 379 (1994).

14 L. Gagnon and J. M. Lina, J. Phys. A 27, 8207 (1994).

25

15 J. P. Modisette, P. Nordlander, J. L. Kinsey, and B. R. Johnson, Chem. Phys. Lett.

250, 485 (1996).

16 A. Askar, A. E. Cetin, and H. Rabitz, J. Phys. Chem. 100, 19165 (1996).

17 S. Wei and M. Y. Chou, Phys. Rev. Lett. 76, 2650 (1996).

18 P. Fischer and M. Defranceschi, SIAM J. Numer. Anal. 35, 1 (1996).

19 S. Goedecker and O. V. Ivanov, Sol. State Comm. 105, 665 (1998).

20 D. J. Kouri, (to be published).

21 W. Sweldens and R. Piessens, SIAM J. Numer. Anal. 31, 1240 (1994).

22 W. Sweldens and R. Piessens, Numerische Mathematik 68, 377 (1994).

23 A. Latto, H. L. Resnikoff, and E. Tenenbaum, Compactly Supported Wavelets,

AD91078, Aware, Inc. (1991).

24 W. Dahmen and C. A. Micchelli, SIAM J. Numer. Anal. 30, 507 (1993).

25 O. V. Vasilyev and S. Paolucci, J. Comp. Phys. 125, 498 (1996).

26 A. Cohen, I. Daubechies, and J. C. Feauveau, Comm. Pure Appl. Math. 55, 485

(1992).

27 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables (U. S. Government Printing Office,

Washington, D.C., 1972).

28 A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, Second ed.

(McGraw-Hill, New York, 1978).

29 H. R. Schwarz and J. Waldvogel, Numerical Analysis: A Comprehensive

Introduction (Wiley, Chichester, 1989).

26

30 G. Deslauriers and S. Dubuc, Constr. Approx. 5, 49 (1989).

31 M. J. Shensa, IEEE Trans. Signal Processing 40, 2464 (1992).

32 H. Rutishauser, Lectures on Numerical Mathematics (Birkhaüser, Boston, 1990).

33 R. A. Gopinath and C. S. Burris, in Proceedings of the ISCAS-92 (San Diego,

1992).

34 G. Beylkin, R. Coifman, and V. Rokhlin, Comm. Pure and Appl. Math 44, 141

(1991).

35 J. Tian and R. O. Wells, Jr., Vanishing Moments and Wavelet Approximation,

Computer Mathematics Laboratory Report CML9501, Rice University (1995).

36 W. Sweldens, in Wavelet Applications in Signal and Image Processing III, Proc.

SPIE 2569, edited by A. F. Laine and M. Unser (1995), pp. 68.

37 W. Sweldens, SIAM J. Math. Anal. 29, 511 (1998).

38 B. R. Johnson and J. L. Kinsey, Phys. Rev. Lett. 62, 1607 (1989).

39 B. R. Johnson and J. L. Kinsey, J. Chem. Phys. 91, 7638 (1989).

40 B. R. Johnson, B.-Y. Chang, C.-W. Hsiao, L. Le, and J. L. Kinsey, J. Chem. Phys.

108, 7670 (1998).

41 C. Leforestier, F. LeQuéré, K. Yamashita, and K. Morokuma, J. Chem. Phys. 101,

3806 (1994).

42 N. Balakrishnan and G. D. Billing, J. Chem. Phys. 101, 2968 (1994).

43 C. Leforestier, (to be published).

44 L. D. Le, C.-W. Hsiao, E. S. Lotfi, C. Kittrell, B. R. Johnson, and J. L. Kinsey,

(in progress).

27

45 D. C. Rouben and G. S. Ezra, J. Chem. Phys. 103, 1375 (1995).

46 K. Hirai, E. J. Heller, and P. Gaspard, J. Chem. Phys. 103, 5970 (1995).

47 J. Brion, A. Chakir, D. Daumont, J. Malicet, and C. Parisse, Chem. Phys. Lett.

213, 610 (1993).

48 F. Remacle, R. D. Levine, and J. L. Kinsey, Chem. Phys. Lett. 205, 267 (1993).

49 F. Remacle and R. D. Levine, J. Chem. Phys. 99, 4908 (1993).

50 F. Remacle and R. D. Levine, J. Phys. Chem. 97, 12553 (1993).

51 S.-Y. Lee, Z. W. Feng, and R. C. K. Yeo, J. Raman Spec. 28, 411 (1997).

52 G. Beylkin, SIAM J. Numer. Anal. 6, 1716 (1992).

53 B. R. Johnson and J. L. Kinsey, J. Chem. Phys. 99, 7267 (1993).

54 B. R. Johnson and J. L. Kinsey, in Femtosecond Chemistry, edited by J. Manz and

L. Wöste (Verlag Chemie, New York, 1995), pp. 353.

55 M. Shapiro, Chem. Phys. Lett. 242, 548 (1995).

56 X.-P. Peng, L.-S. Tian, and Y.-N. Peng, IEEE Trans. Signal Processing 44, 129

(1996).

57 P. Abry and P. Flandrin, IEEE Sig. Proc. Lett. 1, 32 (1994).

58 R. Glowinski, W. M. Lawton, M. Ravachol, and E. Tenenbaum, Wavelet solutions

of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space

dimension, AD890527, Aware, Inc. (1989).

59 J. Ko, A. J. Kurdila, R. O. Wells, and X. Zhou, Comm. Num. Methods Eng. 12,

281 (1996).

60 J. R. Williams and K. Amaratunga, J. Comp. Phys. 122, 30 (1995).

28

61 W. Dahmen, A. Kunoth, and K. Urban, Computing 56, 259 (1996).

62 W. P. Reinhardt, Int. J. Quant. Chem. 21, 133 (1982).

63 K. McCormick and P. J. Nordlander, (unpublished).

64 M. Unser, IEEE Trans. Signal Proc. 44, 519 (1996).

29

Tables

Table 1. Scaling coefficients and lowest moments for the L = 10 Daubechies scaling

functions.

k hk mk

0 0.160102397974192914 1.000000000000000001 0.603829269797189671 1.193908018023822952 0.724308528437772928 1.425416355501573153 0.138428145901320732 1.580259805599972984 –0.242294887066382032 1.451304050272140505 –0.322448695846383746e–1 0.8137105292496663066 0.775714938400457135e–1 –0.2295042676756414267 –0.624149021279827427e–2 –0.9549759730451576788 –0.125807519990819995e–1 –1.672306166280536229 0.333572528547377128e–2 9.13925141629235832

10 4.06900434121189231e211 5.86543569280676173e3

Table 2. Convergence of Morse potential matrix element calculations in adaptive

quadrature. For each scale, Max |∆|, the maximum discrepancy between r = 6 and r = 7

evaluations of the matrix elements, occurs for k' = k = 0. Only approximately half of the

samples needed at each scale refinement require new evaluations.

No. of No. of NewScale Max |∆| Tolerance Samples Samples

4 6.76e-2 1.00e-5 11 113 7.19e-3 5.00e-6 24 132 3.25e-4 2.50e-6 34 171 8.86e-6 1.25e-6 42 210 1.84e-7 6.25e-7 38 19

30

Figures

1.0

0.5

0.0

-0.5

76543210x

(x) (x)

Figure 1. Scaling function φ (x ) and wavelet ψ(x ) for Daubechies family with L = 8.

1200

1000

800

600

400

200

0

44x103 40383634Frequency (cm–1)

Figure 2. Hartley absorption band of ozone measured at 218 K by Brion, et al.,47 withsmoothed reconstruction obtained from expansion in scaling functions at scale 3.28 cm–1.

31

-50

0

50

44x103 40383634Frequency (cm–1)

j = 87

6

54

Figure 3. The significant detail components of the Hartley absorption cross-section atscales 3 . 2j obtained by expansion in wavelets ψ k

j and summation over all k contributionsfor each j. Vertical offsets are given to traces other than that for j = 6.

44x103 40383634Frequency (cm–1)

j = 87

6

54

Figure 4. The wavelet coefficients corresponding to the components of the Hartleyabsorption cross-section shown in Fig. 3. For each j and k, the position on the frequencyscale marks the left hand side of the corresponding wavelet. These positions are markedby dots for the less congested scales. Vertical offsets are given to coefficients other thanthose for j = 6.

32

-4

-2

0

2

1086420Quadrature order r

01234

k

Figure 5. Absolute values of the differences for order r and order r + 1 quadratureevaluation of projection integrals φ k

j V for the Morse potential. The L = 10

Daubechies scaling functions of spacing ∆x = 0.4 Å are used. The sample spacing here is

the same as the basis function spacing.

3210x (Å)

V(x)

φkj

jj – 1j – 2j – 3j – 4

Figure 6. Morse potential and L = 10 Daubechies scaling functions for which projectionintegrals are calculated. For each scale j, the + signs represent the left hand endpoints ofscaling functions included in the adaptive quadrature. The greatest detail is required inthe vicinity of the steep repulsive wall.

33

4

3

2

1

0

43210k 2j λ (Å)

Figure 7. Required integrals φ ′ k j V φ k

j from adaptive calculation of the Morse potential

matrix elements. Each symbol represents a matrix element involving the product φ ′ k j φk

j ;the positions along the left and bottom axes represent the initial points of the k' and kscaling functions, respectively. The circles correspond to the target integrals on level j =4, while the black dots correspond to the matrix elements required on finer levels.Symmetry reduces the number of independent integrals required by approximately afactor of 2.