Quadrature integration for orthogonal wavelet systems

Transcription

1 Quadrature integration for orthogonal wavelet 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 Institute Rice University Houston, TX b Department of Physics and Rice Quantum Institute Rice University Houston, TX

2 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. 2

3 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 Daubechies 1,3 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 ust copies widened by factors 2 and translated by steps k 2, where and k are integers, φ k ( x) = 2 /2 φ ( 2 x k ), (1) ψ k ( x) = 2 /2 ψ ( 2 x k ), (2) 3

4 with φ 0 0 = φ, ψ 0 0 = ψ. By construction, the set of scaling functions at a particular resolution level, { φ k }, form a basis which is orthogonal though not complete. A more 1 complete set { φ k } 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, so that 1 { φ k } = { φ k } { ψ k }. This is expressed in terms of the recursion relations, φ k L 1 ( x) = h k φ 1 2 k+ k ( x ), (3) k = 0 ψ k L 1 ( x) = g k φ 1 2 k + k ( x ), (4) k =0 where the constants are related by g k = ( 1) k +1 h L 1 k. 2 Ultimately, every quantity that must be calculated boils down to some function of the h k, so that the latter take on a central role in calculations. An orthonormal multiresolution basis can be comprised of J scaling functions { φ k } on a coarsest level = J and wavelet functions { ψ k } from this coarsest level down to a finest scale = 0 which provide the various additional octaves of detail. The approximate expansion of a function f in this basis then takes the form ( ) ψ k f x J J J f ψ k ( x) + φ k f φ k ( x). (5) = 0 k Detail functions { ψ k } at each scale which are unimportant (have small expansion k coefficients ψ k 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 proection integrals φ k f, ψ k f for the vector basis and the matrix elements φ k f φ k, φ k f ψ k, 4

5 ψ k f ψ k, 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 Daubechies 3 (p. 166) for calculation of the vector proection integrals uses Fourier convolution. While this method is not widely used, Wei and Chou 17 have adapted it to their density functional calculations. Alternatively, a quadrature method with uniformlyspaced 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 semisingular 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 k f and φ k f φ k. 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 5

6 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. Proection Integral Quadrature For a given and k, a change of variables transforms the general vector integral for scaling functions into a standardized form, φ k f = 2 /2 dx φ ( 2 x k ) f x ( ) = 2 / 2 dx φ( x ) f [ 2 ( x + k )]. (6) Thus it is only necessary to consider the = k = 0 case, for which the integral is approximated by quadrature, ( ) dx φ( x ) f ( x ) ω q f x q r. (7) q =1 The only restriction imposed on the sample points is that the x q are assumed to be ordered, x 1 < x 2 < x r. One can specify theω q in terms of the x q by requiring that Eq. (6) be exact for f equal to a polynomial of order r 1 or less: r p ω q x q = m p dx φ( x ) x p, 0 p r 1. (8) q= 1 The general solution to these equations (see Appendix A) is ω q = r 1 p= 0 m p p! ( p ) L r,q ( 0 ), (9) using the Lagrange interpolating polynomials L r, q ( x ) = r x x q (10) q q x q x q 6

7 ( p and their derivatives L ) r, q at x = 0. The Lagrange polynomials are the coefficients for interpolation of smooth functions, 27 r L r,q q= 1 ( ) ( x ) f x q 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 shown 31 to coincide with the autocorrelation function of the compact support scaling functions (see Appendix B). The Lagrange polynomials 29,32 and their derivatives are easily evaluated once the quadrature nodes are specified. As for the moments m p, their calculation may be accomplished by the simple recursive equation derived by Gopinath and Burrus, 33 m p = 1 2 p ( ) 1 p 1 p m p p µ p p, (12) p = 0 where the µ p are discrete moments of the scaling coefficients of Eq. (3), µ p = 1 2 L 1 h k k p. (13) k =0 The leading moment m 0 = 1 as a matter of normalization, providing the starting point for the recursion. (Not all of the resulting moments m p are independent of each other, as is discussed in Appendix B.) The moments are thus determined completely by the scaling coefficients h k, whereas the quadrature coefficients are determined in terms of both the h k 7

8 and the nodes x q. In calculations for a given compact support basis, the moments need only be calculated once. Sweldens and Piessens 21 considered the important case of uniformly-spaced x q. Such an arrangement allows the sample points for scaling functions of adacent 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 quadrature 28 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 x q 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 adacent scaling functions. In the present work, the disposition of the x q is left open for purposes of adaptability. For compact support wavelets, it is known 21,33 that m 2 = m 1 2, although higher moments are not simple powers of m 1 (Appendix B). Introducing the differences m p = m p m 1 p, (14) Eq. (8) can be rearranged, ω q = r 1 p= 0 m 1 p p! ( p) L r, q ( 0)+ r 1 p= 3 m p p! ( p ) L r,q ( 0) ( ) + = L r,q m 1 r 1 p= 3 m p p! ( p ) L r, q ( 0), (15) 8

9 where the first series has been recognized as the Taylor expansion of the Lagrange interpolation polynomial evaluated at m 1. To the extent that the second series can be ignored, Eqs. (7) and (11) show that ( ) dx φ( x ) f ( x ) L r,q ( m 1 ) f x q r f m 1 q= 1 ( ) (16) This directly reflects the fact determined by Gopinath and Burrus that a one-point quadrature using m 1 as the sample point has O( x 3 ) error. Higher order error generally requires a multi-point quadrature. An exception is found in the Coifman wavelet family, 34,35 where m 1 = m 2 = = m M -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 k Eqs. (3) and (4), and ψ k, satisfy their own scaling relations, cf., φ k ( x) = h 1 k k ( x), (17) k φ 2k+ ψ k ( x) = g 1 k k ( x), (18) k φ 2k+ 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 Sweldens 36,37 for 9

10 purposes of adaptable multiresolution analyses. The general expansion of a function in a biorthogonal basis may be expressed as ( ) f x J ψ k f ψ k ( x) + J φ k = 0 k k rather than Eq. (5). Similarly the analogs of Eqs. (7) and (8) are r dx φ ( x ) ( ) f x r ( ) J f φ k ( x) (19) ω q f x q, (20) q =1 ω x p = m q q p dx φ ( x ) x p, 0 p r 1. (21) q =1 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., and references therein), as shown in Fig. 2. Photoexcitation to a steeply repulsive dissociative region of the potential energy surface for the D 1 B 2 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 ( fs) times corresponding, respectively, to direct and indirect dissociation contributions. Dynamical analysis of the indirect contribution, corresponding to Franck-Condon-region exit and 10

11 return, has progressed in recent years with the aid of both classical 40 and quantum 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 oint 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. Gabor 45 and Vibrogram/Husimi 46 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 proection integrals of translated and scaled scaling functions on a level = 0, dν 1 ν φ ν ν 0 ν k σ( ν) ( ) = ν dx φ x r σ ( ν 0 + k ν + x ν ) ( ) ν ω q σ ν 0 + k ν + x q ν. (22) q=1 11

12 An absolutely typical initial complication arises from the fact that the absorption crosssection σ was measured 47 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 φ k 0 and the associated proections 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 proections onto an orthonormal basis J of wavelets { ψ k }, J, and scaling functions { φ k } 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 proections onto the φ k J 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 k proections, representing the differences in detail between different scales. 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 = 5 7. One obtains localized frequency information even from these summed quantities, e.g., it is clear from = 5 that higher-frequency components of the structure are more prominent on the low-frequency side of band center. 12

13 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, subect 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 Transform 3 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 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 determined 53,54 (see also Shapiro 55 ). 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" 13

14 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 approach 3,30 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, ( ) = D e exp 2a( x x e ) V x [ ] 2D e exp[ a( x x e )], (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 = Å 1, x e = Å, D e = ev. Proections are calculated of this function on the scaling functions from the L = 10 Daubechies system, φ k J ( x) = 2 J λ ( ) 1/2 φ x /2 J λ k ( ), (24) with spacing 2 J λ = 0.4 Å (λ and J are not independent parameters here). 14

15 For the sake of simplicity (but not necessity), the uniformly-spaced quadrature associated with the basis functions on each level is used, x q = ( k + q 1) 2 λ. The same quadrature weights are therefore used for each functionφ k, 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 φ k '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 subected to refinement. Convergence may be measured either by (i) comparison of the numerical results using quadrature at level with those of quadrature at 15

16 level 1 followed by recursion back to, or (ii) direct quadrature at level 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 coarsestlevel 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 = 4. On the finest level required, = 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 φ k for which the integral is recalculated using refinement requires the integrals for φ 1 2k, φ 2k ,, φ 2k +L 1, but there is a multiplexing advantage sinceφ k ±1 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. 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. 16

17 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 = 0 case, 0 φ k f φ k 0 ( ) f x = dxφ x k ( )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 x q,k and weights ω q,k for each of these L 1 values of k, dx φ( x ) f ( x )φ( x k ) ω q,k f x q, k r k. (26) q= 1 ( ) 17

18 The x q,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 r k 1 leads to r k ω q, k x p q,k = m p,k dx φ x q =1 ( ) x p φ( x k), 0 p r k 1. (27) While the moments are different from before, the structure of the equations is identical. Therefore the weights are given by ω q, k = r k 1 m p,k L ( p ) p! rk,q,k i =0 ( 0). (28) The particular moments in Eq. (27) may be obtained in the manner used by Beylkin 52 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, p m p,k = 2 p p L 1 m p p,2 k + k a p p, k. (29) p = 0 k =1 L The a p,k are the easily calculated discrete sums a p, k = h n +k h n n p, (30) n where the scaling function coefficients h n 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 m p, k = p p =0 p 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. 18

19 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 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 L 2 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 2 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. Convergence may again be monitored by comparing results for adacent orders r k and r k + 1. All matrix elements with the same starting samples, i.e., the same max(k', k ), are regarded ointly; 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 19

20 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 and of quadrature at level 1 with recursion to level. 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 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 V φ k 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. 20

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. 21

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, r n L r,q ( x ) x q = x n. (A1) q=1 Differentiating this with respect to x p times and setting x equal to zero yields r ( L p ) r,q q= 1 n ( 0) x q = p!δ p, n, (A2) or, multiplying both sides by m p / p!, m p r p! L ( p ) r,q ( 0 ) x n q = m p δ p,n. (A3) q =1 If this is summed over p up to r 1, the Kronecker delta is unity for p = n and zero otherwise. Thus, r r 1 m p p! L ( p ) r,q ( 0) p = 0 x n q = m n. (A4) q=1 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 scheme 30 ) ( ) = dx Φ y φ( x )φ ( x + y ). (B1) 22

23 The moments of Φ( y ) may be determined after a little algebra in terms of the moments m p of the scaling function defined in Eq. (8), 64 M n = dy y n Φ y n n ( ) = ( 1) n p p =0 m p n p m p (B2) On the other hand, Beylkin 52 and Tian and Wells 35 have shown that M n = 0 for n = 0, 1,, L 1. For odd n this yields no new information; for even n, however, one finds m 2 = m 2 1, L 4, (B3) m 4 = 4 m 3 m 1 3m 4 1, L 6, (B4) m 6 = 6m 5 m 1 +10m m 3 m m 6 1, L 8, (B5) m 8 = 8m 7 m 1 +56m 5 m 3 168m 5 m m 3 m m 2 3 m m 8 1, L 10, (B6) etc. The first of these relations was derived by Gopinath and Burrus 33 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. 23

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29 Tables Table 1. Scaling coefficients and lowest moments for the L = 10 Daubechies scaling functions. k h k m k e e e e e e e3 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 New Scale Max Tolerance Samples Samples e e e e e e e e e e

30 Figures (x) (x) x Figure 1. Scaling function φ (x ) and wavelet ψ(x ) for Daubechies family with L = x10 3 Frequency (cm 1 ) Figure 2. Hartley absorption band of ozone measured at 218 K by Brion, et al., 47 with smoothed reconstruction obtained from expansion in scaling functions at scale cm 1. 30

31 = x10 3 Frequency (cm 1 ) Figure 3. The significant detail components of the Hartley absorption cross-section at scales 3. 2 obtained by expansion in wavelets ψ k and summation over all k contributions for each. Vertical offsets are given to traces other than that for = 6. = Frequency (cm 1 ) 44x10 3 Figure 4. The wavelet coefficients corresponding to the components of the Hartley absorption cross-section shown in Fig. 3. For each and k, the position on the frequency scale marks the left hand side of the corresponding wavelet. These positions are marked by dots for the less congested scales. Vertical offsets are given to coefficients other than those for = 6. 31

32 k Quadrature order r Figure 5. Absolute values of the differences for order r and order r + 1 quadrature evaluation of proection integrals φ k 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. V(x) φ k x (Å) 2 3 Figure 6. Morse potential and L = 10 Daubechies scaling functions for which proection integrals are calculated. For each scale, the + signs represent the left hand endpoints of scaling functions included in the adaptive quadrature. The greatest detail is required in the vicinity of the steep repulsive wall. 32

33 k 2 λ (Å) 3 4 Figure 7. Required integrals φ k V φ k from adaptive calculation of the Morse potential matrix elements. Each symbol represents a matrix element involving the product φ k φ k ; the positions along the left and bottom axes represent the initial points of the k' and k scaling functions, respectively. The circles correspond to the target integrals on level = 4, while the black dots correspond to the matrix elements required on finer levels. Symmetry reduces the number of independent integrals required by approximately a factor of 2. 33