ITAMP Workshop

Wavelets and Applications in Physics and Astrophysics

Organizers: Alfred Msezane (CTSPS, Clark Atlanta Univ.), Romain Murenzi (CTSPS, Clark Atlanta Univ.), Jean-Pierre Antoine (FYMA, Catholic University of Louvain)

October 8-10, 1997

Sponsored Jointly by the Institute for Theoretical Atomic and Molecular Physics at the Harvard-Smithsonian Center for Astrophysics

and the Center for Theoretical Studies of Physical Systems at Clark Atlanta University

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 Participants

Prof. Syed T. Ali
Department of Mathematics and Statistics
Concordia University
7141 Sherbrooke St. West
Montreal, Quebec, Canada H4B 1R6
stali@neumann.concordia.ca

Dr. Jean-Pierre Antoine
Université Catholique de Louvain
Institut de Physique Théorique
2, chemin du Cyclotron
B-1348 Louvain-la-Neuve, Belgium
Phone: 32.10.47.32.83
antoine@fyma.ucl.ac.be

Prof. Tomás A. Arias
MIT, Department of Physics, Rm. 12-110
77 Mass. Ave.
Cambridge, MA 02139
muchomas@mit.edu

Prof. Alain Arneodo
Centre de Recherche Paul Pascal
Avenue Dr. A. Shweitzer
33600 Pessac, France
arneodo@axpp.crpp.u-bordeaux.fr

Prof. Raymond G. M. Brummelhuis
Université de Reims
Département de mathématiques
BP 1039, 51687 Reims CEDEX 2, France
raymond.brummelhuis@univ-reims.fr

Prof. Shih-I Chu
University of Kansas
Dept. of Chemistry
Lawrence, Kansas 66045-0046
sichu@kuhub.cc.ukans.edu


Prof. Ingrid C. Daubechies
Princeton University
Department of Mathematics
218 Fine Hall
Princeton, NJ 08544
ingrid@math.princeton.edu

Dr. Jonathan Edwards
Harvard University
Physics Department
Cambridge, MA 02138
edwards@physics.harvard.edu
edwards@bnpcn.com

Mr. Natnael B. Embaye
Clark Atlanta University
CTSPS
James P. Brawley Dr.
Atlanta, GA 30314
natnael@hubble.cau.edu

Dr. Brett Esry
Harvard-Smithsonian Center for Astrophysics
ITAMP
60 Garden Street, MS 14
Cambridge, MA 02138
besry@cfa.harvard.edu

Dr. Zineb Felfli
CTSPS
Clark Atlanta University
James P. Brawley Dr.
Atlanta, GA 30314
zfelfli@ctsps.cau.edu

Dr. Jean-Pierre Gazeau
LPTMC
Université Paris 7 - Denis-Diderot,
2 Place Jussieu,
75251 Paris Cedex 05, France
gazeau@ccr.jussieu.fr


Prof. Alex Grossmann
Centre de Physique Theorique (CPT)
CNRS-Luminy, Case 907
13288 Marseille Cedex 9, France
Alex.Grossmann@genetique.uvsq.fr

Dr. Carlos R. Handy
Clark Atlanta University
CTSPS
223 James P. Brawley Dr., S.W.
Atlanta, GA 30314
handy@pioneer.cau.edu

Prof. Eric J. Heller
Harvard University
Department of Physics
Cambridge, MA 02138
heller@physics.harvard.edu


Prof. John R. Klauder
University of Florida
Department of Physics
P. O. Box 118440
Gainesville, FL 32611-8440
klauder@phys.ufl.edu


Prof. Donald J. Kouri
University of Houston
Department of Physics
Houston, TX 77204-5506
kouri@uh.edu


Mr. Michal Kovacic
Charles University
Department of Theoretical Physics
V Holesovickach 2,
180 00, Praha 8, Czech Republic
mkov4220@mbox.troja.mff.cuni.cz


Prof. Joseph H. Macek
University of Tennessee
Department of Physics
200 S. College
Knoxville, TN 37996-1501
jmacek@utk.edu

Prof. Steven T. Manson
Georgia State University
Department of Physics & Astronomy
Atlanta, GA 30303
smanson@gsu.edu

Prof. Alfred Z. Msezane
Clark Atlanta University
CTSPS
223 James P. Brawley Dr.
Atlanta, GA 30314
amsezane@pegasus.cau.edu


Prof. Romain Murenzi
Clark Atlanta University
CTSPS
223 James P. Brawley Dr., S.W.
Atlanta, GA 30314
rmurenzi@hubble.cau.edu


Dr. Stephen S. Murray
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, MS 02
Cambridge, MA 02138
smurray@cfa.harvard.edu


Thierry Paul
Université Paris-Dauphine
CEREMADE
Place du Maréchal de Lattre de Tassigny
75775 Paris Cedex 16, France
paulth@ceremade.dauphine.fr


Dr. Heinrich Röder
MS B221, Group T-1
Los Alamos National Laboratory
Los Alamos, NM 87545
hro@lanl.gov


Prof. Mary Beth Ruskai
University of Massachusetts, Lowell
Department of Mathematics
Lowell, MA 01854
bruskai@cs.uml.edu

Dr. Hossein Sadeghpour
Harvard-Smithsonian Center for Astrophysics
ITAMP
60 Garden Street, MS 14
Cambridge, MA 02138
hsadeghpour@cfa.harvard.edu


Prof. Allan D. Stauffer
York University
Department of Physics & Astronomy
4700 Keele St.
Toronto, Ontario M3J IP3, Canada
stauffer@yorku.ca


Dr. Bruno Torresani
LATP, CMI,
Universite de Provence,
39 rue F. Joliot-Curie,
13453 Marseille Cedex 13, France.
Bruno.Torresani@sophia.inria.fr


Dr. Christopher J. Tymczak
MS B221, Group T-1
Los Alamos National Laboratory
Los Alamos, NM 87545
tymczak@lanl.gov


Dr. Xiao-Qian Wang
CTSPS
Clark Atlanta University
James P. Brawley Drive
Atlanta, Georgia 30314
wang@hubble.cau.edu

Prof. Charles A. Weatherford
Florida A&M University
Physics Department
205 Jones Hall
Tallahassee, FL 32307
weatherf@cennas.nhmfl.gov

Mr. Koichiro Yamaguchi
Institute for Chemical Research
Kyoto University, Uji
Kyoto 611-0011, Japan
koichiro@elec.kuicr.kyoto-u.ac.jp



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 Abstracts

Ali

Antoine

Arias

Arneodo

Brummelhuis

Chu

Daubechies

Esry

Grossman

Handy

Klauder

Kouri

Manson

Paul

Torresani

Tymczak

Wang

Yamaguchi

A Generalized Wigner Transform and Its Relation to the Wavelet Transform

Syed T. Ali

Department of Mathematics and Statistics

Concordia University

7141 Sherbrooke St. West

Montreal, Quebec, Canada H4B 1R6

The well-known Wigner transform, mapping states to square-integrable functions on phase space (quasi-probability distributions), can be extended and reformulated as a linear map between the space of Hilbert-Schmidt operators, on the Hilbert space of a representation of the Weyl-Heisenberg group, and the space of all square-integrable functions on phase space. We propose a generalization of this concept to groups which have representations in the discrete series (i.e., square-integrable representations). When so generalized, the Wigner transform appears as a linear map between the space of Hilbert-Schmidt operators of the representation space of the group in question and spaces of square integrable functions on its coadjoint orbits (with respect to the natural invariant measure). Applied to the affine group, our analysis yields an interesting relationship between the Wigner and the wavelet transforms -- a relation which again holds quite generally.

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The Continuous Wavelet Transform: An Overview

J.-P. Antoine

Institut de Physique Théorique

Université Catholique de Louvain

B-1348 Louvain-la-Neuve, Belgium

In this talk, which is aimed at the nonspecialists, we will review the general properties of the continuous wavelet transform, in one or more dimensions. We will describe it first at a pedestrian level, then make the connection with the underlying group-theoretical structure and the theory of coherent states, showing how it allows a straightforward extension to more general situations, such as higher dimensions, wavelets on the sphere or time-dependent wavelets. Finally we will give describe a number of physical applications.

 

Time-Frequency Analysis in Laser-Atom Interaction: Wavelets vs. Gabor

J-P. Antoine,1 Ph. Antoine2 and B. Piraux2

1Inst. Phys. Théor.

2Lab. Phys. Atom. Mol.

Univ. Cath. Louvain,

B-1348 Louvain-la-Neuve, Belgium

When an atom is exposed to a strong laser pulse, it may emit light, in the form of odd harmonics of the driving e.m. field. This harmonic emission has a strong temporal structure (a characteristic plateau, followed by a sharp cut-off), hence its understanding requires a time-frequency analysis, wavelets or Gabor. The energy radiated in harmonic emission is measured by the acceleration of the dipole moment created by the electron oscillating in the laser field. Hence one calculates (numerically) the dipole acceleration a(t) from the time-dependent Schrödinger equation. In the case of a hydrogen atom in a 1s state, a Gabor analysis of a(t) yields the time profile of each harmonic separately, and a wavelet analysis reveals its fine structure. The intensity of each harmonic may be controlled by varying in time the polarization of the driving field. If the atom is in a 2s state, a wavelet analysis reveals the close correlation between the intensity of a given harmonic and the 1s population. Finally the instntaneous frequency of emission, measured by the time derivative of the phase of the Gabor coefficient, presents a blue-shift effect.

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Multiresolution Analysis of Electronic Structure

T.A. Arias

Department of Physics

Massachusetts Institute of Technology

Cambridge, Massachusetts 02139

In recent years, an entirely new avenue for scientific inquiry beyond traditional theoretical and experimental work has opened. With developments in many-body physics and advances in computation it is now possible to predict ab initio, purely from first principles with only fundamental constants as experimental input, the behavior of complex solid and chemical systems from the well-understood underlying quantum mechanics of electrons and nuclei. This ab initio approach has the advantage of providing, with a high degree of reliability, both a degree of control and detailed level of information not available in experiment.

Present approaches to the electronic structure of condensed matter, however, require great care and experience to produce accurate results. In particular, they are all somewhat ad hoc in their treatment of the wide range of length-scales needed to properly account for interactions between the core electrons of atoms and the valence electrons, which control the behavior of matter. Multiresolution analysis provides the first opportunity to develop a systematic treatment of these interactions.

The demands of the calculation of electronic structure have many commonalities with the demands of other problems in the physical sciences. In particular, ab initio calculations require economy of representation for physical fields exhibiting behavior over a wide range of length scales which interact through differential operators and local, non-linear interactions.

This talk will explore how to tailor a multiresolution analysis to the above demands by the introduction of a special property which we refer to as "semicardinality" [1]. We shall describe specialized algorithms for transforms and the application of differential operators which give such semicardinal bases significant advantages [2] and shall present results obtained using this approach [1,3].

----------

1. "Multiresolution analysis of electronic structure: semicardinal and orthogonal wavelet bases," T.A. Arias, Reviews of Modern Physics, in press for April, 1999.

Preprint: http://xxx.lanl.gov/abs/cond-mat/9805262

2. "Multiscale computation with interpolating wavelets," by Ross A. Lippert, T.A. Arias and Alan Edelman, Journal of Computational Physics 140, 278 (1998).

Preprint: http://xxx.lanl.gov/abs/cond-mat/9805283

3. "Wavelet transform representation of the electronic structure of materials," T.A. Arias and K.J. Cho and Pui Lam and M.P. Teter inProceedings of the '94 Mardi Gras Conference: Toward Teraflop Computing and New Grand Challenge Applications, Rajiv K. Kalia and Priya Vashishta, Ed., Nova Science Publishers, Commack New York, (1995).

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Phase-Space Localization for Atoms in Strong Magnetic Fields

R. Brummelhuis

Université de Reims

Département de mathématiques

BP 1039, 51687 Reims CEDEX 2, France

As is well known, atoms in very strong magnetic fields behave essentially as one-dimensional atoms. In this talk we consider some simplified one-dimensional models for such atoms, and discuss the problem of obtaining bounds for their maximal negative ionization. Three-dimensional methods based on configuration-space localization techniques do not work very well, and phase-space localization seems to be needed.

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Wavelet Analysis of the Time-Frequency Profile of Multiple

High-Order Harmonics Generated by Atoms in Intense Laser

Fields

 

Shih-I Chu

 

University of Kansas

Dept. of Chemistry

Lawrence, Kansas 66045

 
 
Multiple high-order harmonic generation (HHG) is one of the most rapidly developing topics in strong-field atomic and molecular physics. The generation of harmonics of orders well in excess of 100 from noble gas targets has been recently demonstrated by several experiments using high-intensity pump lasers. Coherent soft-x-ray harmonics, at wavelengths as short as 2.7 nm (460 eV) [1] and 2.5 nm (0.5 keV) [2], have been reported. The HHG phenomenon provides a feasible and powerful new route for the production of compact (table-top) coherent x-ray-laser light source in the future. Despite these recent technological developments, our understanding of the underlying physical mechanisms and the coherent control of HHG and other strong field processes is still far from completeness. The bottleneck for the theoretical investigation stems from the difficulty in performing accurate solution of the time-dependent Schrödinger equation of many-electron quantum systems and the detailed multiphoton dynamics in strong fields.
 
In this talk we first discuss a generalized pseudospectral time-dependent method recently developed in our group for accurate and efficient solution of time-dependent Schrödinger equation [3] and time-dependent Kohn-Sham-like equations [4] for atomic and molecular systems in intense laser fields. We then use the wavelet transform of the field-induced dipole moment and acceleration to obtain the time-frequency profiles of various harmonics. A theoretical analysis of the time-frequency profile of various harmonics, which is sensitive to laser-atom (molecule) interactions, reveals new insights regarding the multiphoton dynamics and the physical mechanisms responsible for the generation of harmonics in different energy regime [5]. Detailed results and their application to the coherent control of HHG processes will be reported.
 
REFERENCES
 
[1] Z. Chang et al., Phys. Rev. Lett. 79 (1997) 2967.
[2] M. Schnurer et al., Phys. Rev. Lett. 80 (1998) 3236.
[3] X. M. Tong and S. I. Chu, Chem. Phys. 217 (1997) 117.
[4] X. M. Tong and S. I. Chu, Phys. Rev. A57 (1998) 452.
[5] X. M. Tong, T. F. Jiang, and S. I. Chu, to be published.

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Quantizing Frame Coefficients

 

Ingrid Daubechies

 

Princeton University

Department of Mathematics

Princeton, NJ 08544

 
In many applications where wavelets or the windowed Fourier transform are used to analyze a signal, it is well known that an approach using a redundant representation is often the right answer: from the corresponding redundant time-scale or time-frequency representation it is often easier to extract salient features than if an orthonormal basis were used. It is less obvious how to discretize frame coefficients (as opposed to orthonormal basis coefficients) in an efficient way. It turns out that there exists a setting in which frames, and very coarsely quantized frame coefficients, are used in practice: "sigma-delta quantization" is an algorithm used in practical analog-to-digital conversion schemes in which bandlimited signals are heavily oversampled; from these samples a very coarse quantization is then computed, which nevertheless leads to very accurate reconstruction. Surprisingly little mathematical work has been done in this area, however. The talk will present some joint work with Ron DeVore, proving mathematically that for at least some concrete schemes, accuracy to arbitrary inverse polynomial order can be achieved.

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Numerical Methods for Solving the Schrödinger

Equation for One Body or One Million

 

Brett Esry

 

ITAMP

Harvard-Smithsonian Center for Astrophysics

60 Garden Street, MS 14

Cambridge, MA 02138

 
Solving the N-body Schrödinger equation is the main occupation of theorists in atomic and molecular physics. It is a 3N dimensional partial differential equation that can be reduced to 3N-6 dimensions for N>2, and is separable in only a very few cases. In the bulk of its applications, then, numerical methods must be utilized. A survey of some current problems of interest in atomic and molecular physics will be presented along with a selection of the numerical methods used to solve them. In particular, the utility of finite elements and basis splines will be discussed.

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Linear Algebra Methods in the Study of

Biological Sequences

 

C. Devauchelle, A.Henaut, J.L.Risler

 

Genome et Informatique,

Universite de Versailles-Saint-Quentin

Versailles, France

 

A. Grossmann, M. Holschneider, B. Torresani

 

CPT, CNRS-Luminy

Marseille, France

 
The information contained in a set of alignments of biological sequences can be usefully represented by a cloud of points in a suitable space. We describe methods for constructing and analyzing such clouds.

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Moment-Wavelet Quantization of Schrödinger Hamiltonians

 

Carlos R. Handy and Romain Murenzi

 

Department of Physics

and

Center for Theoretical Studies of

Clark Atlanta University

223 James P. Brawley Drive

Atlanta, Georgia 30314, U.S.A.

 
We have developed a new quantization formalism based on the use of appropriately rescaled and translated power moments. This approach defines a multiscale quantization procedure in which infinite scale information (i.e. the energy and certain moments) is used to systematically generate smaller scale quantities, including the recovery of the wavefunction. The latter results from solving a finite set of coupled, linear, first order differential equations in the inverse scale variable. This formalism enables the exact transformation and inversion of Schrödinger Hamiltonian problems within the context of a continuous wavelet transform (CWT) representation, without the need for any Galerkin type of approximation. The formalism and its application are presented, including the development of a complete and straightforward dyadic-CWT analysis based on our moments' perspective.
 
*Supported through the National Science Foundation (HRD 9450386) and the ARL-Fed Labs Consortium (DAAL01-96-2-0001)

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Coherent States: Principles and Selected Applications

 

John Klauder

 

University of Florida

Department of Physics

Gainesville, FL 32611-8440

 
A contemporary introduction to various coherent state families and their applications is presented. Attention is focused on the role of coherent states in the: (1) classical-quantum correspondence; (2) semi-classical approximation; (3) path integral construction (including constraints) and (4) combination of geometry and probability to yield quantization. Some current research directions are also indicated.
 

 

Coherent States for the Hydrogen Atom (Again)

Systems with a general energy spectrum admit coherent states having several desirable physical properties. It is shown how such states may be used in analyses related to the hydrogen atom.

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Distributed Approximating Functionals, Wavelets, Science, and Engineering

 

Donald J. Kouri

 

University of Houston

Department of Physics

Houston, TX 77204-5506

 
Distributed approximating functionals (DAFs) can be viewed as "scale functions" in the context of wavelet theory, and they provide a means of obtaining powerful new classes of wavelets. In addition, DAFs satisfy the properties of "window functions" and as such, can be used in all types of signal processing (either in a wavelet-based context or in a DAF-based context). Since the solution of partial differential equations amounts to deducing the true signal (i.e., the solution of the PDE) from a finite, discrete sampling, it is simply another example of signal processing. As a result, advances in one type of signal processing immediately imply advances in all other areas of signal processing. At least two classes of DAFs (interpolative and non-interpolative) exist and have been developed and used for solving quantum scattering and eigenvalue problems, for solving various nonlinear partial differential equations of science and engineering (e.g., the Kuramoto-Sivishinsky equation describing pattern formation in flame fronts and in other physical and chemical processes, the Navier-Stokes equations of fluid dynamics, various Fokker-Planck equations from kinetic theory and statistical mecahnics, the Sine-Gordon, Klein-Gordon, and Korteweg de Vries equations, and others), for denoising a wide variety of experimental data (e.g., 2-D images such as photographs, mamograms, MRI, target acquisition under low light conditions or sonor detection in the midst of clutter, etc.), for data padding and periodic extension of functions in multi-dimensions (with applications in inversion of spectra, medical imaging and diagnostics, etc.), for time series prediction, for data compression and signal reconstruction, and others. Some important features of DAFs are:
 
1) they are infinitely smooth
 
2) they decay exponentially in both physical and Fourier domains
 
3) they can be applied using Fast Convolution methods, leading to highly efficient algorithms
 
4) they can be made to achieve arbitrarily high accuracy
 
5) they can be used with any type sampling scheme, including those based on equal or unequally spaced Newton-Cotes quadrature, random or Monte Carlo sampling, number theoretic sampling, non-product sampling for multi-dimensional problems, etc.
 
6) they can be used in any of the "dynamical propagation" methods of denoising with "edge protection".

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Problems in Atomic Collision Physics: Can Wavelets Help?

 

Steven T. Manson

 

Department of Physics and Astronomy

Georgia State University

Altanta, GA 30303

 
The theoretical calculation of atomic processes is often performed via the expansion of the wave functions of the initial and final states in a series of some sort of basis states. For the discrete wave functions of bound states, a number of basis sets for expanding the wave functions have been employed--B-splines, Slater-type orbitals, etc. These techniques are not very well-suited to the expansion of atomic continuum wave functions, which occur in ionizing transitions, owing the infinite extent of such wave functions. Thus, we search for a new type of expansion technique for these continuum wave functions. Could wavelets be the answer?

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Coherent States and Semiclassical Approximation

 

Thierry Paul

 

Université Paris-Dauphine

CEREMADE

Place du Maréchal de Lattre de Tassigny

75775 Paris Cedex 16, France

 
A review on recent results in semiclassical asymptotics for the linear Schrödinger equation will be presented, with emphasis on the situation where the underlying classical system presents some chaotic features.

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Wavelet-Based Chirp Detection, and Application to the Detection of Gravitational Waves at Interferometric Detectors

 

Bruno Torresani

 

LATP, CMI,

Université de Provence

39 rue F. Joliot-Curie,

13453 Marseilles Cedex 13, France

 
Current experiments for gravitational waves detection at interferometric detectors raise the problem of detection and parameter estimation for frequency modulated signals with power law instantaneous frequency, embedded in strong experimental noise. We describe a wavelet based method for simultaneous detection and parameter estimation of such signals. We also describe the possible application to the gravitational waves detection problem, and compare its performances to matched filtering strategies.

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Non-separable Wavelets in Two and Three Dimensions

 

C. J. Tymczak and Heinrich Röder

 

MS B221, Group T-1

Los Alamos National Laboratory

Los Alamos NM 87545

 
 
We report on a fast local method to construct non-separable two and three dimensional wavelets. The approach is based upon the solutions of Lagrange interpolation by polynomials in Rd [1]. Expanding upon the work of Sweldens [2], we have
succeeded in constructing two and three dimensional interpolating wavelets up to eighth order. These wavelets are stable, have compact support, are isotropic, and automatically satisfy certain Strang-Fix conditions. Several examples are presented.
 
1. Carl de Boor and Amos Ron, Computational aspects of polynomial interpolation in several variables, Math Comp., V58, 198, April 1992.
 
2. Jelena Kovacevic and Wim Sweldens, Wavelet families of increasing order in arbitrary dimensions, IEEE Trans. Signal Proc., December 1997.

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Wavelet Bases for Quantum Mechanical Calculations

 

Xiao-Qian Wang

 

Department of Physics and

Center for Theoretical Studies of Physical Systems

Clark Atlanta University

Atlanta, GA 30314

 
We have devised an approach [1] utilizing compactly supported, orthonormal wavelet bases for quantum molecular dynamics (Car-Parrinello) algorithms. A wavelet selection scheme is developed and tested for prototype problems, such as the three-dimensional harmonic oscillator, the hydrogen atom, and the local density approximation to atomic and molecular systems. Our method shows systematic convergence with increased grid size, along with improvement on compression rates; thereby yielding an optimal grid for self-consistent electronic structure calculations. The application of this method to large systems is in progress.
 
On the other hand, we have discovered an efficient scheme [2] of applying non-orthogonal wavelets to quantum chemistry calculations. This is based on the observation that in a power-series type expansion of the electronic wave function with a properly-chosen reference function (such as Gaussion or Slater functions), the converging zeros of the coefficient functions approximate the exact energies of atomic and molecular levels. This approach is shown to be equivalent to a convergent variational determinant quantization procedure. This method has been applied, with remarkable success, to various quantum mechanical problems. The application of this method to quantum-chemistry calculations will be discussed.
 
[1] C. J. Tymczak and Xiao-Qian Wang, Phys. Rev. Lett. 78 (1997) 3654.
[2] C. J. Tymczak, G. S. Japaridze, C. R. Handy, and Xiao-Qian Wang, Phys. Rev. Lett. 80 (1998) 3673.

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Numerical Solution of the Basic XAFS Equation by the Wavelet-Galerkin Method

 

Koichiro Yamaguchi*, Yoshiaki Ito*, Takeshi Mukoyama*, Masao Takahashi** and Shuichi Emura**

 

*Institute for Chemical Research

Kyoto University

Uji, Kyoto 611, Japan

 

**The I.S.I.R.

Osaka University

Mihogaoka 8-1,

Ibaraki, Osaka 567, Japan

 
Currently, the basic XAFS equation is commonly solved with the Fourier transform. The radial distribution functions given by the Fourier transform are generally complex valued and often severely deformed from the real one. Techniques to solve linear inverse problems are preferable to the Fourier transform to obtain the real valued optimized solution of the basic XAFS equation. We have constructed a linear inverse problem algorithm based on wavelet-Galerkin method to solve the single-component basic XAFS equation. Our method is examined by using synthesized model K-edge Cu XAFS spectra and experimentally observed one.

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Analysis of Random Cascades using Wavelet Analysis: From Theoretical Concepts to Experimental Applications

 

A. Arneodo

 

Centre de Recherche Paul Pascal

Avenue Schweitzer, 33600 Pessac, France

 
Multiplicative cascade models have enjoyed an increasing interest in recent years as the paradigm of multifractal objects1-4. The notion of cascade actually refers to a self-similar process whose properties are defined multiplicatively from coarse to fine scales. In that respect, it occupies a central place in the statistical theory of turbulence3,5. Since Richardson's famous poem, the turbulent cascade picture has been often invoked to account for the intermittency phenomenon observed in fully developed turbulent flows5,6: energy is transfered from large eddies down to small scales (where it is dissipated) through a cascade process in which the transfer rate at a given scale is not spatially homogeneous, as supposed in the theory developed by Kolmogorov7 in 1941, but undergoes local intermittent fluctuations5. Over the past thirty years, refined models including the log-normal model of Kolmogorov8 and Obukhov9, multiplicative hierarchical cascade models like the random beta-model, the alpha-model, the p-model (for review, see Ref. [3]), the log-stable models10 and more recently the log-infinitely divisible cascade models11-14 with the rather popular log-Poisson model advocated by She and Leveque15, have grown in the literature as reasonable models to mimic the energy cascading process in turbulent flows. On a very general ground, a self-similar cascade is defined by the way the scales are refined and by the statistics of the multiplicative factors at each step of the process3,4,10. One can thus distinguish discrete cascades that involve discrete scale ratios leading to log-periodic corrections to scaling (discrete scale invariance16), from continous cascades without preferable scale factors (continuous scale invariance). As far as the fragmentation process is concerned, one can specify whether some conservation laws are operating or not4; in particular, one can discriminate between conservative (the measure is conserved at each cascade step) and non conservative (only some fraction of the measure
is transfered at each step) cascades. More fundamentally, there are two main classes of self-similar cascade processes: deterministic cascades that generally correspond to solvable models and random cascades that are likely to provide more realistic models but for which some theoretical care is required as far as their multifractal limit and some basic multifractal properties (including multifractal phase transitions) are concerned4. As a notable member of the later class, the independent random cascades introduced by Mandelbrot6 as a general model of random curdling in fully developed turbulence, have a special status since they are the main cascade model for which deep mathematical results have been obtained17,18. More recently, the concept of self-similar cascades leading to multifractal measures has been generalized to the construction of scale-invariant signals using orthonormal wavelet basis19-22. Instead of reditributing the measure over sub-intervals with multiplicative weights, one allocates the wavelet coefficients in a multiplicative way on a dyadic grid. This method allows us to generate multifractal functions from a given deterministic or probabilistic multiplicative process.
 
The main goal of this paper is to emphasize the continuous wavelet transform as a very efficient tool to get deep insight into the hierarchical structural complexity of multifractal objects23,24. Our point is to show that the main information about the underlying cascading process can actually be extracted from the wavelet transform skeleton defined by the wavelet transform modulus maxima25 (WTMM). We will discuss two situations which are typical, namely the case of deterministic discrete cascades and the case of random continuous cascades. For both cases, we will illustrate our purpose with some applications to experimental data ranging from fractal growth phenomena to fully developed turbulence and to financial time-series.
 
Deterministic discrete cascades: In many cases, the self-similarity properties of fractal objects can be expressed in terms of a dynamical system which leaves the object invariant. The inverse problem consists in recovering this dynamical system (or its main characteristics) from the data representing the fractal object. This problem has been previously approached within the theory of Iterated Function Systems26 (IFS). But the methods developed in this context are based on the search for a "best-fit" within a prescribed class of IFS attractors (mainly linear homogeneous attractors). In that sense, they approximate the self-similarity properties more than they reveal them. We show that, in many situations, the WTMM skeleton can be used to extract some one-dimensional (1D) map which accounts for its construction process27,28. We illustrate our theoretical considerations on pedagogical examples including Bernoulli invariant measures of linear and nonlinear expanding Markov maps as well as the invariant measure of period-doubling dynamical systems at the onset of chaos. We apply this wavelet based technique to analyze the fractal properties of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off\--lattice Diffusion\--Limited\--Aggregates (DLA) with a circle28-30. This study clearly reveals the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology31. These results demonstrate the statistical relevance of the golden mean arithmetic to the fractal hierarchy of DLA azimuthal Cantor sets.
 
Random continuous cascades: We start introducing various statistical quantities such as i) the statistical singularity spectrum which quantifies the statistical contribution of each singularity in the signals22; ii) the self-similarity kernel14,20,21,32,33 which, from a statistical point of view, characterizes the nature of the cascade process from a given scale to a finer scale and iii) the space-scale correlation functions34 which can be proved to follow a power-law behavior when varying the spatial distance of the two wavelet coefficients. We show mathematically and check numerically on various computer synthetized signals22,35 that these statistical quantities can be extracted directly from the considered fractal function using its WTMM skeleton with an arbitrary analyzing wavelet. This mathematical study actually provides algorithms that are readily applicable to experimental situations. We report on recent applications of our methodology in the context of fully-developed turbulence20,21,34,35. Our results show that the commonly accepted multifractal description of the intermittency phenomenon is not valid for finite Reynolds number flows. In particular, the search for a cascade process reveals a lack of scale-invariance for both the velocity and the dissipation fields. The statistical study of the WTMM yields a very convincing log-normal law on a well defined range of scales that can be further used as an objective definition of the inertial range. But in contrast to the Kolmogorov-Obukhov log-normal cascade model8,9, we find that the number of cascade steps does not evolve logarithmically as a function of the scale which is the signature of the breaking of scale-invariance20,21. We further comment on the possible asymptotic validity of the log-normal multifractal description.
 
These very first applications are very promising as fas as further experimental investigations of multilplicative cascade processes are concerned. It is likely that similar wavelet-based statistical analysis will lead to significant progress in other fields than fractal growth phenomena and fully developed turbulence. To conclude, we report on preliminary results of a similar investigation of financial time series36. Underlying the fluctuations of volatility (standard deviation) of price variations, we have discovered the existence of a causal information cascade from large to small time scales that can be visualized with the wavelet representation. Let us emphasize that the fact the variations of prices over one month scale influence in the future the daily prices variations, is likely to be extraordinarily rich in consequences and this, not only for the fundamental understanding of the nature of financial markets, but also (and may be more important) for practical applications. Indeed, the nature of the correlations across scales that are implied by this causal cascade have profound implications on the market risk, a problem of upmost concern for all financial institutions as well as individuals.
 
References
 
[1] B.B. Mandelbrot, Fractals: Form, Chance and Dimension Freeman, San Francisco, 1977); The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
 
[2] G. Paladin et A. Vulpiani, Phys. Rep. 156 (1987) 148.
 
[3] C. Meneveau and K.R. Sreenivasan, J. Fluid Mech. 224 (1991) 429.
 
[4] H.G.E. Hentschel, Phys. Rev. E 50 (1994) 243.
 
[5] U. Frisch, Turbulence (Cambridge Univ. Press, Cambridge, 1995).
 
[6] B.B. Mandelbrot, J. Fluid Mech. 62 (1974) 331.
 
[7] A.N. Kolmogorov, C.R. Acad. Sci. USSR 30 (1941) 301.
 
[8] A.N. Kolmogorov, J. Fluid Mech. 13 (1962) 82.
 
[9] A.M. Obukhov, J. Fluid Mech. 13 (1962) 77.
 
[10] D. Schertzer, S. Lovejoy, F. Schmitt, Y. Chigirinskaya and D. Marsan, Fractals 5 (1997) 427.
 
[11] E.A. Novikov, Phys. Fluids A 2 (1990) 814; Phys. Rev. E 50 (1995) 3303.
 
[12] Z.S. She and E.C. Waymire, Phys. Rev. Lett. 74 (1995) 262.
 
[13] B. Dubrulle, Phys. Rev. Lett. 73 (1994) 959; J. Phys. II France 6 (1996) 1825.
 
[14] B. Castaing and B. Dubrulle, J. Phys. II France 5 (1995) 895.
 
[15] Z.S. She and E. Leveque, Phys. Rev. Lett. 72 (1994) 336.
 
[16] D. Sornette, in Scale Invariance and Beyond, B. Dubrulle, F. Graner and D.Sornette, eds (EDP Sciences, Les Ulis & Springer-Verlag, Berlin, 1997), pp. 235.
 
[17] J.P. Kahane and J. Peyrière, Advances in Mathematics 22 (1976) 131.
 
[18] G.M. Molchan, Comm. Math. Phys. 179 (1996) 681.
 
[19] R. Benzi, L. Biferale, A. Crisanti, G. Paladin, M. Vergassola and A. Vulpiani, Physica D 65 (1993) 352.
 
[20] A. Arneodo, J.F. Muzy and S.G. Roux, J. Phys II France 7 (1997) 363.
 
[21] A. Arneodo, S. Manneville and J.F. Muzy, Eur. Phys. J. B 1 (1998) 129.
 
[22] A. Arneodo, E. Bacry and J.F. Muzy, "Random cascades on wavelet dyadic trees", J. Math. Phys. (1998), to appear.
 
[23] J.F. Muzy, E. Bacry and A. Arneodo, Int. J. of Bifurcation and Chaos 4 (1994) 245.
 
[24] A. Arneodo, E. Bacry and J.F. Muzy, Physica A 213 (1994) 232.
 
[25] S. Mallat, A Wavelet Tour in Signal Processing (Academic Press, New York, 1998).
 
[26] M.F. Barnsley, Fractals Everywhere (Academic Press, New York, 1988).
 
[27] A. Arneodo, E. Bacry and J.F. Muzy, Europhys. Lett. 25 (1994) 479.
 
[28] A. Arneodo, F. Argoul, J.F. Muzy, M. Tabard and E. Bacry, Fractals 1 (1993) 629.
 
[29] A. Arneodo, F. Argoul, E. Bacry, J.F. Muzy and M. Tabard, Phys. Rev. Lett. 68 (1992) 3456; J. Diff. Eq. & Appl. 1 (1995) 117.
 
[30] A. Kuhn, F. Argoul, J.F. Muzy and A. Arneodo, Phys. Rev. Lett. 73 (1994) 2998.
 
[31] A. Arneodo, F. Argoul, J.F. Muzy and M. Tabard, Phys. Lett. 171 A (1992) 31; Physica 188 A (1992) 217.
 
[32] B. Castaing, Y. Gagne and E.J. Hopfinger, Physica D 46 (1990) 177.
 
[33] B. Castaing, in Scale Invariance and Beyond, B. Dubrulle, F. Graner and D. Sornette, eds (EDP Sciences, Les Ulis & Springer-Verlag, Berlin 1997), pp. 225.
 
[34] A. Arneodo, E. Bacry, S. Manneville and J.F. Muzy, Phys. Rev. Lett. 80 (1998) 708.
 
[35] S.G. Roux, Thesis, University of Aix-Marseille II (1996).
 
[36] A. Arneodo, J.F. Muzy and D. Sornette, Eur. Phys. J. B 1 (1998), to appear.

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 Schedule of Talks

Thursday, October 8, 1998 [Pratt Conference Room]

 8:45 a.m. Welcome coffee; pick up workshop packets and nametags; introductory remarks

Session I: Introduction to Wavelets

Chair: Romain Murenzi

 9:15 a.m. J.-P. Antoine: "The continuous wavelet transform: an overview"
 10:05 a.m. Coffee
10:30 a.m. M.B. Ruskai: "Overcoming the uncertainty principle: wavelets and their offshoots"
11:20 a.m. J. R. Klauder: "Coherent states: principles and selected applications"
12:10 p.m. Lunch

Session II: Application of Wavelets in Atomic and Molecular Physics

Chair: Alfred Msezane

[Classroom, Building A]

 2:00 p.m. T.A. Arias: "Multiresolution analysis of electronic structure"
 2:35 p.m. D. Kouri: "Distributed approximating functionals, wavelets, science, and engineering"
 3:10 p.m. Refreshments
3:30 p.m. X.-Q. Wang: "Wavelet bases for quantum mechanical calculations"
 4:05 p.m. C.R. Handy: "Moment-wavelet quantization of Schrödinger Hamiltonians"
 4:40 p.m. K. Yamaguchi: "Numerical solution of the basic XAFS equation by the Wavelet-Galerkin method"

 5:30-6:30 p.m.: Reception in Perkin Lobby

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Friday, October 9, 1998 [Phillips Auditorium]

Session III: Multi-disciplinary Application of Wavelets

Chair: Jean-Pierre Antoine

 9:00 a.m. A. Arneodo: "Analysis of random cascades using wavelet analysis: from theoretical concepts to experimental applications."
 9:35  a.m. A. Grossman: "Linear algebra methods in the study of biological  sequences"
 10:10 a.m. Coffee
 10:30 a.m. B. Torresani: "Wavelet-based chirp detection, and application to the detection ofgravitational waves at interferometric detectors"
 11:05 a.m. J.P. Gazeau: "An algebraic construction of affine wavelets for Penrose-Robinson tilings in the complex plane"
 11:40 a.m. C. J. Tymczak: "Non-separable wavelets in two and three dimensions"
 12:15 p.m. J.-P. Antoine: "Time-frequency analysis in laser-atom interaction: wavelets vs. Gabor"
 12:50 p.m. Lunch

Session IV: Numerical Schemes in Atomic and Molecular Physics

Chair: Hossein Sadeghpour

 2:00  p.m. B. Esry: "Numerical methods for solving the Schrödinger equation for one body or one million"
 2:35  p.m. S.T. Manson: "Problems in atomic collision physics: can wavelets help?"
 3:10  p.m. Refreshments
 3:25 p.m. S-I. Chu: "Wavelet analysis of the time-frequency profile of multiple high-order harmonics generated by atoms in intense laser fields"
 4:00 p.m. Round Table Discussion

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Saturday, October 10, 1998 [Phillips Auditorium]

Session V: Wavelets and More

Chair: Alex Grossman

 9:00 a.m. I. Daubechies: "Quantizing frame coefficients"
 9:45 a.m. J. Edwards: "Wavelet analysis of two dimensional quantum scattering"
 10:20 a.m. Coffee
 10:35 a.m. T. Paul: "Coherent states and semiclassical approximation"
 11:10 a.m. S.T. Ali: "A generalized Wigner transform and its relation to the wavelet transform"
 11:45 a.m. J. R. Klauder: "Coherent states for the hydrogen atom (again)"
 12:20 p.m. R. Brummelhuis: "Phase-space localization for atoms in strong magnetic fields"
1:00 p.m. End of Workshop

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