Citation
Demanet, Laurent (2006) Curvelets, wave atoms, and wave equations. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd05262006133555
Abstract
We argue that two specific wave packet familiescurvelets and wave atomsprovide powerful tools for representing linear systems of hyperbolic differential equations with smooth, timeindependent coefficients. In both cases, we prove that the matrix representation of the Green's function is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e., faster than any negative polynomial), and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals, whose location is predicted by geometrical optics.
This result holds only when the basis elements obey a precise parabolic balance between oscillations and support size, shared by curvelets and wave atoms but not wavelets, Gabor atoms, or any other such transform.
A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.
We also provide fast digital implementations of tight frames of curvelets and wave atoms in two dimensions. In both cases the complexity is O(N^2 log N) flops for NbyN Cartesian arrays, for forward as well as inverse transforms.
Finally, we present a geometric strategy based on wave atoms for the numerical solution of wave equations in smoothly varying, 2D timeindependent periodic media. Our algorithm is based on sparsity of the matrix representation of Green's function, as above, and also exploits its lowrank block structure after separation of the spatial indices. As a result, it becomes realistic to accurately build the full matrix exponential using repeated squaring, up to some time which is much larger than the CFL timestep. Once available, the wave atom representation of the Green's function can be used to perform 'upscaled' timestepping.
We show numerical examples and prove complexity results based on a priori estimates of sparsity and separation ranks. They beat the O(N^3) bottleneck on an NbyN grid, for a wide range of physically relevant situations. In practice, the current wave atom solver can become competitive over a pseudospectral method in the regime when the wave equation should be solved several times with different initial conditions, as in reflection seismology.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  harmonic analysis; microlocal analysis; numerical analysis; wave propagation 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Applied And Computational Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  19 May 2006 
Record Number:  CaltechETD:etd05262006133555 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd05262006133555 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2112 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  01 Jun 2006 
Last Modified:  26 Dec 2012 02:47 
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