Advanced

A multi-scale approach to hyperbolic evolution equations with limited smoothness

Andersson, Fredrik LU ; De Hoop, Maarten V; Smith, Hart F and Uhlmann, Gunther (2008) In Communications in Partial Differential Equations 33(6). p.988-1017
Abstract
We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Holder... (More)
We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Holder regularity s >= 2, and (iii) suggests a novel computational algorithm that requires solving for the mentioned geometry on the one hand and solving a matrix Volterra integral equation of the second kind on the other hand. The Volterra equation can be solved by recursionas in the computation of certain multiple scattering seriesrevealing a curvelet-curvelet interaction. We give precise estimates expressing the degree of concentration of curvelets following the propagation of singularities. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
pseudodifferential evolution equations, decomposition, paradifferential, curvelets, dyadic parabolic decomposition
in
Communications in Partial Differential Equations
volume
33
issue
6
pages
988 - 1017
publisher
Taylor & Francis
external identifiers
  • wos:000257078500003
  • scopus:45949095172
ISSN
0360-5302
DOI
10.1080/03605300701629393
language
English
LU publication?
yes
id
95157408-d995-4609-890e-18c5e461586f (old id 1187020)
date added to LUP
2008-09-04 10:49:37
date last changed
2017-01-01 05:00:47
@article{95157408-d995-4609-890e-18c5e461586f,
  abstract     = {We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Holder regularity s >= 2, and (iii) suggests a novel computational algorithm that requires solving for the mentioned geometry on the one hand and solving a matrix Volterra integral equation of the second kind on the other hand. The Volterra equation can be solved by recursionas in the computation of certain multiple scattering seriesrevealing a curvelet-curvelet interaction. We give precise estimates expressing the degree of concentration of curvelets following the propagation of singularities.},
  author       = {Andersson, Fredrik and De Hoop, Maarten V and Smith, Hart F and Uhlmann, Gunther},
  issn         = {0360-5302},
  keyword      = {pseudodifferential evolution equations,decomposition,paradifferential,curvelets,dyadic parabolic decomposition},
  language     = {eng},
  number       = {6},
  pages        = {988--1017},
  publisher    = {Taylor & Francis},
  series       = {Communications in Partial Differential Equations},
  title        = {A multi-scale approach to hyperbolic evolution equations with limited smoothness},
  url          = {http://dx.doi.org/10.1080/03605300701629393},
  volume       = {33},
  year         = {2008},
}