NDLI: Compressive Wave Computation

Content Provider	Springer Nature Link
Author	Demanet, Laurent Peyré, Gabriel
Copyright Year	2011
Abstract	This paper presents a method for computing the solution to the time-dependent wave equation from the knowledge of a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen at random. While a linear superposition of eigenfunctions can fail to properly synthesize the solution if a single term is missing, it is shown that solving a sparsity-promoting ℓ 1 minimization problem can vastly enhance the quality of recovery. This phenomenon may be seen as “compressive sampling in the Helmholtz domain.”An error bound is formulated for the one-dimensional wave equation with coefficients of small bounded variation. Under suitable assumptions, it is shown that the number of eigenfunctions needed to evolve a sparse wavefield defined on N points, accurately with very high probability, is bounded by $$C(\eta) \cdot\log N \cdot\log\log N,$$ where C(η) is related to the desired accuracy η and can be made to grow at a much slower rate than N when the solution is sparse. To the authors’ knowledge, the partial differential equation estimates that underlie this result are new and may be of independent mathematical interest. They include an L 1 estimate for the wave equation, an L ∞−L 2 estimate of the extension of eigenfunctions, and a bound for eigenvalue gaps in Sturm–Liouville problems.In practice, the compressive strategy is highly parallelizable, and may eventually lead to memory savings for certain inverse problems involving the wave equation. Numerical experiments illustrate these properties in one spatial dimension.
Ending Page	303
Page Count	47
Starting Page	257
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	3
Volume Number	11
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-02-24
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Wave propagation Economics general Sparsity Numerical Analysis Computer Science Linear and Multilinear Algebras, Matrix Theory Computational harmonic analysis Applications of Mathematics Math Applications in Computer Science Compressed sensing
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Scattering in Flatland: Efficient Representations via Wave Atoms

Remote Sensing via ℓ 1-Minimization

Zigzag Persistence

Joint Invariant Signatures

Generalized Sampling and Infinite-Dimensional Compressed Sensing

Optimal Stability and Eigenvalue Multiplicity

Projective Plane and Planar Quantum Codes

Poly-Locality in Quantum Computing

Erratum to: Numerical Solution of Riemann–Hilbert Problems: Painlevé II

Compressive Wave Computation

Similar Documents

Scattering in Flatland: Efficient Representations via Wave Atoms

Remote Sensing via ℓ 1-Minimization

Zigzag Persistence

Joint Invariant Signatures

Generalized Sampling and Infinite-Dimensional Compressed Sensing

Optimal Stability and Eigenvalue Multiplicity

Projective Plane and Planar Quantum Codes

Poly-Locality in Quantum Computing

Erratum to: Numerical Solution of Riemann–Hilbert Problems: Painlevé II

Compressive Wave Computation