caa a22 4500 445331569 CHVBK 20180317142739.0 cr unu---uuuuu 170323e20110601xx s 000 0 eng 10.1007/s10208-011-9085-5 doi (NATIONALLICENCE)springer-10.1007/s10208-011-9085-5 Compressive Wave Computation [Elektronische Daten] [Laurent Demanet, Gabriel Peyré] 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. SFoCM, 2011 Wave propagation nationallicence Sparsity nationallicence Compressed sensing nationallicence Computational harmonic analysis nationallicence Demanet Laurent Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA aut Peyré Gabriel CNRS and Ceremade, Université Paris-Dauphine, 75775, Paris Cedex 16, France aut Foundations of Computational Mathematics Springer-Verlag 11/3(2011-06-01), 257-303 1615-3375 11:3<257 2011 11 10208 https://doi.org/10.1007/s10208-011-9085-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10208-011-9085-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Demanet Laurent Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA aut NATIONALLICENCE 700 1- Peyré Gabriel CNRS and Ceremade, Université Paris-Dauphine, 75775, Paris Cedex 16, France aut NATIONALLICENCE 773 0- Foundations of Computational Mathematics Springer-Verlag 11/3(2011-06-01), 257-303 1615-3375 11:3<257 2011 11 10208 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer