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Sampling and reconstruction of solutions to the Helmholtz equation

By Gilles Chardon, Albert Cohen and Laurent Daudet

Abstract

International audienceWe consider the problem of reconstructing general solutionsto the Helmholtz equation $\Delta u+\lambda^2 u=0$,for some fixed $\lambda>0$, on some domain $\Omega\subset \R^2$from the data of these functions at scattered points$x_1,\dots,x_n\subset \Omega$.This problem typically arises when sampling acoustic fields with $n$ microphones for the purpose ofreconstructing this field over a region of interest $\Omega$ that is contained in a larger domain $D$ (i.e., a room).% in which the acoustic field is defined.In many applied settings, the boundary conditionssatisfied by the acoustic field on $\partial D$ are unknownas well as the exact shape of $D$.Our reconstruction method is based onthe approximation of a general solution $u$by linear combinations of Fourier-Bessel functions or plane waves$e_{\bk}(x):=e^{i \bk \cdot x}$ with $|\bk|=\lambda$.We study two different ways of discretizingthe infinite dimensional space $V_\lambda$ of solutions to the Helmholtz equation, leading to two differenttypes of finite dimensional approximation subspaces, and we analyze the convergence of the least squares estimates to $u$ in these subspaces basedon the samples $(u(x_i))_{i=1,\dots,n}$. Our analysis describes the amount of regularization that is needed to guarantee the convergence of the least squares estimatetowards $u$, in terms of a condition that depends on thedimension of the approximation subspace andthe sample size $n$. This condition also involvesthe distribution of the samples and reveals theadvantage of using non-uniform distributionsthat have more points near or on the boundary of $\Omega$.Numerical illustrations show that our approach comparesfavorably with reconstruction methodsusing other basis functions, and other typesof regularization

Topics: [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA]
Publisher: Sampling publishing
Year: 2014
OAI identifier: oai:HAL:hal-01350609v1
Provided by: Hal-Diderot

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