20,334 research outputs found
Analysis of new direct sampling indicators for far-field measurements
This article focuses on the analysis of three direct sampling indicators
which can be used for recovering scatterers from the far-field pattern of
time-harmonic acoustic measurements. These methods fall under the category of
sampling methods where an indicator function is constructed using the far-field
operator. Motivated by some recent work, we study the standard indicator using
the far-field operator and two indicators derived from the factorization
method. We show the equivalence of two indicators previously studied as well as
propose a new indicator based on the Tikhonov regularization applied to the
far-field equation for the factorization method. Finally, we give some
numerical examples to show how the reconstructions compare to other direct
sampling methods
Nearfield Acoustic Holography using sparsity and compressive sampling principles
Regularization of the inverse problem is a complex issue when using
Near-field Acoustic Holography (NAH) techniques to identify the vibrating
sources. This paper shows that, for convex homogeneous plates with arbitrary
boundary conditions, new regularization schemes can be developed, based on the
sparsity of the normal velocity of the plate in a well-designed basis, i.e. the
possibility to approximate it as a weighted sum of few elementary basis
functions. In particular, these new techniques can handle discontinuities of
the velocity field at the boundaries, which can be problematic with standard
techniques. This comes at the cost of a higher computational complexity to
solve the associated optimization problem, though it remains easily tractable
with out-of-the-box software. Furthermore, this sparsity framework allows us to
take advantage of the concept of Compressive Sampling: under some conditions on
the sampling process (here, the design of a random array, which can be
numerically and experimentally validated), it is possible to reconstruct the
sparse signals with significantly less measurements (i.e., microphones) than
classically required. After introducing the different concepts, this paper
presents numerical and experimental results of NAH with two plate geometries,
and compares the advantages and limitations of these sparsity-based techniques
over standard Tikhonov regularization.Comment: Journal of the Acoustical Society of America (2012
Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues
We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a
bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound
the distance between an arbitrary parameter and the spectrum
in terms of the boundary -norm of a normalized trial solution of the
Helmholtz equation . We also bound the -norm of the
error of this trial solution from an eigenfunction. Both of these results are
sharp up to constants, hold for all greater than a small constant, and
improve upon the best-known bounds of Moler--Payne by a factor of the
wavenumber . One application is to the solution of eigenvalue
problems at high frequency, via, for example, the method of particular
solutions. In the case of planar, strictly star-shaped domains we give an
inclusion bound where the constant is also sharp. We give explicit constants in
the theorems, and show a numerical example where an eigenvalue around the
2500th is computed to 14 digits of relative accuracy. The proof makes use of a
new quasi-orthogonality property of the boundary normal derivatives of the
eigenmodes, of interest in its own right.Comment: 18 pages, 3 figure
The Virtual Element Method with curved edges
In this paper we initiate the investigation of Virtual Elements with curved
faces. We consider the case of a fixed curved boundary in two dimensions, as it
happens in the approximation of problems posed on a curved domain or with a
curved interface. While an approximation of the domain with polygons leads, for
degree of accuracy , to a sub-optimal rate of convergence, we show
(both theoretically and numerically) that the proposed curved VEM lead to an
optimal rate of convergence
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