Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sampling and reconstruction of solutions to the Helmholtz equation   We consider the inverse problem of reconstructing general solutions to the
Helmholtz equation on some domain $\Omega$ from their values at scattered
points $x_1,\dots,x_n\subset \Omega$. This problem typically arises when
sampling acoustic fields with $n$ microphones for the purpose of reconstructing
this field over a region of interest $\Omega$ contained in a larger domain $D$
in which the acoustic field propagates. In many applied settings, the shape of
$D$ and the boundary conditions on its border are unknown. Our reconstruction
method is based on the approximation of a general solution $u$ by linear
combinations of Fourier-Bessel functions or plane waves. We analyze the
convergence of the least-squares estimates to $u$ using these families of
functions based on the samples $(u(x_i))_{i=1,\dots,n}$. Our analysis describes
the amount of regularization needed to guarantee the convergence of the least
squares estimate towards $u$, in terms of a condition that depends on the
dimension of the approximation subspace, the sample size $n$ and the
distribution of the samples. It reveals the advantage of using non-uniform
distributions that have more points on the boundary of $\Omega$. Numerical
illustrations show that our approach compares favorably with reconstruction
methods using other basis functions, and other types of regularization.

30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics