A1 Journal article (refereed), original research

Inverse Problem for the Wave Equation with a White Noise Source


Open Access publication

LUT Authors / Editors

Publication Details
Authors: Helin Tapio, Lassas Matti, Oksanen Lauri
Publisher: Springer Verlag (Germany)
Publication year: 2014
Language: English
Related Journal or Series Information: Communications in Mathematical Physics
Journal name in source: Communications in Mathematical Physics
Volume number: 332
Issue number: 3
Start page: 933
End page: 953
Number of pages: 21
ISSN: 0010-3616
JUFO-Level of this publication: 3
Open Access: Open Access publication
Location of the parallel saved publication: https://arxiv.org/pdf/1308.4879.pdf

Abstract


We consider a smooth Riemannian metric tensor g on Rn and study the stochastic
wave equation for the Laplace-Beltrami operator 2t u_gu =
F. Here, F =F(t, x, ω) is a random source that has
white noise distribution supported on the boundary of some smooth compact
domain M
Rn. We study the following
formally posed inverse problem with only one measurement. Suppose that g is known only outside of a compact
subset of Mint and that a solution u(t, x, ω0) is produced by a single
realization of the source F(t, x, ω0). We ask what information
regarding g can be recovered by measuring
u(t, x, ω0) on R+ ×M? We prove that such
measurement together with the realization of the source determine the
scattering relation of the Riemannian manifold (M, g) with probability one. That
is, for all geodesics passing through M, the travel times together
with the entering and exit points and directions are determined. In particular,
if (M, g) is a simple Riemannian
manifold and g is conformally Euclidian in M, the measurement determines
the metric g in M.


Last updated on 2021-09-04 at 11:05