980 research outputs found
A novel time-domain direct sampling approach for inverse scattering problems in acoustics
This work is concerned with an inverse scattering problem of determining
unknown scatterers from time-dependent acoustic measurements. A novel
time-domain direct sampling method is developed to efficiently determine both
the locations and shapes of inhomogeneous media. In particular, our approach is
very easy to implement since only cheap space-time integrations are involved in
the evaluation of the imaging functionals. Based on the Fourier-Laplace
transform, we establish an inherent connection between the time-domain and
frequency-domain direct sampling method. Moreover, rigorous theoretical
justifications and numerical experiments are provided to verify the validity
and feasibility of the proposed method
Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise
In this paper, we consider a new approach for semi-discretization in time and
spatial discretization of a class of semi-linear stochastic partial
differential equations (SPDEs) with multiplicative noise. The drift term of the
SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the
diffusion term is assumed to be globally Lipschitz continuous. Our new strategy
for time discretization is based on the Milstein method from stochastic
differential equations. We use the energy method for its error analysis and
show a strong convergence order of nearly for the approximate solution. The
proof is based on new H\"older continuity estimates of the SPDE solution and
the nonlinear term. For the general polynomial-type drift term, there are
difficulties in deriving even the stability of the numerical solutions. We
propose an interpolation-based finite element method for spatial discretization
to overcome the difficulties. Then we obtain stability, higher moment
stability, stability, and higher moment stability results
using numerical and stochastic techniques. The nearly optimal convergence
orders in time and space are hence obtained by coupling all previous results.
Numerical experiments are presented to implement the proposed numerical scheme
and to validate the theoretical results.Comment: 28 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1811.0502
Multipolar Acoustic Source Reconstruction from Sparse Far-Field Data using ALOHA
The reconstruction of multipolar acoustic or electromagnetic sources from
their far-field signature plays a crucial role in numerous applications. Most
of the existing techniques require dense multi-frequency data at the Nyquist
sampling rate. The availability of a sub-sampled grid contributes to the null
space of the inverse source-to-data operator, which causes significant imaging
artifacts. For this purpose, additional knowledge about the source or
regularization is required. In this letter, we propose a novel two-stage
strategy for multipolar source reconstruction from sub-sampled sparse data that
takes advantage of the sparsity of the sources in the physical domain. The data
at the Nyquist sampling rate is recovered from sub-sampled data and then a
conventional inversion algorithm is used to reconstruct sources. The data
recovery problem is linked to a spectrum recovery problem for the signal with
the \textit{finite rate of innovations} (FIR) that is solved using an
annihilating filter-based structured Hankel matrix completion approach (ALOHA).
For an accurate reconstruction, a Fourier inversion algorithm is used. The
suitability of the approach is supported by experiments.Comment: 11 pages, 2 figure
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