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

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 $1$ 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 $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ 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