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Inverse source problems are central to many applications in acoustics, geophysics, non-destructive testing, and more. Traditional imaging methods suffer from the resolution limit, preventing distinction of sources separated by less than the emitted wavelength. In this work we propose a method based on physically informed neural-networks for solving the source refocusing problem, constructing a novel loss term which promotes super-resolving capabilities of the network and is based on the physics of wave propagation. We demonstrate the approach in the setup of imaging an a priori unknown number of point sources in a two-dimensional rectangular waveguide from measurements of wavefield recordings along a vertical cross section. The results show the ability of the method to approximate the locations of sources with high accuracy, even when placed close to each other. [ABSTRACT FROM AUTHOR]

We consider the problem of stable recovery of sparse signals of the form $$\begin{equation*}F(x)=\sum_{j=1}^d a_j\delta(x-x_j),\quad x_j\in\mathbb{R},\;a_j\in\mathbb{C}, \end{equation*}$$ from their spectral measurements, known in a bandwidth |$\varOmega $| with absolute error not exceeding |$\epsilon>0$|⁠. We consider the case when at most |$p\leqslant d$| nodes |$\{x_j\}$| of |$F$| form a cluster whose extent is smaller than the Rayleigh limit |${1\over \varOmega }$|⁠ , while the rest of the nodes is well separated. Provided that |$\epsilon \lessapprox \operatorname{SRF}^{-2p+1}$|⁠ , where |$\operatorname{SRF}=(\varOmega \varDelta)^{-1}$| and |$\varDelta $| is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order |${1\over \varOmega }\operatorname{SRF}^{2p-1}\epsilon $|⁠ , while for recovering the corresponding amplitudes |$\{a_j\}$| the rate is of the order |$\operatorname{SRF}^{2p-1}\epsilon $|⁠. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are |${\epsilon \over \varOmega }$| and |$\epsilon $|⁠ , respectively. These results suggest that stable super-resolution is possible in much more general situations than previously thought. Our numerical experiments show that the well-known matrix pencil method achieves the above accuracy bounds. [ABSTRACT FROM AUTHOR]

We prove sharp lower bounds for the smallest singular value of a partial Fourier matrix with arbitrary "off the grid"" nodes (equivalently, a rectangular Vandermonde matrix with the nodes on the unit circle) in the case when some of the nodes are separated by less than the inverse bandwidth. The bound is polynomial in the reciprocal of the so-called superresolution factor, while the exponent is controlled by the maximal number of nodes which are clustered together. As a corollary, we obtain sharp minimax bounds for the problem of sparse superresolution on a grid under the partial clustering assumptions. [ABSTRACT FROM AUTHOR]

In this paper we provide a reconstruction algorithm for piecewisesmooth functions with a priori known smoothness and a number of discontinuities, from their Fourier coefficients, possessing the maximal possible asymptotic rate of convergence-including the positions of the discontinuities and the pointwise values of the function. This algorithm is a modification of our earlier method, which is in turn based on the algebraic method of K. Eckhoff proposed in the 1990s. The key ingredient of the new algorithm is to use a different set of Eckhoff's equations for reconstructing the location of each discontinuity. Instead of consecutive Fourier samples, we propose to use a "decimated" set which is evenly spread throughout the spectrum. [ABSTRACT FROM AUTHOR]

Let Q be a relatively compact subset in a Hilbert space V. For a given ε > 0 let N(ε, Q) be the minimal number of linear measurements, sufficient to reconstruct any x ∈ Q with the accuracy ε. We call N(ε, Q) the sampling ε-entropy of Q. Using dimensionality reduction, as provided by the Johnson-Lindenstrauss lemma, we show that, in an appropriate probabilistic setting, N(ε, Q) is bounded from above by Kolmogorov's ε-entropy H(ε, Q), defined as H(ε, Q) = log M(ε, Q), with M(ε, Q) being the minimal number of ε-balls covering Q. As the main application, we show that piecewise smooth (piecewise analytic) functions in one and several variables can be sampled with essentially the same accuracy rate as their regular counterparts. For univariate piecewise Ck-smooth functions this result, which settles the so-called Eckhoff conjecture, was recently established in D. Batenkov, Complete Algebraic Reconstruction of Piecewise-smooth Functions from Fourier Data, arXiv:1211.0680, 2012 via a deterministic "algebraic reconstruction" algorithm. [ABSTRACT FROM AUTHOR]

In this paper we consider several nonlinear systems of algebraic equations which can be called "Prony-type." These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of "maximal possible accuracy" of solving Prony-type systems, putting stress on the "local" behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that "global" solution techniques such as Prony's algorithm and the ESPRIT method are suboptimal when compared to this theoretical "best local" behavior. [ABSTRACT FROM AUTHOR]