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We investigate stationary and non-stationary asymptotical regularization methods for statistical inverse problems in Banach spaces. The mean-squared errors (MSE) of both methods are estimated without the conventional assumption of the commutativity between the solution variance and the noise variance. Moreover, the a priori smoothness of solution is characterized in terms of real interpolation, which is strictly weaker than the classical settings. The sufficient and necessary conditions of optimal convergence rates of MSE are proven. Our results extend theoretical findings in Hilbert settings given in the recent paper by Lu et al. [SIAM/ASA J. Uncertain., 9 (2021), pp. 1488-1525]. Two examples of integral equations with white noise and inverse source problems in elliptic partial differential equations are demonstrated to show the usefulness of our theoretical results. [ABSTRACT FROM AUTHOR]

We propose an inverse scattering scheme of recovering a polyhedral obstacle in R n , n = 2 , 3 , by only a few high-frequency acoustic backscattering measurements. The obstacle could be sound-soft or sound-hard. It is shown that the modulus of the far-field pattern in the backscattering aperture possesses a certain local maximum behavior, from which one can determine the exterior normal directions of the front sides/faces. Then by using the phaseless backscattering data corresponding to a few incident plane waves with suitably chosen incident directions, one can determine the exterior unit normal vector of each side/face of the obstacle. After the determination of the exterior unit normals, the recovery is reduced to a finite-dimensional problem of determining a location point of the obstacle and the distance of each side/face away from the location point. For the latter reconstruction, we need to make use of the far-field data with phases. Numerical experiments are also presented to illustrate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract: We develop two enhanced techniques for the multilevel linear sampling method (MLSM) proposed in [32] for inverse scattering problems. Under some practical situations, the MLSM suffers certain undesirable “breakage cells” problem. We propose to avoid the curse of “breakage cells” by incorporating “expanding” and “searching” techniques. The new techniques are shown to significantly improve the robustness of the MLSM, and meanwhile they possess the same optimal computational complexity as the MLSM. Numerical experiments are presented to illustrate the promising features of the enhanced MLSMs. [Copyright &y& Elsevier]

In this paper, we propose a numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for the interface-resolved finite element approximation in both, the physical and the stochastic dimension. In particular, a fast finite difference scheme is proposed to compute the variance of random solutions by using a low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to validate and quantify the method. [ABSTRACT FROM AUTHOR]

In this article we investigate the analysis of a finite element method for solving H( curl; Ω)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H( curl; Ω)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter δ that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H( curl; Ω)-norm are obtained for the first time. The analysis is based on a δ-strip argument, a new extension theorem for H( curl; Ω)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H( curl; Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel method of reconstructing acoustic obstacles in R2, which follows a similar spirit of the linear sampling method originated by Colton and Kirsch. The reconstruction scheme makes use of the near-field measurements encoded into the boundary Dirichlet-to-Neumann map or the Neumann-to-Dirichlet map. Both the plane waves and cylindrical waves are shown to meet the reconstruction purpose. Rigorous mathematical justification of the reconstruction scheme is established. The mapping properties of the newly introduced function operators involved in the reconstruction scheme are established. These results are of significant mathematical interests for their own sake. Moreover, due to the distinct properties of the function operators, the indictor function in the proposed reconstruction scheme exhibits completely different behaviors from those having been established for the indictor function in the original linear sampling method for inverse scattering problems. Numerical experiments are presented to illustrate the effectiveness of the proposed reconstruction scheme. [ABSTRACT FROM AUTHOR]

We consider approximate cloaking from a regularization viewpoint introduced in Kohn et al (2008 Inverse Problems 24 015016) for EIT and further investigated in Kohn et al (2010 Commun. Pure Appl. Math. 63 0973-1016) and Liu (2009 Inverse Problems 25 045006) for the Helmholtz equation. The cloaking schemes given by Kohn et al and Liu are shown to be (optimally) within | ln ρ|-1 in 2D and ρ in 3D of perfect cloaking, where ρ denotes the regularization parameter. In this paper, we show that by employing a soundhard layer right outside the cloaked region, one could (optimally) achieve ρN in RN, N ≥ 2, which significantly enhances the near-cloak. We then develop a cloaking scheme by making use of a lossy layer with well-chosen parameters. The lossy-layer cloaking scheme is shown to possess the same cloaking performance as the one with a sound-hard layer. Moreover, it is shown that the lossy layer could be taken as a finite realization of the sound-hard layer. Numerical experiments are also presented to assess the cloaking performances of all the cloaking schemes for comparisons. [ABSTRACT FROM AUTHOR]

The presence of magnetized anomalies in the shell of the Earth interrupts its geomagnetic field. We consider the inverse problem of identifying the anomalies by monitoring the variation of the geomagnetic field. Motivated by the theoretical unique identifiability result in [7] , we develop a novel numerical scheme of locating multiple magnetized anomalies. In our study, we do not assume the source that generates the geomagnetic field, and the medium configurations of the Earth's core and the magnetized anomalies are a-priori known. The core of the reconstruction scheme is a novel imaging functional whose quantitative behaviours can be used to identify the anomalies. Both rigorous analysis and extensive numerical experiments are provided to verify the effectiveness and promising features of the proposed reconstruction scheme. • Magnetic anomalies detection from geomagnetic monitoring is practically important and challenging. • Based on a sophisticated geomagnetic model, a novel reconstruction scheme is proposed and developed. • In our method, the geomagnetic source and the medium configuration of the Earth's core are both not a-priori known. • The core is a novel imaging functional whose promising features are both theoretically and computationally verified. [ABSTRACT FROM AUTHOR]