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Author: Hu, Guanghui × End date: 2019 × Start date: 2010 × DDC: 510 × WGL Institute: WIAS ×

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Now showing 1 - 10 of 32
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    Near-field imaging of scattering obstacles with the factorization method
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Hu, Guanghui; Yang, Jiaqing; Zhang, Bo; Zhang, Haiwen
    In this paper we establish a factorization method for recovering the location and shape of an acoustic bounded obstacle with using the near-field data, corresponding to infinitely many incident point sources. The obstacle is allowed to be an impenetrable scatterer of sound-soft, sound-hard or impedance type or a penetrable scatterer. An outgoing-to-incoming operator is constructed for facilitating the factorization of the near-field operator, which can be easily implemented numerically. Numerical examples are presented to demonstrate the feasibility and effectiveness of our inversion algorithm, including the case where limited aperture near-field data are available only.
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    A time domain sampling method for inverse acoustic scattering problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Guo, Yukun; Hömberg, Dietmar; Hu, Guanghui; Li, Jingzhi; Liu, Hongyu
    This work concerns the inverse scattering problems of imaging unknown/inaccessible scatterers by transient acoustic near-field measurements. Based on the analysis of the migration method, we propose efficient and effective sampling schemes for imaging small and extended scatterers from knowledge of time-dependent scattered data due to incident impulsive point sources. Though the inverse scattering problems are known to be nonlinear and ill-posed, the proposed imaging algorithms are totally direct involving only integral calculations on the measurement surface. Theoretical justifications are presented and numerical experiments are conducted to demonstrate the effectiveness and robustness of our methods. In particular, the proposed static imaging functionals enhance the performance of the total focusing method (TFM) and the dynamic imaging functionals show analogous behavior to the time reversal inversion but without solving time-dependent wave equations.
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    Elastic scattering by finitely many point-like obstacles
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hu, Guanghui; Sini, Mourad
    This paper is concerned with the time-harmonic elastic scattering by a finite number N of point-like obstacles in Rn (n = 2, 3). We analyze the N-point interactions model in elasticity and derive the associated Green's tensor (integral kernel) in terms of the point positions and the scattering coefficients attached to them, following the approach in quantum mechanics for modeling N-particle interactions. In particular, explicit expressions are given for the scattered near and far fields corresponding to elastic plane waves or point-source incidences. As a result, we rigorously justify the Foldy method for modeling the multiple scattering by finitely many point-like obstacles for the Lame model. The arguments are based on the Fourier analysis and the Weinstein-Aronszajn inversion formula of the resolvent for the finite rank perturbations of closed operators in Hilbert spaces.
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    Unique determination of balls and polyhedral scatterers with a single point source wave
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Hu, Guanghui; Liu, Xiaodong
    In this paper, we prove uniqueness in determining a sound-soft ball or polyhedral scatterer in the inverse acoustic scattering problem with a single incident point source wave in RN (N = 2, 3). Our proofs rely on the reflection principle for the Helmholtz equation with respect to a Dirichlet hyperplane or sphere, which is essentially a 'point-to-point extension formula. The method has been adapted to proving uniqueness in inverse scattering from sound-soft cavities with interior measurement data incited by a single point source. The corresponding uniqueness for sound-hard balls or polyhedral scatterers has also been discussed.
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    Scattering of plane elastic waves by three-dimensional diffraction gratings
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    The reflection and transmission of a time-harmonic plane wave in an isotropic elastic medium by a three-dimensional diffraction grating is investigated. If the diffractive structure involves an impenetrable surface, we study the first, second, third and fourth kind boundary value problems for the Navier equation in an unbounded domain by the variational approach. Based on the Rayleigh expansions, a radiation condition for quasi-periodic solutions is proposed. Existence of solutions in Sobolev spaces is established if the grating profile is a two dimensional Lipschitz surface, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Similar solvability results are obtained for multilayered transmission gratings in the case of an incident pressure wave. Moreover, by a periodic Rellich identity, uniqueness of the solution to the first kind (Dirichlet) boundary value problem is established for all frequencies under the assumption that the impenetrable surface is given by the graph of a Lipschitz function
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    An optimization method in inverse elastic scattering for one-dimensional grating profiles
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Elschner, Johannes; Hu, Guanghui
    Consider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure. We formulate the inverse problem as a least squares optimization problem, following the two-step algorithm by G. Bruckner and J. Elschner (Inverse Problems (2003) 19, 315-329) for electromagnetic diffraction gratings. Such a method is based on the Kirsch-Kress optimization scheme and consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed one. We apply this method to both smooth ($C^2$) and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation. Numerical reconstructions from exact and noisy data illustrate the feasibility of the method.
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    Some inverse problems arising from elastic scattering by rigid obstacles
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hu, Guanghui; Kirsch, Andreas; Sini, Mourad
    In the first part, it is proved that a C2-regular rigid scatterer in R3 can be uniquely identified by the shear part (i.e. S-part) of the far-field pattern corresponding to all incident shear waves at any fixed frequency. The proof is short and it is based on a kind of decoupling of the S-part of scattered wave from its pressure part (i.e. P-part) on the boundary of the scatterer. Moreover, uniqueness using the S-part of the far-field pattern corresponding to only one incident plane shear wave holds for a ball or a convex Lipschitz polyhedron. In the second part, we adapt the factorization method to recover the shape of a rigid body from the scattered S-waves (resp. P-waves) corresponding to all incident plane shear (resp. pressure) waves. Numerical examples illustrate the accuracy of our reconstruction in R2. In particular, the factorization method also leads to some uniqueness results for all frequencies excluding possibly a discrete set.
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    An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Hu, Guanghui; Yang, Jiaqing; Zhang, Bo
    This paper is concerned with uniqueness for reconstructing a periodic inhomogeneous medium covered on a perfectly conducting plate. We deal with the problem in the frame of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation for two refractive indices is obtained, and then inspired by Kirsch's idea, the refractive index can be identified by utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm-Liouville eigenvalue problem.
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    Multiple scattering of electromagnetic waves by a finite number of point-like obstacles
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Challa, Durga Prasad; Hu, Guanghui; Sini, Mourad
    This paper is concerned with the time-harmonic electromagnetic scattering problem for a finite number M of point-like obstacles in R^3. First, we give a rigorous justification of the Foldy method and describe the intermediate levels of scattering between the Born and Foldy models. Second, we study the problem of detecting the scatterers and the scattering strengths from the far-field measurements and discuss the effect of multiple scattering related to each of these models.
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    Recovering complex elastic scatterers by a single far-field pattern
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Hu, Guanghui; Li, Jingzhi; Liu, Hongyu
    We consider the inverse scattering problem of reconstructing multiple impenetrable bodies embedded in an unbounded, homogeneous and isotropic elastic medium. The inverse problem is nonlinear and ill-posed. Our study is conducted in an extremely general and practical setting: the number of scatterers is unknown in advance; and each scatterer could be either a rigid body or a cavity which is not required to be known in advance; and moreover there might be components of multiscale sizes presented simultaneously. We develop several locating schemes by making use of only a single far-field pattern, which is widely known to be challenging in the literature. The inverse scattering schemes are of a totally direct"nature without any inversion involved. For the recovery of multiple small scatterers, the nonlinear inverse problem is linearized and to that end, we derive sharp asymptotic expansion of the elastic far-field pattern in terms of the relative size of the cavities. The asymptotic expansion is based on the boundary-layer-potential technique and the result obtained is of significant mathematical interest for its own sake. The recovery of regular-size/extended scatterers is based on projecting the measured far-field pattern into an admissible solution space. With a local tuning technique, we can further recover multiple multiscale elastic scatterers.