arXiv subjects

Jiguang Sun

Publications and source records attributed to Jiguang Sun.

At least 37 records · Page 2Linked to original sources

arXiv2020

An extended sampling-ensemble Kalman filter approach for partial data inverse elastic problems

Inverse problems are more challenging when only partial data are available in general. In this paper, we propose a two-step approach combining the extended sampling method and the ensemble Kalman filter to reconstruct an elastic rigid obstacle using partial data. In the first step, the approximate location of the unknown obstacle is obtained by the extended sampling method. In the second step, the ensemble Kalman filter is employed to reconstruct the shape. The location obtained in the first step guides the construction of the initial particles of the ensemble Kalman filter, which is critical to the performance of the second step. Both steps are based on the same physical model and use the same scattering data. Numerical examples are shown to illustrate the effectiveness of the proposed method.

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arXiv2020

Finite Element Calculation of Photonic Band Structures for Frequency Dependent Materials

We consider the calculation of the band structure of frequency dependent photonic crystals. The associated eigenvalue problem is nonlinear and it is challenging to develop effective convergent numerical methods. In this paper, the band structure problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Lagrange finite elements are used to discretize the operator function. Then the convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions. A spectral indicator method is developed to practically compute the eigenvalues. Numerical examples are presented to validate the theory and show the effectiveness of the proposed method.

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arXiv2020

A priori and a posteriori error estimates for the quad-curl eigenvalue problem

In this paper, we propose a new family of H(curl^2)-conforming elements for the quad-curl eigenvalue problem in 2D. The accuracy of this family is one order higher than that in [32]. We prove a priori and a posteriori error estimates. The a priori estimate of the eigenvalue with a convergence order 2(s-1) is obtained if the eigenvector u\in H^{s+1}(Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the eigenvector in an energy norm and an upper bound for the eigenvalues. Numerical examples are presented for validation.

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arXiv2020

A Multilevel Spectral Indicator Method for Eigenvalues of Large Non-Hermitian Matrices

Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed. Given a region on the complex plane, SIMs first compute an indicator by the spectral projection. The indicator is used to test if the region contains eigenvalue(s). Then the region containing eigenvalues(s) is subdivided and tested. The procedure is repeated until the eigenvalues are identified within a specified precision. In this paper, using Cayley transformation and Krylov subspaces, a memory efficient multilevel eigensolver is proposed. The method uses less memory compared with the early versions of SIMs and is particularly suitable to compute many eigenvalues of large sparse (non-Hermitian) matrices. Several examples are presented for demonstration.

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arXiv2020

Finite Element Approximation of the Modified Maxwell's Stekloff Eigenvalues

The modified Maxwell's Stekloff eigenvalue problem arises recently from the inverse electromagnetic scattering theory for inhomogeneous media. This paper contains a rigorous analysis of both the eigenvalue problem and the associated source problem on Lipschitz polyhedra. A new finite element method is proposed to compute Stekloff eigenvalues. By applying the Babuska-Osborn theory, we prove an error estimate without additional regularity assumptions. Numerical results are presented for validation.

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arXiv2020

Quality-Bayesian approach to inverse acoustic source problems with partial data

A quality-Bayesian approach, combining the direct sampling method and the Bayesian inversion, is proposed to reconstruct the locations and intensities of the unknown acoustic sources using partial data. First, we extend the direct sampling method by constructing a new indicator function to obtain the approximate locations of the sources. The behavior of the indicator is analyzed. Second, the inverse problem is formulated as a statistical inference problem using the Bayes' formula. The well-posedness of the posterior distribution is proved. The source locations obtained in the first step are coded in the priors. Then an Metropolis-Hastings MCMC algorithm is used to explore the posterior density. Both steps use the same physical model and measured data. Numerical experiments show that the proposed method is effective even with partial data.

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arXiv2020

A new finite element approach for the Dirichlet eigenvalue problem

In this paper, we propose a new finite element approach, which is different than the classic Babuska-Osborn theory, to approximate Dirichlet eigenvalues. The Dirichlet eigenvalue problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Using conforming finite elements, the convergence is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed to compute the eigenvalues. A numerical example is presented to validate the theory.

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arXiv2020

Finite Element Approximation of Transmission Eigenvalues for Anisotropic Media

The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media and has important applications in many qualitative methods. The problem is posted as a system of two second order partial differential equations and is essentially nonlinear, non-selfadjoint, and of higher order. It is nontrivial to develop effective numerical methods and the proof of convergence is challenging. In this paper, we formulate the transmission eigenvalue problem for anisotropic media as an eigenvalue problem of a holomorphic Fredholm operator function of index zero. The Lagrange finite elements are used for discretization and the convergence is proved using the abstract approximation theory for holomorphic operator functions. A spectral indicator method is developed to compute the eigenvalues. Numerical examples are presented for validation.

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arXiv2019

Revisiting the Jones eigenproblem in fluid-structure interaction

The Jones eigenvalue problem first described by D.S. Jones in 1983 concerns unusual modes in bounded elastic bodies: time-harmonic displacements whose tractions and normal components are both identically zero on the boundary. This problem is usually associated with a lack of unique solvability for certain models of fluid-structure interaction. The boundary conditions in this problem appear, at first glance, to rule out {\it any} non-trivial modes unless the domain possesses significant geometric symmetries. Indeed, Jones modes were shown to not be possible in most $C^\infty$ domains (see article by T. Hargé 1990). However, we should in this paper that while the existence of Jones modes sensitively depends on the domain geometry, such modes {\it do} exist in a broad class of domains. This paper presents the first detailed theoretical and computational investigation of this eigenvalue problem in Lipschitz domains. We also analytically demonstrate Jones modes on some simple geometries.

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arXiv2019

Limited Aperture Inverse Scattering Problems using Bayesian Approach and Extended Sampling Method

Inverse scattering problems have many important applications. In this paper, given limited aperture data, we propose a Bayesian method for the inverse acoustic scattering to reconstruct the shape of an obstacle. The inverse problem is formulated as a statistical model using the Baye's formula. The well-posedness is proved in the sense of the Hellinger metric. The extended sampling method is modified to provide the initial guess of the target location, which is critical to the fast convergence of the MCMC algorithm. An extensive numerical study is presented to illustrate the performance of the proposed method.

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arXiv2019

An inverse medium problem using Stekloff eigenvalues and a Bayesian approach

This paper studies the reconstruction of Stekloff eigenvalues and the index of refraction of an inhomogeneous medium from Cauchy data. The inverse spectrum problem to reconstruct Stekloff eigenvalues is investigated using a new integral equation for the reciprocity gap method. Given reconstructed eigenvalues, a Bayesian approach is proposed to estimate the index of refraction. Moreover, since it is impossible to know the multiplicities of the reconstructed eigenvalues and since the eigenvalues can be complex, we employ the recently developed spectral indicator method to compute Stekloff eigenvalues. Numerical experiments validate the effectiveness of the proposed methods.

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arXiv2018

Extended Sampling Method in Inverse Scattering

A new sampling method for inverse scattering problems is proposed to process far field data of one incident wave. As the linear sampling method, the method sets up ill-posed integral equations and uses the (approximate) solutions to reconstruct the target. In contrast, the kernels of the associated integral operators are the far field patterns of sound soft balls. The measured data is moved to right hand sides of the equations, which gives the method the ability to process limit aperture data. Furthermore, a multilevel technique is employed to improve the reconstruction. Numerical examples show that the method can effectively determine the location and approximate the support with little a priori information of the unknown target.

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arXiv2018

Spectral Indicator Method for A Non-selfadjoint Steklov Eigenvalue Problem

We propose an efficient numerical method for a non-selfadjoint Steklov eigenvalue problem. The Lagrange finite element is used for discretization. The convergence is proved using the spectral perturbation theory for compact operators. The non-sefadjointness of the problem leads to non-Hermitian matrix eigenvalue problem. Due to the existence of complex eigenvalues and lack of a priori spectral information, we propose a modified version of the recently developed spectral indicator method to compute (complex) eigenvalues in a given region on the complex plane. In particular, to reduce computational cost, the problem is transformed into a much smaller matrix eigenvalue problem involving the unknowns only on the boundary of the domain. Numerical examples are presented to validate the effectiveness of the proposed method.

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arXiv2018

Computation of Transmission Eigenvalues for Elastic Waves

The goal of this paper is to develop numerical methods computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. We construct a nonlinear function whose values are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. Using an $H^2$-conforming finite element for the self-adjoint fourth order eigenvalue problems, we employ a secant method to compute the roots of the nonlinear function. The convergence of the proposed method is proved. In addition, a mixed finite element method is developed for the purpose of verification. Numerical examples are presented to verify the theory and demonstrate the effectiveness of the two methods.

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arXiv2018

Data recovery: from limited-aperture to full-aperture

The inverse scattering problems have been popular for the past thirty years. While very successful in many cases, progress has lagged when only {\em limited-aperture} measurement is available. In this paper, we perform some elementary study to recover data that can not be measured directly. To be precise, we aim at recovering the {\em full-aperture} far field data from {\em limited-aperture} measurement. Due to the reciprocity relation, the multi-static response matrix (MSR) has a symmetric structure. Using the Green's formula and single layer potential, we propose two schemes to recover {\em full-aperture} MSR. The recovered data is tested by a recently proposed direct sampling method and the factorization method. The numerical results show the possibility to recover, at least partially, the missing data and consequently improve the reconstruction of the scatterer.

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arXiv2017

Fast acoustic source imaging using multi-frequency sparse data

We consider the acoustic source imaging problems using multiple frequency data. Using the data of one observation direction/point, we prove that some information (size and location) of the source support can be recovered. A non-iterative method is then proposed to image the source for the Helmholtz equation using multiple frequency far field data of one or several observation directions. The method is simple to implement and extremely fast since it only computes an indicator function on the interested domain using only matrix vector multiplications. Numerical examples are presented to validate the effectiveness of the method.

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arXiv2016

Direct and Inverse Elastic Scattering From Anisotropic Media

Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show the uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the boundary of the scatterer is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are presented to show the validity and accuracy of our methods.

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arXiv2016

A spectral projection method for transmission eigenvalues

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.

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