arXiv subjects

Jun Zou

Publications and source records attributed to Jun Zou.

At least 73 records · Page 4Linked to original sources

arXiv2020

Receive Antenna Selection for Secure Pre-coding Aided Spatial Modulation

In this paper, we make an investigation of receive antenna selection (RAS) strategies in the secure pre-coding aided spatial modulation (PSM) system with the aid of artificial noise. Due to a lack of the closed-form expression for secrecy rate (SR) in secure PSM systems, it is hard to optimize the RAS. To address this issue, the cut-off rate is used as an approximation of the SR. Further, two low-complexity RAS schemes for maximizing SR, called Max-SR-L and Max-SR-H, are derived in the low and high signal-to-noise ratio (SNR) regions, respectively. Due to the fact that the former works well in the low SNR region but becomes worse in the medium and high SNR regions while the latter also has the similar problem, a novel RAS strategy Max-SR-A is proposed to cover all SNR regions. Simulation results show that the proposed Max-SR-H and Max-SR-L schemes approach the optimal SR performances of the exhaustive search (ES) in the high and low SNR regions, respectively. In particular, the SR performance of the proposed Max-SR-A is close to that of the optimal ES and better than that of the random method in almost all SNR regions.

cs.IT↗

arXiv2020

Super-resolution in recovering embedded electromagnetic sources in high contrast media

The purpose of this work is to provide a rigorous mathematical analysis of the expected super-resolution phenomenon in the time-reversal imaging of electromagnetic (EM) radiating sources embedded in a high contrast medium. It is known that the resolution limit is essentially determined by the sharpness of the imaginary part of the EM Green's tensor for the associated background. We first establish the close connection between the resolution and the material parameters and the resolvent of the electric integral operator, via the Lippmann-Schwinger representation formula. We then present an insightful characterization of the spectral structure of the integral operator for a general bounded domain and derive the pole-pencil decomposition of its resolvent in the high contrast regime. For the special case of a spherical domain, we provide some quantitative asymptotic behavior of the eigenvalues and eigenfunctions. These mathematical findings shall enable us to provide a concise and rigorous illustration of the super-resolution in the EM source reconstruction in high contrast media. Some numerical examples are also presented to verify our main theoretical results.

math.AP↗

arXiv2019

Convergence Rates of Tikhonov Regularizations for Elliptic and Parabolic Inverse Radiativity Problems

We shall study in this paper the Lipschitz type stabilities and convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions. The Lipschitz type stability estimates are derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions for the considered inverse radiativity problems in high dimensional spaces, some new variational source conditions are proposed. The conditions are rigorously verified in general dimensional spaces under the Lipschitz type stability estimates and the reasonable convergence rates are achieved.

math.AP↗

arXiv2019

Low-Complexity Leakage-based Secure Precise Wireless Transmission with Hybrid Beamforming

In conventional secure precise wireless transmission (SPWT), fully digital beamforming (FDB) has a high secrecy performance in transmit antenna system, but results in a huge RF-chain circuit budget for medium-scale and large-scale systems. To reduce the complexity, this letter considers a hybrid digital and analog (HDA) structure with random frequency mapped into the RF-chains to achieve SPWT. Then, a hybrid SPWT scheme based on maximizing signal-to-leakage-and-noise ratio (SLNR) and artificial-noise-to-leakage-and-noise ratio (ANLNR) (M-SLNR-ANLNR) is proposed. Compared to the FDB scheme, the proposed scheme reduces the circuit budget with low computational complexity and comparable secrecy performance.

eess.SP↗

arXiv2019

On nodal and generalized singular structures of Laplacian eigenfunctions and applications in $\mathbb{R}^3$

This paper is a continuation and an extension of our recent work [3] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In [3], the analytic behaviour of the Laplacian eigenfunctions is investigated at a point where two nodal or generalised singular lines intersect. It reveals a certain intriguing property that the vanishing order of the eigenfunction at the intersecting point is related to the rationality of the intersecting angle. In this paper, we consider the 3D counterpart of such a study by studying the analytic behaviours of the Laplacian eigenfunctions at places where nodal or generalised singular planes are intersect. Compared to the 2D case, the geometric situation is more complicated: the intersection of two planes generates an edge corner, whereas the intersection of more than three planes generates a vertex corner. We provide a comprehensive characterisation for all of those cases. Moreover, we apply the spectral results to establish some novel unique identifiability results for the geometric inverse problems of recovering the shape as well as the (possible) surface impedance parameter by the associated scattering far-field measurements.

math.AP↗

arXiv2019

Mathematical and numerical study of a three-dimensional inverse eddy current problem

We study an inverse problem associated with an eddy current model. We first address the ill-posedness of the inverse problem by proving the compactness of the forward map with respect to the conductivity and the non-uniqueness of the recovery process. Then by virtue of non-radiating source conceptions, we establish a regularity result for the tangential trace of the true solution on the boundary, which is necessary to justify our subsequent mathematical formulation. After that, we formulate the inverse problem as a constrained optimization problem with an appropriate regularization and prove the existence and stability of the regularized minimizers. To facilitate the numerical solution of the nonlinear non-convex constrained optimization, we introduce a feasible Lagrangian and its discrete variant. Then the gradient of the objective functional is derived using the adjoint technique. By means of the gradient, a nonlinear conjugate gradient method is formulated for solving the optimization system, and a Sobolev gradient is incorporated to accelerate the iterative process. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm.

math.AP↗

arXiv2019

Mathematical analysis of electromagnetic plasmonic metasurfaces

We study the anomalous electromagnetic scattering in the homogenization regime, by a subwavelength thin layer of periodically distributed plasmonic nanoparticles on a perfect conducting plane. By using layer potential techniques, we derive the asymptotic expansion of the electromagnetic field away from the thin layer and quantitatively analyze the field enhancement due to the mixed collective plasmonic resonances, which can be characterized by the spectra of periodic Neumann-Poincaré type operators. Based on the asymptotic behavior of the scattered field in the macroscopic scale, we further demonstrate that the optical effect of this thin layer can be effectively approximated by a Leontovich boundary condition, which is uniformly valid no matter whether the incident frequency is near the resonant range but varies with the magnetic property of the plasmonic nanoparticles. The quantitative approximation clearly shows the blow-up of the field energy and the conversion of polarization when resonance occurs, resulting in a significant change of the reflection property of the conducting plane. These results confirm essential physical changes of electromagnetic metasurface at resonances mathematically, whose occurrence was verified earlier for the acoustic case \cite{ammari2017bubble} and the transverse magnetic case \cite{ammari2016mathematical}.

math.AP↗

arXiv2019

On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method randomly chooses one equation from the nonlinear system to obtain an unbiased stochastic estimate of the gradient, and then performs a descent step with the estimated gradient. It is a randomized version of the classical Landweber method for nonlinear inverse problems, and it is highly scalable to the problem size and holds significant potentials for solving large-scale inverse problems. Under the canonical tangential cone condition, we prove the regularizing property for a priori stopping rules, and then establish the convergence rates under suitable sourcewise condition and range invariance condition.

math.OC↗

arXiv2019

On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems

In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisation of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is {\it irrational}, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field patterns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory.

math.AP↗

arXiv2019

Mathematical and numerical analysis of a nonlocal Drude model in nanoplasmonics

In this paper, we consider the frequency-domain Maxwell's equations coupled to a nonlocal Drude model which describes the nonlocal optical response in metallic nanostructures. We prove the existence and uniqueness of weak solutions to the coupled equations. A Galerkin finite element method based on the Raviart--Thomas and Nédélec elements is proposed to solve the equations and the associated error estimates are given. This is the first work on the mathematical and numerical analysis of this model. Numerical examples are presented to verify our theoretical analysis.

math.NA↗

arXiv2019

Variational source conditions for the reconstruction of distributed fluxes

This paper is devoted to the inverse problem of recovering the unknown distributed flux on an inaccessible part of boundary using measurement data on the accessible part. We establish and verify a variational source condition for this inverse problem, leading to a logarithmic-type convergence rate for the corresponding Tikhonov regularization method under a low Sobolev regularity assumption on the distributed flux. Our proof is based on the conditional stability and Carleman estimates together with the complex interpolation theory on a proper Gelfand triple.

math.AP↗

arXiv2018

Unique determination of a penetrable scatterer of rectangular type for inverse Maxwell equations by a single incoming wave

This work is concerned with an inverse electromagnetic scattering problem in two dimensions. We prove that in the TE polarization case, the knowledge of the electric far-field pattern incited by a single incoming wave is sufficient to uniquely determine the shape of a penetrable scatterer of rectangular type. As a by-product, the uniqueness is also confirmed to inverse transmission problems modelled by scalar Helmholtz equations with discontinuous normal derivatives at the scattering interface. Keywords: Uniqueness, inverse medium scattering, Maxwell equations, one incoming wave, shape identification, right corners

math.AP↗

arXiv2018

Long-time behavior of numerical solutions to nonlinear fractional ODEs

In this work, we study the long time behaviors, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By mens of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grunwald-Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including some second order F-BDFs and $3-α$ order method, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

math.NA↗

arXiv2018

Fully Discrete Schemes and Their Analyses for Forward-Backward Stochastic Differential Equations

We propose some numerical schemes for forward-backward stochastic differential equations (FBSDEs) based on a new fundamental concept of transposition solutions. These schemes exploit time-splitting methods for the variation of constants formula of the associated partial differential equations and a discrete representation of the transition semigroups. The convergence of the schemes is established for FBSDEs with uniformly Lipschitz drivers, locally Lipschitz and maximal monotone drivers. Numerical experiments are presented for several nonlinear financial derivative pricing problems to demonstrate the adaptivity and effectiveness of the new schemes. The ideas here can be applied to construct high-order schemes for FBSDEs with general Markov forward processes.

math.NA↗

arXiv2017

Frequency offset tolerant synchronization signal design in NB-IoT

Timing detection is the first step and very important in wireless communication systems. Timing detection performance is usually affected by the frequency offset. Therefore, it is a challenge to design the synchronization signal in massive narrowband Internet of Things (NB-IoT) scenarios where the frequency offset is usually large due to the low cost requirement. In this paper, we firstly proposed a new general synchronization signal structure with a couple of sequences which are conjugated to remove the potential timing error arose from large frequency offset. Then, we analyze the suitable sequence for our proposed synchronization signal structure and discuss a special ZC sequence as an example. Finally, the simulation results demonstrate our proposed synchronization signal can work well when the frequency offset is large. It means that our proposed synchronization signal design is very suitable for the massive NB-IoT.

eess.SP↗

arXiv2016

Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions

We consider the ill-posed operator equation $Ax=y$ with an injective and bounded linear operator $A$ mapping between $\ell^2$ and a Hilbert space $Y$, possessing the unique solution \linebreak $x^†=\{x^†_k\}_{k=1}^\infty$. For the cases that sparsity $x^†\in \ell^0$ is expected but often slightly violated in practice, we investigate in comparison with the $\ell^1$-regularization the elastic-net regularization, where the penalty is a weighted superposition of the $\ell^1$-norm and the $\ell^2$-norm square, under the assumption that $x^†\in \ell^1$. There occur two positive parameters in this approach, the weight parameter $η$ and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in $\ell^1$-regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator $A$ and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay $x^†_k \to 0$ for $k \to \infty$ and the classical smoothness properties of $x^†$ as an element in $\ell^2$.

math.FA↗

arXiv2016

Preconditioners and Their Analyses for Edge Element Saddle-point Systems Arising from Time-harmonic Maxwell Equations

We shall propose and analyze some new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell equations in three dimensions. We will first consider the saddle-point systems with vanishing wave number, for which we present an important relation between the solutions of the singular curl-curl system and the non-singular saddle-point system, then demonstrate that the saddle-point system can be efficiently solved by the Hiptmair-Xu solver. For the saddle-point systems with non-vanishing wave numbers, we will show that the PCG with a new preconditioner can apply for the non-singular system when wave numbers are small, while the methods like preconditioned MINRES may apply for some existing and new preconditioners when wave numbers are large. The spectral behaviors of the resulting preconditioned systems for the existing and new preconditioners are analyzed and compared, and numerical experiments are presented to demonstrate and compare the efficiencies of these preconditioners.

math.NA↗

arXiv2016

A Convergent Adaptive Finite Element Method for Electrical Impedance Tomography

In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage measurements. The reconstruction technique is based on Tikhonov regularization with a Sobolev smoothness penalty and discretizing the forward model using continuous piecewise linear finite elements. We derive an adaptive finite element algorithm with an a posteriori error estimator involving the concerned state and adjoint variables and the recovered conductivity. The convergence of the algorithm is established, in the sense that the sequence of discrete solutions contains a convergent subsequence to a solution of the optimality system for the continuous formulation. Numerical results are presented to verify the convergence and efficiency of the algorithm.

math.NA↗