arXiv subjects

Jun Zou

Publications and source records attributed to Jun Zou.

At least 91 records · Page 5Linked to original sources

arXiv2016

A Multilevel Sampling Method for Detecting Sources in a Stratified Ocean Waveguide

In the reconstruction process of sound waves in a 3D stratified waveguide, a key technique is to effectively reduce the huge computational demand. In this work, we propose an efficient and simple multilevel reconstruction method to help locate the accurate position of a point source in a stratified ocean. The proposed method can be viewed as a direct sampling method since no solutions of optimizations or linear systems are involved. The novel method exhibits several strengths: fast convergence, robustness against noise, advantages in computational complexity and applicability for a very small number of receivers.

math.NA↗

arXiv2016

Cellular Communications on License-Exempt Spectrum: A Tutorial

A traditional cellular system (e.g., LTE) operates only on the licensed spectrum. This tutorial explains the concept of cellular communications on both licensed and license-exempt spectrum under a unified architecture. The purpose to extend a cellular system into the bandwidth-rich license-exempt spectrum is to form a larger cellular network for all spectrum types. This would result in an ultimate mobile converged cellular network. This tutorial examines the benefits of this concept, the technical challenges, and provides a conceptual LTE-based design example that helps to show how a traditional cellular system like the LTE can adapt itself to a different spectrum type, conform to the regulatory requirements, and harmoniously co-exist with the incumbent systems such as Wi-Fi. In order to cope with the interference and regulation rules on license-exempt spectrum, a special medium access mechanism is introduced into the existing LTE transmission frame structure to exploit the full benefits of coordinated and managed cellular architecture.

cs.NI↗

arXiv2015

Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients

In this work we shall review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We perform sensitivity, resolution and stability analysis of both phased and phaseless problems and compare the degree of ill-posedness of the phased and phaseless reconstructions. The phaseless reconstruction is highly nonlinear and much more severely ill-posed. Algorithms are provided to solve both the phased and phaseless reconstructions in the linearized case. Stability is studied by estimating the condition number of the inversion process for both the phased and phaseless cases. An optimal strategy is suggested to attain the infimum of the condition numbers of the phaseless reconstruction, which may provide an important guidance for efficient phaseless measurements in practical applications. To the best of our knowledge, the stability analysis in terms of condition numbers are new for the phased and phaseless inverse scattering problems, and are very important to help us understand the degree of ill-posedness of these inverse problems. Numerical experiments are provided to illustrate the theoretical asymptotic behavior, as well as the effectiveness and robustness of the phaseless reconstruction algorithm.

math.NA↗

arXiv2015

A parallel space-time domain decomposition method for unsteady source inversion problems

In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknowns. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications.

math.OC↗

arXiv2015

Two-level space-time domain decomposition methods for unsteady inverse problems

As the number of processor cores on supercomputers becomes larger and larger, algorithms with high degree of parallelism attract more attention. In this work, we propose a novel space-time coupled algorithm for solving an inverse problem associated with the time-dependent convection-diffusion equation in three dimensions. We introduce a mixed finite element/finite difference method and a one-level and a two-level space-time parallel domain decomposition preconditioner for the Karush-Kuhn-Tucker (KKT) system induced from reformulating the inverse problem as an output least-squares optimization problem in the space-time domain. The new full space approach eliminates the sequential steps of the optimization outer loop and the inner forward and backward time marching processes, thus achieves high degree of parallelism. Numerical experiments validate that this approach is effective and robust for recovering unsteady moving sources. We report strong scalability results obtained on a supercomputer with more than 1,000 processors.

math.NA↗

arXiv2015

Analysis on Non-negative Factorizations and Applications

In this work we perform some mathematical analysis on non-negative matrix factorizations (NMF) and apply NMF to some imaging and inverse problems. We will propose a sparse low-rank approximation of big positive data and images in terms of tensor products of positive vectors, and investigate its effectiveness in terms of the number of tensor products to be used in the approximation. A new concept of multi-level analysis (MLA) framework is also suggested to extract major components in the matrix representing structures of different resolutions, but still preserving the positivity of the basis and sparsity of the approximation. We will also propose a semi-smooth Newton method based on primal-dual active sets for the non-negative factorization. Numerical results are given to demonstrate the effectiveness of the proposed method to capture features in images and structures of inverse problems under no a-priori assumption on the data structure, as well as to provide a sparse low-rank representation of the data.

math.OC↗

arXiv2015

Well-posedness of a Pulsed Electric Field Model in Biological Media and its Finite Element Approximation

This work aims at providing a mathematical and numerical framework for the analysis on the effects of pulsed electric fields on biological media. Biological tissues and cell suspensions are described as having a heteregeneous permittivity and a heteregeneous conductivity. Well-posedness of the model problem and the regularity of its solution are established. A fully discrete finite element scheme is proposed for the numerical approximation of the potential distribution as a function of time and space simultaneously for an arbitrary shaped pulse, and it is demonstrated to enjoy the optimal convergence order in both space and time. The proposed numerical scheme has potential applications in the fields of medicine, food sciences, and biotechnology.

math.AP↗

arXiv2014

Randomized Algorithms for Large-scale Inverse Problems with General Regularizations

We shall investigate randomized algorithms for solving large-scale linear inverse problems with general regularizations. We first present some techniques to transform inverse problems of general form into the ones of standard form, then apply randomized algorithms to reduce large-scale systems of standard form to much smaller-scale systems and seek their regularized solutions in combination with some popular choice rules for regularization parameters. Then we will propose a second approach to solve large-scale ill-posed systems with general regularizations. This involves a new randomized generalized SVD algorithm that can essentially reduce the size of the original large-scale ill-posed systems. The reduced systems can provide approximate regularized solutions with about the same accuracy as the ones by the classical generalized SVD, and more importantly, the new approach gains obvious robustness, stability and computational time as it needs only to work on problems of much smaller size. Numerical results are given to demonstrated the efficiency of the algorithms.

math.NA↗

arXiv2014

Super-resolution in Imaging High Contrast Targets from the Perspective of Scattering Coefficients

In this paper we consider the inverse scattering problem for high-contrast targets. We mathematically analyze the experimentally-observed phenomenon of super-resolution in imaging the target shape. This is the first time that a mathematical theory of super-resolution has been established in the context of imaging high contrast inclusions. We illustrate our main findings with a variety of numerical examples. Our analysis is based on the novel concept of scattering coefficients. These findings may help in developing resonant structures for resolution enhancement.

math-ph↗

arXiv2014

Direct Sampling Method for Diffusive Optical Tomography

In this work, we are concerned with the diffusive optical tomography (DOT) problem in the case when only one or two pairs of Cauchy data is available. We propose a simple and efficient direct sampling method (DSM) to locate inhomogeneities inside a homogeneous background and solve the DOT problem in both full and limited aperture cases. This new method is easy to implement and less expensive computationally. Numerical experiments demonstrate its effectiveness and robustness against noise in the data. This provides a new promising numerical strategy for the DOT problem.

math-ph↗

arXiv2014

An Inexact Uzawa Algorithm for Generalized Saddle-Point Problems and Its Convergence

We propose an inexact Uzawa algorithm with two variable relaxation parameters for solving the generalized saddle-point system. The saddle-point problems can be found in a wide class of applications, such as the augmented Lagrangian formulation of the constrained minimization, the mixed finite element method, the mortar domain decomposition method and the discretization of elliptic and parabolic interface problems. The two variable parameters can be updated at each iteration, requiring no a priori estimates on the spectrum of two preconditioned subsystems involved. The convergence and convergence rate of the algorithm are analysed. Both symmetric and nonsymmetric saddle-point systems are discussed, and numerical experiments are presented to demonstrate the robustness and effectiveness of the algorithm.

math.NA↗

arXiv2013

The Concept of Heterogeneous Scattering Coefficients and Its Application in Inverse Medium Scattering

This work investigates the scattering coefficients for inverse medium scattering problems. It shows some fundamental properties of the coefficients such as symmetry and tensorial properties. The relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to changes in the permittivity and permeability distributions is investigated. In the linearized case, explicit formulas for reconstructing permittivity and permeability distributions from the scattering coefficients is proposed. They relate the exponentially ill-posed character of the inverse medium scattering problem at a fixed frequency to the exponential decay of the scattering coefficients. Moreover, they show the stability of the reconstruction from multifrequency measurements. This provides a new direction for solving inverse medium scattering problems.

math.AP↗

arXiv2013

Optimal Shape Design by Partial Spectral Data

In this paper, we are concerned with a shape design problem, in which our target is to design, up to rigid transformations and scaling, the shape of an object given either its polarization tensor at multiple contrasts or the partial eigenvalues of its Neumann-Poincaré operator, which are known as the Fredholm eigenvalues. We begin by proposing to recover the eigenvalues of the Neumann-Poincaré operator from the polarization tensor by means of the holomorphic functional calculus. Then we develop a regularized Gauss-Newton optimization method for the shape reconstruction process. We present numerical results to demonstrate the effectiveness of the proposed methods and to illustrate important properties of the Fredholm eigenvalues and their associated eigenfunctions. Our results are expected to have important applications in the design of plasmon resonances in nanoparticles as well as in the multifrequency or pulsed imaging of small anomalies.

math.OC↗

arXiv2013

Convergence of an Adaptive Finite Element Method for Distributed Flux Reconstruction

We shall establish the convergence of an adaptive conforming finite element method for the reconstruction of the distributed flux in a diffusion system. The adaptive method is based on a posteriori error estimators for the distributed flux, state and costate variables. The sequence of discrete solutions produced by the adaptive algorithm is proved to converge to the true triplet satisfying the optimality conditions in the energy norm and the corresponding error estimator converges to zero asymptotically.

math.NA↗

arXiv2013

Nearly Cloaking the Full Maxwell Equations: Cloaking Active Contents with General Conducting Layers

The regularized near-cloak via the transformation optics approach in the time-harmonic electromagnetic scattering is considered. This work extends the existing studies mainly in two aspects. First, it presents a near-cloak construction by incorporating a much more general conducting layer between the cloaked and cloaking regions. This might be of significant practical importance when production fluctuations occur. Second, it allows the cloaked contents to be both passive and active with an applied current inside. In assessing the near-cloaking performance, comprehensive and sharp estimates are derived for the scattering amplitude in terms of the asymptotic regularization parameter and the material tensors of the conducting layer. The scattering estimates are independent of the passive/active contents being cloaked, which implies that one could nearly cloak arbitrary contents by using the proposed near-cloak construction.

math.AP↗

arXiv2013

A New Splitting Method for Time-dependent Convection-dominated Diffusion Problems

We present a new splitting method for time-dependent convection-dominated diffusion problems. The original convection diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. The scheme has the following nice features: the convection subproblem is solved explicitly and a multistep technique is introduced to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; the diffusion subproblem is always self-adjoint and coercive so that it can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme is then extended for Navier-Stokes equations, where the nonlinear convection is resolved by a linear explicit multistep scheme at the convection step, and only a generalized Stokes problem is needed to solve at the diffusion step with the resulting stiffness matrix being invariant in the time marching process. The new schemes are all free from tuning some stabilization parameters for the convection-dominated diffusion problems. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.

math-ph↗

arXiv2012

A Direct Sampling Method for Inverse Scattering Using Far-Field Data

This work is concerned with a direct sampling method (DSM) for inverse acoustic scattering problems using far-field data. The method characterizes some unknown obstacles, inhomogeneous media or cracks, directly through an indicator function computed from the measured data. Using one or very few incident waves, the DSM provides quite reasonable profiles of scatterers in time-harmonic inverse acoustic scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. We shall first derive the DSM using far-field data, then carry out a systematic evaluation of the performances and distinctions of the DSM using both near-field and far-field data. The numerical simulations are shown to demonstrate interesting and promising potentials of the DSM: a) ability to identify not only medium scatterers, but also obstacles, and even cracks, using measurement data from one or few incident directions, b) robustness with respect to large noise, and c) computational efficiency with only inner products involved.

math-ph↗

arXiv2012

A Direct Sampling Method for Inverse Electromagnetic Medium Scattering

In this paper, we study the inverse electromagnetic medium scattering problem of estimating the support and shape of medium scatterers from scattered electric or magnetic near-field data. We shall develop a novel direct sampling method based on an analysis of electromagnetic scattering and the behavior of the fundamental solution. The method is applicable even with one incident field and needs only to compute inner products of the measured scattered field with the fundamental solutions located at sampling points. Hence it is strictly direct, computationally very efficient, and highly tolerant to the presence of noise in the data. Two- and three-dimensional numerical experiments indicate that it can provide reliable support estimates of one single and multiple scatterers in case of both exact and highly noisy data.

math.NA↗