arXiv subjects

Jun Zou

Publications and source records attributed to Jun Zou.

At least 55 records · Page 3Linked to original sources

arXiv2022

Asymptotic Analysis for a Nonlinear Reaction-Diffusion System Modeling an Infectious Disease

In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as those for cholera. One of the significant features in this model is that a certain portion of the recovered human hosts may lose a lifetime immunity and could be infected again. Another important feature in the model is that the mobility for each species is allowed to be dependent upon both the location and time. With the whole population assumed to be susceptible with the bacteria, the model is a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on the model parameters and the given data. Moreover, the long-time behavior and stability analysis for the solutions are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability of all steady-state solutions. These new results provide the answers to several open questions raised in the literature.

math.AP↗

arXiv2022

Mittag--Leffler stability of numerical solutions to time fractional ODEs

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3 (1990), 216-240), we are able to prove that both $\mathcal{L}$1 scheme and strong $A$-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong $A$-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $α$-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation and the decay rate of the continuous fractional resolvent operator. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations, fractional linear system and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

math.NA↗

arXiv2022

Least-Squares Method for Inverse Medium Problems

We present a two-stage least-squares method to inverse medium problems of reconstructing multiple unknown coefficients simultaneously from noisy data. A direct sampling method is applied to detect the location of the inhomogeneity in the first stage, while a total least-squares method with mixed regularization is used to recover the medium profile in the second stage. The total least-squares method is designed to minimize the residual of the model equation and the data fitting, along with an appropriate regularization, in an attempt to significantly improve the accuracy of the approximation obtained from the first stage. We shall also present an analysis on the well-posedness and convergence of this algorithm. Numerical experiments are carried out to verify the accuracies and robustness of this novel two-stage least-squares algorithm, with great tolerance of noise.

math.NA↗

arXiv2021

An Analysis of Stochastic Variance Reduced Gradient for Linear Inverse Problems

Stochastic variance reduced gradient (SVRG) is a popular variance reduction technique for accelerating stochastic gradient descent (SGD). We provide a first analysis of the method for solving a class of linear inverse problems in the lens of the classical regularization theory. We prove that for a suitable constant step size schedule, the method can achieve an optimal convergence rate in terms of the noise level (under suitable regularity condition) and the variance of the SVRG iterate error is smaller than that by SGD. These theoretical findings are corroborated by a set of numerical experiments.

math.NA↗

arXiv2021

Two unique Identifiability results for inverse scattering problems within polyhedral geometries

We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in $\mathbb{R}^3$ by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [8,9]. In this paper, we derive novel geometric properties that generalize and extend the related results in [9], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.

math.AP↗

arXiv2021

Stochastic convergence of regularized solutions and their finite element approximations to inverse source problems

In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and establish the stochastic convergence and optimal finite element convergence rates of these solutions, under pointwise measurement data with random noise. Unlike most existing regularization theories, the regularization error estimates are derived without any source conditions, while the error estimates of finite element solutions show their explicit dependence on the noise level, regularization parameter, mesh size, and time step size, which can guide practical choices among these key parameters in real applications. The error estimates also suggest an iterative algorithm for determining an optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the analytical results.

math.NA↗

arXiv2021

Silicon photonics integrated dynamic polarization controller for continuous-variable quantum key distribution

We designed and demonstrated experimentally a silicon photonics integrated dynamic polarization controller which is a crucial component of a continuous-variable quantum key distribution system. By using a variable step simulated annealing approach, we achieve a dynamic polarization extinction ratio greater than 25 dB. The dynamic polarization controller can be utilized in silicon photonics integrated continuous-variable quantum key distribution system to minimize the size and decrease the cost further.

quant-ph↗

arXiv2021

Mathematical analysis of electromagnetic scattering by dielectric nanoparticles with high refractive indices

In this work, we are concerned with the mathematical modeling of the electromagnetic (EM) scattering by arbitrarily shaped non-magnetic nanoparticles with high refractive indices. When illuminated by visible light, such particles can exhibit a very strong isotropic magnetic response, resulting from the coupling of the incident wave with the circular displacement currents of the EM fields. The main aim of this work is to mathematically illustrate this phenomenon. We shall first introduce the EM scattering resolvent and the concept of dielectric subwavelength resonances. Then we derive the a priori estimates for the subwavelength resonances and the associated resonant modes. We also show the existence of resonances and obtain their asymptotic expansions in terms of the small particle size and the high contrast parameter. After that, we investigate the enhancement of the scattering amplitude and the cross sections when the resonances occur. In doing so, we develop a novel multipole radiation framework that directly separates the electric and magnetic multipole moments and allows us to clearly see their orders of magnitude and blow--up rates. We prove that at the dielectric subwavelength resonant frequencies, the nanoparticles with high refractive indices behave like the sum of the electric dipole and the resonant magnetic dipole. Some explicit calculations and numerical experiments are also provided to validate our general results and formulas.

math-ph↗

arXiv2021

On the Saturation Phenomenon of Stochastic Gradient Descent for Linear Inverse Problems

Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. The current mathematical theory in the lens of regularization theory predicts that SGD with a polynomially decaying stepsize schedule may suffer from an undesirable saturation phenomenon, i.e., the convergence rate does not further improve with the solution regularity index when it is beyond a certain range. In this work, we present a refined convergence rate analysis of SGD, and prove that saturation actually does not occur if the initial stepsize of the schedule is sufficiently small. Several numerical experiments are provided to complement the analysis.

math.OC↗

arXiv2021

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

In this work, an adaptive edge element method is developed for an H(curl)-elliptic constrained optimal control problem. We use the lowest-order Nedelec's edge elements of first family and the piecewise (element-wise) constant functions to approximate the state and the control, respectively, and propose a new adaptive algorithm with error estimators involving both residual-type error estimators and lower-order data oscillations. By using a local regular decomposition for H(curl)-functions and the standard bubble function techniques, we derive the a posteriori error estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonal $L^2$-projections and the mesh-size functions to demonstrate that the sequences of the discrete states and controls generated by the adaptive algorithm converge strongly to the exact solutions of the state and control in the energy-norm and $L^2$-norm, respectively, by first achieving the strong convergence towards the solution to a limiting control problem. Three-dimensional numerical experiments are also presented to confirm our theoretical results and the quasi-optimality of the adaptive edge element method.

math.NA↗

arXiv2021

Fast Ambiguous DOA Elimination Method of DOA Measurement for Hybrid Massive MIMO Receiver

DOA estimation for massive multiple-input multiple-output (MIMO) system can provide ultra-high-resolution angle estimation. However, due to the high computational complexity and cost of all digital MIMO systems, a hybrid analog digital (HAD) structure MIMO was proposed. In this paper, a fast ambiguous phase elimination method is proposed to solve the problem of direction-finding ambiguity caused by the HAD MIMO. Only two-data-blocks are used to realize DOA estimation. Simulation results show that the proposed method can greatly reduce the estimation delay with a slight performance loss.

cs.IT↗

arXiv2021

Minnaert resonances for bubbles in soft elastic materials

Minnaert resonance is a well known physical phenomenon and it has many important applications, in particular in the effective realisation of acoustic metamaterials using bubbly mediums in recent years. In this paper, motived by the Minnaert resonance, we consider the low-frequency resonance for acoustic bubbles embedded in soft elastic materials. This is a hybrid physical process that couples the acoustic and elastic wave propagations. By delicately and subtly balancing the acoustic and elastic parameters as well as the geometry of the bubble, we show that Minnaert resonance can (approximately) occur for rather general constructions. Our study poses a great potential for the effective realisation of negative elastic materials by using bubbly elastic mediums.

math.AP↗

arXiv2020

Solving Two Dimensional H(curl)-elliptic Interface Systems with Optimal Convergence On Unfitted Meshes

In this article, we develop and analyze a finite element method with the first family Nédélec elements of the lowest degree for solving a Maxwell interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use $\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface elements while keep using the standard Nédélec functions on all the non-interface elements. We establish a few important properties for the IFE functions including the unisolvence according to the edge degrees of freedom, the exact sequence relating to the $H^1$ IFE functions and the optimal approximation capabilities. In order to achieve the optimal convergence rates, we employ a Petrov-Galerkin method in which the IFE functions are only used as the trial functions and the standard Nédélec functions are used as the test functions which can eliminate the non-conformity errors. We analyze the inf-sup conditions under certain conditions and show the optimal convergence rates which are also validated by numerical experiments.

math.NA↗

arXiv2020

A Direct Sampling Method for the Inversion of the Radon Transform

We propose a novel direct sampling method (DSM) for the effective and stable inversion of the Radon transform. The DSM is based on a generalization of the important almost orthogonality property in classical DSMs to fractional order Sobolev duality products and to a new family of probing functions. The fractional order duality product proves to be able to greatly enhance the robustness of the reconstructions in some practically important but severely ill-posed inverse problems associated with the Radon transform. We present a detailed analysis to better understand the performance of the new probing and index functions, which are crucial to stable and effective numerical reconstructions. The DSM can be computed in a very fast and highly parallel manner. Numerical experiments are carried out to compare the DSM with a popular existing method, and to illustrate the efficiency, stability, and accuracy of the DSM.

math.NA↗

arXiv2020

A Direct Sampling Method for Simultaneously Recovering Inhomogeneous Inclusions of Different Nature

In this work, we investigate a class of elliptic inverse problems and aim to simultaneously recover multiple inhomogeneous inclusions arising from two different physical parameters, using very limited boundary Cauchy data collected only at one or two measurement events. We propose a new fast, stable and highly parallelable direct sampling method (DSM) for the simultaneous reconstruction process. Two groups of probing and index functions are constructed, and their desired properties are analyzed. In order to identify and decouple the multiple inhomogeneous inclusions of different physical nature, we introduce a new concept of mutually almost orthogonality property that generalizes the important concept of almost orthogonality property in classical DSMs for inhomogeneous inclusions of same physical nature. With the help of this new concept, we develop a reliable strategy to distinguish two different types of inhomogeneous inclusions with noisy data collected at one or two measurement events. We further improve the decoupling effect by choosing an appropriate boundary influx. Numerical experiments are presented to illustrate the robustness and efficiency of the proposed method.

math.NA↗

arXiv2020

Unique continuation from a generalized impedance edge-corner for Maxwell's system and applications to inverse problems

We consider the time-harmonic Maxwell system in a domain with a generalized impedance edge-corner, namely the presence of two generalized impedance planes that intersect at an edge. The impedance parameter can be $0, \infty$ or a finite non-identically vanishing variable function. We establish an accurate relationship between the vanishing order of the solutions to the Maxwell system and the dihedral angle of the edge-corner. In particular, if the angle is irrational, the vanishing order is infinity, i.e. strong unique continuation holds from the edge-corner. The establishment of those new quantitative results involve a highly intricate and subtle algebraic argument. The unique continuation study is strongly motivated by our study of a longstanding inverse electromagnetic scattering problem. As a significant application, we derive several novel unique identifiability results in determining a polyhedral obstacle as well as it surface impedance by a single far-field measurement. We also discuss another potential and interesting application of our result in the inverse scattering theory related to the information encoding.

math.AP↗

arXiv2020

Uniqueness for time-dependent inverse problems with single dynamical data

In this work, we investigate the shape identification and coefficient determination associated with two time-dependent partial differential equations in two dimensions. We consider the inverse problems of determining a convex polygonal obstacle and the coefficient appearing in the wave and Schrödinger equations from a single dynamical data along with the time. With the far field data, we first prove that the sound speed of the wave equation together with its contrast support of convex-polygon type can be uniquely determined, then establish a uniqueness result for recovering an electric potential as well as its support appearing in the Schrödinger equation. As a consequence of these results, we demonstrate a uniqueness result for recovering the refractive index of a medium from a single far field pattern at a fixed frequency in the time-harmonic regime.

math.AP↗

arXiv2020

Sequential and Incremental Precoder Design for Joint Transmission Network MIMO Systems with Imperfect Backhaul

In this paper, we propose a sequential and incremental precoder design for downlink joint transmission (JT) network MIMO systems with imperfect backhaul links. The objective of our design is to minimize the maximum of the sub-stream mean square errors (MSE), which dominates the average bit error rate (BER) performance of the system. In the proposed scheme,we first optimize the precoder at the serving base station (BS), and then sequentially optimize the precoders of non-serving BSs in the JT set according to the descending order of their probabilities of participating in JT. The BS-wise sequential optimization process can improve the system performance when some BSs have to temporarily quit the JT operations because of poor instant backhaul conditions. Besides, the precoder of an additional BS is derived in an incremental way, i.e., the sequentially optimized precoders of previous BSs are fixed, thus the additional precoder plays an incremental part in the multi-BS JT operations. An iterative algorithm is designed to jointly optimize the sub-stream precoder and sub-stream power allocation for each additional BS in the proposed sequential and incremental optimization scheme. Simulations show that, under the practical backhaul link conditions, our scheme significantly outperforms the autonomous global precoding (AGP) scheme in terms of BER performance.

eess.SP↗