arXiv subjects

Jun Zou

Publications and source records attributed to Jun Zou.

At least 37 records · Page 2Linked to original sources

arXiv2024

Adaptive Finite Element Method for a Nonlinear Helmholtz Equation with High Wave Number

A nonlinear Helmholtz (NLH) equation with high frequencies and corner singularities is discretized by the linear finite element method (FEM). After deriving some wave-number-explicit stability estimates and the singularity decomposition for the NLH problem, a priori stability and error estimates are established for the FEM on shape regular meshes including the case of locally refined meshes. Then a posteriori upper and lower bounds using a new residual-type error estimator, which is equivalent to the standard one, are derived for the FE solutions to the NLH problem. These a posteriori estimates have confirmed a significant fact that is also valid for the NLH problem, namely the residual-type estimator seriously underestimates the error of the FE solution in the preasymptotic regime, which was first observed by Babuška et al. [Int J Numer Methods Eng 40 (1997)] for a one-dimensional linear problem. Based on the new a posteriori error estimator, both the convergence and the quasi-optimality of the resulting adaptive finite element algorithm are proved the first time for the NLH problem, when the initial mesh size lying in the preasymptotic regime. Finally, numerical examples are presented to validate the theoretical findings and demonstrate that applying the continuous interior penalty (CIP) technique with appropriate penalty parameters can reduce the pollution errors efficiently. In particular, the nonlinear phenomenon of optical bistability with Gaussian incident waves is successfully simulated by the adaptive CIPFEM.

math.NA↗

arXiv2024

On the convergence of discrete dynamic unbalanced transport models

A generalized unbalanced optimal transport distance ${\rm WB}_Λ$ on matrix-valued measures $\mathcal{M}(Ω,\mathbb{S}_+^n)$ was defined in [arXiv:2011.05845] à la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with ${\rm WB}_Λ$. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, under the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Moreover, thanks to the static formulation, we show that such an assumption can be removed for the Wasserstein-Fisher-Rao distance.

math.NA↗

arXiv2024

A Novel Tree Model-based DNN to Achieve a High-Resolution DOA Estimation via Massive MIMO receive array

To satisfy the high-resolution requirements of direction-of-arrival (DOA) estimation, conventional deep neural network (DNN)-based methods using grid idea need to significantly increase the number of output classifications and also produce a huge high model complexity. To address this problem, a multi-level tree-based DNN model (TDNN) is proposed as an alternative, where each level takes small-scale multi-layer neural networks (MLNNs) as nodes to divide the target angular interval into multiple sub-intervals, and each output class is associated to a MLNN at the next level. Then the number of MLNNs is gradually increasing from the first level to the last level, and so increasing the depth of tree will dramatically raise the number of output classes to improve the estimation accuracy. More importantly, this network is extended to make a multi-emitter DOA estimation. Simulation results show that the proposed TDNN performs much better than conventional DNN and root-MUSIC at extremely low signal-to-noise ratio (SNR), and can achieve Cramer-Rao lower bound (CRLB). Additionally, in the multi-emitter scenario, the proposed Q-TDNN has also made a substantial performance enhancement over DNN and Root-MUSIC, and this gain grows as the number of emitters increases.

eess.SP↗

arXiv2024

Compact quantum random number generator based on a laser diode and silicon photonics integrated hybrid chip

In this study, a compact and low-power-consumption quantum random number generator (QRNG) based on a laser diode and silicon photonics integrated hybrid chip is proposed and verified experimentally. The hybrid chip's size is 8.8*2.6*1 mm3, and the power of entropy source is 80 mW. A common mode rejection ratio greater than 40 dB was achieved using an optimized 1*2 multimode interferometer structure. A method for optimizing the quantum-to-classical noise ratio is presented. A quantum-to-classical noise ratio of approximately 9 dB was achieved when the photoelectron current is 1 microampere using a balance homodyne detector with a high dark current GeSi photodiode. The proposed QRNG has the potential for use in scenarios of moderate MHz random number generation speed, with low power, small volume, and low cost prioritized.

quant-ph↗

arXiv2023

Calibration System and Algorithm Design for a Soft Hinged Micro Scanning Mirror with a Triaxial Hall Effect Sensor

Micro scanning mirrors (MSM) extend the range and field of view of LiDARs, medical imaging devices, and laser projectors. However, a new class of soft-hinged MSMs contains out-of-plane translation in addition to the 2 degree-of-freedom rotations, which presents a cabliration challenge. We report a new calibration system and algorithm design to address the challenge. In the calibration system, a new low-cost calibration rig design employs a minimal 2-laser beam approach. The new new algorithm builds on the reflection principle and an optimization approach to precisely measure MSM poses. To establish the mapping between Hall sensor readings and MSM poses, we propose a self-synchronizing periodicity-based model fitting calibration approach. We achieve an MSM poses estimation accuracy of 0.020° with a standard deviation of 0.011°.

cs.RO↗

arXiv2023

A Fingertip Sensor and Algorithms for Pre-touch Distance Ranging and Material Detection in Robotic Grasping

To enhance robotic grasping capabilities, we are developing new contactless fingertip sensors to measure distance in close proximity and simultaneously detect the type of material and the interior structure. These sensors are referred to as pre-touch dual-modal and dual-mechanism (PDM$^2$) sensors, and they operate using both pulse-echo ultrasound (US) and optoacoustic (OA) modalities. We present the design of a PDM$^2$ sensor that utilizes a pulsed laser beam and a customized ultrasound transceiver with a wide acoustic bandwidth for ranging and sensing. Both US and OA signals are collected simultaneously, triggered by the same laser pulse. To validate our design, we have fabricated a prototype of the PDM$^2$ sensor and integrated it into an object scanning system. We have also developed algorithms to enable the sensor, including time-of-flight (ToF) auto estimation, ranging rectification, sensor and system calibration, distance ranging, material/structure detection, and object contour detection and reconstruction. The experimental results demonstrate that the new PDM$^2$ sensor and its algorithms effectively enable the object scanning system to achieve satisfactory ranging and contour reconstruction performances, along with satisfying material/structure detection capabilities. In conclusion, the PDM$^2$ sensor offers a practical and powerful solution to improve grasping of unknown objects with the robotic gripper by providing advanced perception capabilities.

cs.RO↗

arXiv2023

Mathematical theory on dipolar resonances of hard inclusions within a soft elastic material

The current study is motivated by the paper [Z. Liu, et al., {\it Science}, 289(5485), 2000], which investigates the incorporation of hard inclusions within a soft elastic matrix (HISE). The objective is to attain a negative mass density, which is caused by sub-wavelength dipolar resonances. This paper offers a comprehensive and mathematically rigorous understanding of the dipolar resonances within the HISE structure. Firstly, the original formula for the resonant frequencies of arbitrarily shaped hard inclusions is derived explicitly for the first time. Additionally, we demonstrate that the resonances arise due to the high contrast in Lamé parameters between various materials, while the sub-wavelength characteristics of the resonances are attributed to the significant disparities in material densities. Furthermore, the dipolar characterization of the resonances is rigorously justified. To validate our findings, we investigate resonant phenomena within a spherical geometry, offering both an alternative method for calculating resonant frequencies and a confirmation of the results presented in this paper.

math.AP↗

arXiv2023

On a general matrix-valued unbalanced optimal transport problem

We introduce a general class of transport distances ${\rm WB}_Λ$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(Ω,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a distance is defined via a generalized Benamou-Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimization problem. Some recently proposed models, including the Kantorovich-Bures distance and the Wasserstein-Fisher-Rao distance, can naturally fit into ours. We give a complete characterization of the minimizer and explore the topological and geometrical properties of the space $(\mathcal{M}(Ω,\mathbb{S}_+^n),{\rm WB}_Λ)$. In particular, we show that $(\mathcal{M}(Ω,\mathbb{S}_+^n),{\rm WB}_Λ)$ is a complete geodesic space and exhibits a conic structure.

math.NA↗

arXiv2023

Fano resonances in all-dielectric electromagnetic metasurfaces

We are interested in the resonant electromagnetic (EM) scattering by all-dielectric metasurfaces made of a two-dimensional lattice of nanoparticles with high refractive indices. In [Ammari et al., Trans. AMS, 376 (2023), 39-90], it has been shown that a single high-index nanoresonator can couple with the incident wave and exhibit a strong magnetic dipole response. Recent physics experiments reveal that when the particles are arranged in certain periodic configurations, they may have different anomalous scattering effects in the macroscopic scale, compared to the single-particle case. In this work, we shall develop a rigorous mathematical framework for analyzing the resonant behaviors of all-dielectric metasurfaces. We start with the characterization of subwavelength scattering resonances in such a periodic setting and their asymptotic expansions in terms of the refractive index of the nanoparticles. Then we show that real resonances always exist below the essential spectrum of the periodic Maxwell operator and that they are the simple poles of the scattering resolvent with the exponentially decaying resonant modes. By using group theory, we discuss the implications of the symmetry of the metasurface on the subwavelength band functions and their associated eigenfunctions. For the symmetric metasurfaces with the normal incidence, we use a variational method to show the existence of embedded eigenvalues (i.e., real subwavelength resonances embedded in the continuous radiation spectrum). Furthermore, we break the configuration symmetry either by introducing a small deformation of particles or by slightly deviating from the normal incidence and prove that Fano-type reflection and transmission anomalies can arise in both of these scenarios.

physics.optics↗

arXiv2023

Weak Galerkin Method for Electrical Impedance Tomography

In this work, we propose and analyse a weak Galerkin method for the electrical impedance tomography based on a bounded variation regularization. We use the complete electrode model as the forward system that is approximated by a weak Galerkin method with lowest order. The error estimates are studied for the forward problem, which are used to establish the convergence of this weak Galerkin algorithm for the inverse problem. Numerical examples are presented to verify the effectiveness and efficiency of the weak Galerkin algorithm for the electrical impedance tomography.

math.NA↗

arXiv2023

Acoustic scattering and field enhancement through a single aperture

We study the acoustic field enhancement through a circular hole in a hard plate of finite thickness in this work. We derive the boundary-integral equations corresponding to the acoustic scattering to investigate the Fabry-Perot type resonant frequencies that induce the field enhancement. The asymptotic expansions of Fabry-Perot type resonances with respect to the size of the hole are derived, and the quantitative analysis of the field enhancement caused by the configuration at the resonant frequencies is presented with both the enhancement order and the shapes of resonant modes characterized. The transmission in the nonresonant quasi-static regime is also investigated.

math.AP↗

arXiv2023

On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in several applications. The universal approximation properties of INNs have also been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems.

math.NA↗

arXiv2023

Silicon photonics-integrated time-domain balanced homodyne detector in continuous-variable quantum key distribution

We designed and experimentally demonstrated a silicon photonics-integrated time-domain balanced homodyne detector (TBHD), whose optical part has dimensions of 1.5 mm * 0.4 mm. To automatically and accurately balance the detector, new variable optical attenuators were used, and a common mode rejection ratio of 86.9 dB could be achieved. In the quantum tomography experiment, the density matrix and Wigner function of a coherent state were reconstructed with 99.97 % fidelity. The feasibility of this TBHD in a continuous-variable quantum key distribution (CVQKD) system was also demonstrated. This facilitates the integration of the optical circuits of the CVQKD system based on the GG02 protocol on the silicon photonics chip using TBHD.

quant-ph↗

arXiv2023

A Direct Sampling-Based Deep Learning Approach for Inverse Medium Scattering Problems

In this work, we focus on the inverse medium scattering problem (IMSP), which aims to recover unknown scatterers based on measured scattered data. Motivated by the efficient direct sampling method (DSM) introduced in [23], we propose a novel direct sampling-based deep learning approach (DSM-DL)for reconstructing inhomogeneous scatterers. In particular, we use the U-Net neural network to learn the relation between the index functions and the true contrasts. Our proposed DSM-DL is computationally efficient, robust to noise, easy to implement, and able to naturally incorporate multiple measured data to achieve high-quality reconstructions. Some representative tests are carried out with varying numbers of incident waves and different noise levels to evaluate the performance of the proposed method. The results demonstrate the promising benefits of combining deep learning techniques with the DSM for IMSP.

eess.SP↗

arXiv2022

Finite Element Method for a Nonlinear PML Helmholtz Equation with High Wave Number

A nonlinear Helmholtz equation (NLH) with high wave number and Sommerfeld radiation condition is approximated by the perfectly matched layer (PML) technique and then discretized by the linear finite element method (FEM). Wave-number-explicit stability and regularity estimates and the exponential convergence are proved for the nonlinear truncated PML problem. Preasymptotic error estimates are obtained for the FEM, where the logarithmic factors in h required by the previous results for the NLH with impedance boundary condition are removed in the case of two dimensions. Moreover, local quadratic convergences of the Newton's methods are derived for both the NLH with PML and its FEM. Numerical examples are presented to verify the accuracy of the FEM, which demonstrate that the pollution errors may be greatly reduced by applying the interior penalty technique with proper penalty parameters to the FEM. The nonlinear phenomenon of optical bistability can be successfully simulated.

math.NA↗

arXiv2022

A family of immersed finite element spaces and applications to three dimensional $\mathbf{H}(\text{curl})$ interface problems

Maxwell interface problems are of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in 3D computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on non-conforming approximation spaces, which may cause a loss of accuracy for solving Maxwell equations, and the widely-used penalty techniques in the literature may not help in recovering the optimal convergence. In this article, we provide a remedy by developing Nédélec-type immersed finite element spaces with a Petrov-Galerkin scheme that is able to produce optimal-convergent solutions. To establish a systematic framework, we analyze all the $H^1$, $\mathbf{H}(\text{curl})$ and $\mathbf{H}(\text{div})$ IFE spaces and form a discrete de Rham complex. Based on these fundamental results, we further develop a fast solver using a modified Hiptmair-Xu preconditioner which works for both the GMRES and CG methods.

math.NA↗

arXiv2022

Performance Analysis of Wireless Network Aided by Discrete-Phase-Shifter IRS

Discrete phase shifters of intelligent reflecting surface (IRS) generates phase quantization error (QE) and degrades the receive performance at the receiver. To make an analysis of the performance loss caused by IRS with phase QE, based on the law of large numbers, the closed-form expressions of signal-to-noise ratio (SNR) performance loss (PL), achievable rate (AR), and bit error rate (BER) are successively derived under line-of-sight (LoS) channels and Rayleigh channels. Moreover, based on the Taylor series expansion, the approximate simple closed form of PL of IRS with approximate QE is also given. The simulation results show that the performance losses of SNR and AR decrease as the number of quantization bits increase, while they gradually increase with the number of IRS phase shifter elements increase. Regardless of LoS channels or Rayleigh channels, when the number of quantization bits is larger than or equal to 3, the performance losses of SNR and AR are less than 0.23dB and 0.08bits/s/Hz, respectively, and the BER performance degradation is trivial. In particular, the performance loss difference between IRS with QE and IRS with approximate QE is negligible when the number of quantization bits is not less than 2.

cs.IT↗

arXiv2022

Elastodynamical resonances and cloaking of negative material structures beyond quasistatic approximation

Given the flexibility of choosing negative elastic parameters, we construct material structures that can induce two resonance phenomena, referred to as the elastodynamical resonances. They mimic the emerging plasmon/polariton resonance and anomalous localized resonance in optics for subwavelength particles. However, we study the peculiar resonance phenomena for linear elasticity beyond the subwavelength regime. It is shown that the resonance behaviours possess distinct characters, with some similar to the subwavelength resonances, but some sharply different due to the frequency effect. It is particularly noted that we construct a core-shell material structure that can induce anomalous localized resonance as well as cloaking phenomena beyond the quasi-static limit. The study is boiled down to analyzing the so-called elastic Neumann-Poincáre (N-P) operator in the frequency regime. We provide an in-depth analysis of the spectral properties of the N-P operator beyond the quasi-static approximation,and these results are of independent interest to the spectral theory of layer potential operators.

math-ph↗