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This work studies an inverse scattering problem when limited-aperture data are available that are from just one or a few incident fields. This inverse problem is highly ill-posed due to the limited receivers and a few incident fields employed. Solving inverse scattering problems with limited-aperture data is important in applications as collecting full data is often either unrealistic or too expensive. The direct sampling methods (DSMs) with full-aperture data can effectively and stably estimate the locations and geometric shapes of the unknown scatterers with a very limited number of incident waves. However, a direct application of DSMs to the case of limited receivers would face the resolution limit. To break this limitation, we propose a finite space framework with two specific schemes, and an unsupervised deep learning strategy to construct effective probing functions for the DSMs in the case with limited-aperture data. Several representative numerical experiments are carried out to illustrate and compare the performance of different proposed schemes.

In this work, we propose and analyze two two-level hybrid Schwarz preconditioners for solving the Helmholtz equation with high wave number in two and three dimensions. Both preconditioners are defined over a set of overlapping subdomains, with each preconditioner formed by a global coarse solver and one local solver on each subdomain. The global coarse solver is based on the localized orthogonal decomposition (LOD) technique, which was proposed in [27,28] originally for the discretization schemes for elliptic multiscale problems with heterogeneous and highly oscillating coefficients and Helmholtz problems with high wave number to eliminate the pollution effect. The local subproblems are Helmholtz problems in subdomains with homogeneous boundary conditions (the first preconditioner) or impedance boundary conditions (the second preconditioner). Both preconditioners are shown to be optimal under some reasonable conditions, that is, a uniform upper bound of the preconditioned operator norm and a uniform lower bound of the field of values are established in terms of all the key parameters, such as the fine mesh size, the coarse mesh size, the subdomain size and the wave numbers. It is the first time to show that the LOD solver can be a very effective coarse solver when it is used appropriately in the Schwarz method with multiple overlapping subdomains. Numerical experiments are presented to confirm the optimality and efficiency of the two proposed preconditioners.

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 Babu\v{s}ka 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.

We consider in this work an inverse acoustic scattering problem when only phaseless data is available. The inverse problem is highly nonlinear and ill-posed due to the lack of the phase information. Solving inverse scattering problems with phaseless data is important in applications as the collection of physically acceptable phased data is usually difficult and expensive. A novel direct sampling method (DSM) will be developed to effectively estimate the locations and geometric shapes of the unknown scatterers from phaseless data generated by a very limited number of incident waves. With a careful theoretical analysis of the behavior of the index function and some representative numerical examples, the new DSM is shown to be computationally efficient, easy to implement, robust to large noise, and does not require any prior knowledge of the unknown scatterers. Furthermore, to fully exploit the index functions obtained from the DSM, we also propose to integrate the DSM with a deep learning technique (DSM-DL) to achieve high-quality reconstructions. Several challenging and representative numerical experiments are carried out to demonstrate the accuracy and robustness of reconstructions by DSM-DL. The DSM-DL networks trained by phased data are further theoretically and numerically shown to be able to solve problems with phaseless data. Additionally, our numerical experiments also show the DSM-DL can solve inverse scattering problems with mixed types of scatterers, which renders its applications in many important practical scenarios.

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., Comment: 15 pages, 10 figures

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{\deg} with a standard deviation of 0.011{\deg}.

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.

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.

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 Lam\'e 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.

A generalized unbalanced optimal transport distance ${\rm WB}_{\Lambda}$ on matrix-valued measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$ was defined in [arXiv:2011.05845] \`{a} 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 ${\rm WB}_{\Lambda}$. 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., Comment: revised

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., Comment: 32 pages

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.

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.

We study a mean-field version of Bank-El Karoui's representation theorem of stochastic processes. Under different technical conditions, we establish some existence and uniqueness results. As motivation and first applications, our mean-field representation results provide a unified approach to study different Mean-Field Games (MFGs) in the setting with common noise and multiple populations, including the MFG of timing, the MFG with singular control, etc. As a crucial technical step, we provide a stability result on the classical Bank-El Karoui's representation theorem, which has its own interests and other applications, such as in deriving stability results of the optimizers (in the strong sense) for a class of optimal stopping problems and singular control problems., Comment: 46 pages, 0 figures

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., Comment: revised

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.