A convexity-preserving and perimeter-decreasing parametric finite element method for the area-preserving curve shortening flow

We propose and analyze a semi-discrete parametric finite element scheme for solving the area-preserving curve shortening flow. The scheme is based on Dziuk's approach (SIAM J. Numer. Anal. 36(6): 1808-1830, 1999) for the anisotropic curve shortening flow. We prove that the scheme preserves two fundamental geometric structures of the flow with an initially convex curve: (i) the convexity-preserving property, and (ii) the perimeter-decreasing property. To the best of our knowledge, the convexity-preserving property of numerical schemes which approximate the flow is rigorously proved for the first time. Furthermore, the error estimate of the semi-discrete scheme is established, and numerical results are provided to demonstrate the structure-preserving properties as well as the accuracy of the scheme.Comment: 24 pages, 2 figure

Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay

We make the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions

A second-order in time, BGN-based parametric finite element method for geometric flows of curves

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441{467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which incorporates a mesh regularization technique when necessary, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as a mesh regularization technique when necessary, our proposed second-order scheme exhibits good properties with respect to the mesh distribution.Comment: 35 pages, 9 figure

Tailoring Bulk Photovoltaic Effects in Magnetic Sliding Ferroelectric Materials

The bulk photovoltaic effect that is intimately associated with crystalline symmetry has been extensively studied in various nonmagnetic materials, especially ferroelectrics with a switchable electric polarization. In order to further engineer the symmetry, one could resort to spin-polarized systems possessing an extra magnetic degree of freedom. Here, we investigate the bulk photovoltaic effect in two-dimensional magnetic sliding ferroelectric (MSFE) systems, illustrated in VSe2, FeCl2, and CrI3 bilayers. The transition metal elements in these systems exhibit intrinsic spin polarization, and the stacking mismatch between the two layers produce a finite out-of-plane electric dipole. Through symmetry analyses and first-principles calculations, we show that photoinduced in-plane bulk photovoltaic current can be effectively tuned by their magnetic order and the out-of-plane dipole moment. The underlying mechanism is elucidated from the quantum metric dipole distribution in the reciprocal space. The ease of the fabrication and manipulation of MSFEs guarantee practical optoelectronic applications

Alternative schemes for twin-field quantum key distribution with discrete-phase-randomized sources

The twin-field quantum key distribution (TF-QKD) protocol and its variants can overcome the well-known rate-loss bound without quantum repeaters, which have attracted significant attention. Generally, to ensure the security of these protocols, weak coherent states with continuous randomized phases are always assumed in the test mode. However, this assumption is difficult to meet in practice. To bridge the gap between theory and practice, we propose two alternative discrete-phase-randomized (DPR)-twin-field quantum key distribution protocols, which remove the phase sifting procedure in the code mode. Simulation results show that when compared with previous discrete-phase-randomized-twin-field quantum key distribution protocols, our modified protocols can significantly improve the secret key rate in the low channel loss range, which is very promising for practical twin-field quantum key distribution systems

Ultrathin metasurface devices for phase and polarization control

Metamaterials are artificial materials, which consist of periodic or non-periodic structures of manmade “atoms” with a typical size of subwavelength scale. Benefiting from the freedom to tailor the properties of electromagnetic waves that are unavailable in nature, metamaterials have brought new concepts and new discoveries to the world (e.g., negative refraction, super imaging and cloaking). However, the application of 3D metamaerials in the optical range is limited by current nanofabrication techniques. As 2D counterparts of metamaterials, metasurfaces, which are comprised of a single layer or a stack of several layers of 2D structures, can facilitate manufacturing process and manipulate light propagation in desirable manner. Metsurfaces are ultrathin and ultraflat, which has enabled ultrathin optical devices that can outperform the capabilities of traditional bulky optical elements. The unprecedented capabilities of metasurfaces in the manipulation of amplitude, phase and polarization have led to the development of novel, compact optical devices with specially designed functionalities. In this thesis, we simultaneously control the phase and polarization state of light with one single metasurface, enabling novel multifunctional devices that are not possible with conventional optical devices. To meet the growing requirement of device miniaturization and system integration, it is of great importance and interest to develop ultrathin optical devices that integrate multiple functionalities into one device while preserving their independent functionalities. To increase the functionality density, we develop metasurface devices that can dynamically control the superposition of laser beams with various orbital angular momentum (OAM). This unique approach can arbitrarily realize different functionalities in multiple channels based on a single plasmonic metasurface. As a proof of concept, we experimentally demonstrate an ultrathin optical device that can simultaneously realize polarization-controllable hologram and superposition of OAM beams in multiple channels, which is realized by controlling the polarization state of the incident light. Although human eyes or cameras are sensitive to spatially varying intensity or colour profiles, they are blind to polarization profiles with uniform intensity profiles. We propose and experimentally demonstrate an approach to hide a high-resolution grayscale image in a laser beam. The space-variant polarization profile originates from the superposition of two circularly polarized beams with opposite handedness that come from from a single metasurface device. Upon the illumination of a linearly polarized light beam, we experimentally demonstrate a metasurface device that can generate a light beam with inhomogeneous polarization profile for hiding a quick response (QR) code. The unique measurement technique used here holds great promise for anti-counterfeiting and encryption. This approach is extended to hide a color image through the local control of both polarization and color selectivity based on a high-efficiency transmissive dielectric metasurface, which consists of silicon nanoblocks that can precisely control brightness and contrast. This approach provides an extraordinary capability in additive color mixing and tailoring the polarization of light at the nanoscale. The unprecedented capabilities of optical metasurfaces in the local manipulation of the phase and polarization of light at the subwavelength scale, open a new window for the implementation of a plethora of novel optical components. The unique advantages of simplicity and robustness of our design, ease of fabrication, compactness and unusual functionalities of ultrathin metadevices render optical metasurfaces very attractive for new applications in free-space imaging, information processing, encryption, anticounterfeiting, optical communications, and fundamental physics. The plan for the future work in this amazing field is also discussed in the last chapter

Spatial residual blocks combined parallel network for hyperspectral image classification.

In hyperspectral image (HSI) classification, there are challenges of the spatial variation in spectral features and the lack of labeled samples. In this paper, a novel spatial residual blocks combined parallel network (SRPNet) is proposed for HSI classification. Firstly, the spatial residual blocks extract spatial features from rich spatial contexts information, which can be used to deal with the spatial variation of spectral signatures. Especially, the skip connection in spatial residual blocks is conducive to the backpropagation of gradients and mitigates the declining-accuracy phenomenon in the deep network. Secondly, the parallel structure is employed to extract spectral features. Spectral feature learning on parallel branches contains fewer independent connection weighs through parameter sharing. Thus, fewer parameters of the network require a lesser number of training samples. Furthermore, the feature fusion is conducted on the multi-scale features from different layers in the spectral feature learning part. Extensive experiments of three representative HSI data sets illustrate the effectiveness of the proposed network