arXiv subjects

Jie Shen

Publications and source records attributed to Jie Shen.

At least 181 records · Page 10Linked to original sources

arXiv2020

Log orthogonal functions: approximation properties and applications

We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions and apply them to solve several typical fractional differential equations whose solutions exhibit weak singularities. Our error analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions which have weak singularities at one endpoint, and can lead to exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used.

math.NA↗

arXiv2020

New SAV-pressure correction methods for the Navier-Stokes equations: stability and error analysis

We construct new first- and second-order pressure correction schemes using the scalar auxiliary variable (SAV) approach for the Navier-Stokes equations. These schemes are linear, decoupled and only require a sequence of solving Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous error estimates for the velocity and pressure approximation of the first-order scheme without any condition on the time step.

math.NA↗

arXiv2019

Hard superconducting gap and diffusion-induced superconductors in Ge-Si nanowires

We show a hard induced superconducting gap in a Ge-Si nanowire Josephson transistor up to in-plane magnetic fields of $250$ mT, an important step towards creating and detecting Majorana zero modes in this system. A hard induced gap requires a highly homogeneous tunneling heterointerface between the superconducting contacts and the semiconducting nanowire. This is realized by annealing devices at $180$ $^\circ$C during which aluminium inter-diffuses and replaces the germanium in a section of the nanowire. Next to Al, we find a superconductor with lower critical temperature ($T_\mathrm{C}=0.9$ K) and a higher critical field ($B_\mathrm{C}=0.9-1.2$ T). We can therefore selectively switch either superconductor to the normal state by tuning the temperature and the magnetic field and observe that the additional superconductor induces a proximity supercurrent in the semiconducting part of the nanowire even when the Al is in the normal state. In another device where the diffusion of Al rendered the nanowire completely metallic, a superconductor with a much higher critical temperature ($T_\mathrm{C}=2.9$ K) and critical field ($B_\mathrm{C}=3.4$ T) is found. The small size of diffusion-induced superconductors inside nanowires may be of special interest for applications requiring high magnetic fields in arbitrary direction.

cond-mat.mes-hall↗

arXiv2019

Global constraints preserving SAV schemes for gradient flows

We develop several efficient numerical schemes which preserve exactly the global constraints for constrained gradient flows. Our schemes are based on the SAV approach combined with the Lagrangian multiplier approach. They are as efficient as the SAV schemes for unconstrained gradient flows, i.e., only require solving linear equations with constant coefficients at each time step plus an additional nonlinear algebraic system which can be solved at negligible cost, can be unconditionally energy stable, and preserve exactly the global constraints for constrained gradient flows. Ample numerical results for phase-field vesicle membrane and optimal partition models are presented to validate the effectiveness and accuracy of the proposed numerical schemes.

math.NA↗

arXiv2019

A new Lagrange Multiplier approach for gradient flows

We propose a new Lagrange Multiplier approach to design unconditional energy stable schemes for gradient flows. The new approach leads to unconditionally energy stable schemes that are as accurate and efficient as the recently proposed SAV approach \cite{SAV01}, but enjoys two additional advantages: (i) schemes based on the new approach dissipate the original energy, as opposed to a modified energy in the recently proposed SAV approach \cite{SAV01}; and (ii) they do not require the nonlinear part of the free energy to be bounded from below as is required in the SAV approach. The price we pay for these advantages is that a nonlinear algebraic equation has to be solved to determine the Lagrange multiplier. We present ample numerical results to validate the new approach, and, as a particular example of applications, we consider a coupled Cahn-Hilliard model for block copolymers (BCP), and carry out interesting simulations which are consistent with experiment results.

math.NA↗

arXiv2019

SEWA DB: A Rich Database for Audio-Visual Emotion and Sentiment Research in the Wild

Natural human-computer interaction and audio-visual human behaviour sensing systems, which would achieve robust performance in-the-wild are more needed than ever as digital devices are increasingly becoming an indispensable part of our life. Accurately annotated real-world data are the crux in devising such systems. However, existing databases usually consider controlled settings, low demographic variability, and a single task. In this paper, we introduce the SEWA database of more than 2000 minutes of audio-visual data of 398 people coming from six cultures, 50% female, and uniformly spanning the age range of 18 to 65 years old. Subjects were recorded in two different contexts: while watching adverts and while discussing adverts in a video chat. The database includes rich annotations of the recordings in terms of facial landmarks, facial action units (FAU), various vocalisations, mirroring, and continuously valued valence, arousal, liking, agreement, and prototypic examples of (dis)liking. This database aims to be an extremely valuable resource for researchers in affective computing and automatic human sensing and is expected to push forward the research in human behaviour analysis, including cultural studies. Along with the database, we provide extensive baseline experiments for automatic FAU detection and automatic valence, arousal and (dis)liking intensity estimation.

cs.HC↗

arXiv2019

Towards Pose-invariant Lip-Reading

Lip-reading models have been significantly improved recently thanks to powerful deep learning architectures. However, most works focused on frontal or near frontal views of the mouth. As a consequence, lip-reading performance seriously deteriorates in non-frontal mouth views. In this work, we present a framework for training pose-invariant lip-reading models on synthetic data instead of collecting and annotating non-frontal data which is costly and tedious. The proposed model significantly outperforms previous approaches on non-frontal views while retaining the superior performance on frontal and near frontal mouth views. Specifically, we propose to use a 3D Morphable Model (3DMM) to augment LRW, an existing large-scale but mostly frontal dataset, by generating synthetic facial data in arbitrary poses. The newly derived dataset, is used to train a state-of-the-art neural network for lip-reading. We conducted a cross-database experiment for isolated word recognition on the LRS2 dataset, and reported an absolute improvement of 2.55%. The benefit of the proposed approach becomes clearer in extreme poses where an absolute improvement of up to 20.64% over the baseline is achieved.

cs.CV↗

arXiv2019

Shape Constrained Network for Eye Segmentation in the Wild

Semantic segmentation of eyes has long been a vital pre-processing step in many biometric applications. Majority of the works focus only on high resolution eye images, while little has been done to segment the eyes from low quality images in the wild. However, this is a particularly interesting and meaningful topic, as eyes play a crucial role in conveying the emotional state and mental well-being of a person. In this work, we take two steps toward solving this problem: (1) We collect and annotate a challenging eye segmentation dataset containing 8882 eye patches from 4461 facial images of different resolutions, illumination conditions and head poses; (2) We develop a novel eye segmentation method, Shape Constrained Network (SCN), that incorporates shape prior into the segmentation network training procedure. Specifically, we learn the shape prior from our dataset using VAE-GAN, and leverage the pre-trained encoder and discriminator to regularise the training of SegNet. To improve the accuracy and quality of predicted masks, we replace the loss of SegNet with three new losses: Intersection-over-Union (IoU) loss, shape discriminator loss and shape embedding loss. Extensive experiments shows that our method outperforms state-of-the-art segmentation and landmark detection methods in terms of mean IoU (mIoU) accuracy and the quality of segmentation masks. The eye segmentation database is available at https://www.dropbox.com/s/yvveouvxsvti08x/Eye_Segmentation_Database.zip?dl=0.

cs.CV↗

arXiv2019

An efficient numerical scheme for a 3D spherical dynamo equation

We develop an efficient numerical scheme for the 3D mean-field spherical dynamo equation. The scheme is based on a semi-implicit discretization in time and a spectral method in space based on the divergence-free spherical harmonic functions. A special semi-implicit approach is proposed such that at each time step one only needs to solve a linear system with constant coefficients. Then, using expansion in divergence-free spherical harmonic functions in the transverse directions allows us to reduce the linear system at each time step to a sequence of one-dimensional equations in the radial direction, which can then be efficiently solved by using a spectral-element method. We show that the solution of fully discretized scheme remains bounded independent of the number of unknowns, and present numerical results to validate our scheme.

math.NA↗

arXiv2019

Error analysis of the SAV-MAC scheme for the Navier-Stokes equations

An efficient numerical scheme based on the scalar auxiliary variable (SAV) and marker and cell scheme (MAC) is constructed for the Navier-Stokes equations. A particular feature of the scheme is that the nonlinear term is treated explicitly while being unconditionally energy stable. A rigorous error analysis is carried out to show that both velocity and pressure approximations are second-order accurate in time and space. Numerical experiments are presented to verify the theoretical results.

math.NA↗

arXiv2019

Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time-discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schr{ö}dinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.

math.NA↗

arXiv2019

Stability and Error estimates of the SAV Fourier-spectral method for the Phase Field Crystal Equation

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order $O(Δt^2+N^{-m})$, where $Δt$, $N$ and $m$ are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes.

math.AP↗

arXiv2019

Crossed Andreev Reflection in InSb Flake Josephson Junctions

We study superconducting quantum interference in InSb flake Josephson junctions. An even-odd effect in the amplitude and periodicity of the superconducting quantum interference pattern is found. Interestingly, the occurrence of this pattern coincides with enhanced conduction at both edges of the flake, as is deduced from measuring a SQUID pattern at reduced gate voltages. We identify the specific crystal facet of the edge with enhanced conduction, and confirm this by measuring multiple devices. Furthermore, we argue the even-odd effect is due to crossed Andreev reflection, a process where a Cooper pair splits up over the two edges and recombines at the opposite contact. An entirely $h/e$ periodic SQUID pattern, as well as the observation of both even-odd and odd-even effects, corroborates this conclusion. Crossed Andreev reflection could be harnessed for creating a topological state of matter or performing experiments on the non-local spin-entanglement of spatially separated Cooper pairs.

cond-mat.mes-hall↗

arXiv2019

On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes Phase Field Model

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn-Hilliard-Navier-Stokes phase field model, and carry out stability and error analysis. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase field variable, chemical potential, velocity and pressure in different discrete norms. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the our scheme.

math.AP↗

arXiv2019

Distributional Compatibility for Change of Measures

In this paper, we characterize compatibility of distributions and probability measures on a measurable space. For a set of indices $\mathcal J$, we say that the tuples of probability measures $(Q_i)_{i\in \mathcal J} $ and distributions $(F_i)_{i\in \mathcal J} $ are {compatible} if there exists a random variable having distribution $F_i$ under $Q_i$ for each $i\in \mathcal J$. We first establish an equivalent condition using conditional expectations for general (possibly uncountable) $\mathcal J$. For a finite $n$, it turns out that compatibility of $(Q_1,\dots,Q_n)$ and $(F_1,\dots,F_n)$ depends on the heterogeneity among $Q_1,\dots,Q_n$ compared with that among $F_1,\dots,F_n$. We show that, under an assumption that the measurable space is rich enough, $(Q_1,\dots,Q_n)$ and $(F_1,\dots,F_n)$ are compatible if and only if $(Q_1,\dots,Q_n)$ dominates $(F_1,\dots,F_n)$ in a notion of heterogeneity order, defined via multivariate convex order between the Radon-Nikodym derivatives of $(Q_1,\dots,Q_n)$ and $(F_1,\dots,F_n)$ with respect to some reference measures. We then proceed to generalize our results to stochastic processes, and conclude the paper with an application to portfolio selection problems under multiple constraints.

math.PR↗

arXiv2019

Evidence of Majorana Zero Modes in Josephson Trijunctions

In search of fault-tolerant topological quantum computation (TQC), zero-bias conductance peak as a necessary signature of Majorana zero modes (MZMs) has been observed in a number of solid-state systems. Here, we present the signature of MZMs from a phase-sensitive experiment on Josephson trijunctions constructed on the surface of three-dimensional topological insulators. We observed that the minigap at the center of the trijunction is protected to close over extended regions in phase space, supporting in principle the Majorana phase diagram proposed by Fu and Kane in 2008. Our study paves the way for further braiding MZMs and exploring TQC on a scalable two-dimensional platform.

cond-mat.mes-hall↗

arXiv2019

MobiFace: A Novel Dataset for Mobile Face Tracking in the Wild

Face tracking serves as the crucial initial step in mobile applications trying to analyse target faces over time in mobile settings. However, this problem has received little attention, mainly due to the scarcity of dedicated face tracking benchmarks. In this work, we introduce MobiFace, the first dataset for single face tracking in mobile situations. It consists of 80 unedited live-streaming mobile videos captured by 70 different smartphone users in fully unconstrained environments. Over $95K$ bounding boxes are manually labelled. The videos are carefully selected to cover typical smartphone usage. The videos are also annotated with 14 attributes, including 6 newly proposed attributes and 8 commonly seen in object tracking. 36 state-of-the-art trackers, including facial landmark trackers, generic object trackers and trackers that we have fine-tuned or improved, are evaluated. The results suggest that mobile face tracking cannot be solved through existing approaches. In addition, we show that fine-tuning on the MobiFace training data significantly boosts the performance of deep learning-based trackers, suggesting that MobiFace captures the unique characteristics of mobile face tracking. Our goal is to offer the community a diverse dataset to enable the design and evaluation of mobile face trackers. The dataset, annotations and the evaluation server will be on \url{https://mobiface.github.io/}.

cs.CV↗

arXiv2018

Energy Stability and Convergence of SAV Block-centered Finite Difference Method for Gradient Flows

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

math.NA↗