arXiv subjects

Lei Li

Publications and source records attributed to Lei Li.

At least 775 records · Page 43Linked to original sources

arXiv2018

Some compactness criteria for weak solutions of time fractional PDEs

The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order $γ\in (0,1)$ for functions valued in general Banach spaces, consistent with the traditional definition if the space is $\mathbb{R}^d$ and functions are absolutely continuous. Based on a Volterra type integral form, we establish some time regularity estimates of the functions provided that the weak Caputo derivatives are in certain spaces. The compactness criteria are then established using the time regularity estimates. The existence of weak solutions for a special case of time fractional compressible Navier-Stokes equations with constant density and time fractional Keller-Segel equations in $\mathbb{R}^2$ are then proved as model problems. This work provides a framework for studying weak solutions of nonlinear time fractional PDEs.

math.FA↗

arXiv2018

A generalized definition of Caputo derivatives and its application to fractional ODEs

We extend in this paper the definition of Caputo derivatives of order in $(0,1)$ to a certain class of locally integrable functions using a convolution group. Our strategy is to define a fractional calculus for a certain class of distributions using the convolution group. When acting on causal functions, this fractional calculus agrees with the traditional Riemann-Liouville definition for $t>0$ but includes some singularities at $t=0$ so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in literature while reveals the underlying group structure. Since the new definition is valid for a class of locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs with very weak conditions. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Grönwall inequality (or comparison principle) with the most general conditions. This then provides the essential tools for {\it a priori} energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions.

math.CA↗

arXiv2018

Dense People Counting Using IR-UWB Radar with a Hybrid Feature Extraction Method

People counting is one of the hottest issues in sensing applications. The impulse radio ultra-wideband (IR-UWB) radar has been extensively applied to count people, providing a device-free solution without illumination and privacy concerns. However, performance of current solutions is limited in congested environments due to the superposition and obstruction of signals. In this letter, a hybrid feature extraction method based on curvelet transform and distance bin is proposed. 2-D radar matrix features are extracted in multiple scales and multiple angles by applying the curvelet transform. Furthermore, the distance bin is introduced by dividing each row of the matrix into several bins along the propagating distance to select features. The radar signal dataset in three dense scenarios is constructed, including people randomly walking in the constrained area with densities of 3 and 4 persons per square meter, and queueing with an average distance of 10 centimeters. The number of people is up to 20 in the dataset. Four classifiers including decision tree, AdaBoost, random forest and neural network are compared to validate the hybrid features, and random forest performs the highest accuracies of all above 97% in three dense scenarios. Moreover, to ensure the reliability of the hybrid features, three other features including cluster features, activity features and CNN features are compared. The experimental results reveal that the proposed hybrid feature extraction method exhibits stable performance with significantly superior effectiveness.

eess.SP↗

arXiv2018

BRITS: Bidirectional Recurrent Imputation for Time Series

Time series are widely used as signals in many classification/regression tasks. It is ubiquitous that time series contains many missing values. Given multiple correlated time series data, how to fill in missing values and to predict their class labels? Existing imputation methods often impose strong assumptions of the underlying data generating process, such as linear dynamics in the state space. In this paper, we propose BRITS, a novel method based on recurrent neural networks for missing value imputation in time series data. Our proposed method directly learns the missing values in a bidirectional recurrent dynamical system, without any specific assumption. The imputed values are treated as variables of RNN graph and can be effectively updated during the backpropagation.BRITS has three advantages: (a) it can handle multiple correlated missing values in time series; (b) it generalizes to time series with nonlinear dynamics underlying; (c) it provides a data-driven imputation procedure and applies to general settings with missing data.We evaluate our model on three real-world datasets, including an air quality dataset, a health-care data, and a localization data for human activity. Experiments show that our model outperforms the state-of-the-art methods in both imputation and classification/regression accuracies.

cs.LG↗

arXiv2018

Spin direction controlled electronic band structure in two dimensional ferromagnetic CrI3

Manipulating physical properties using the spin degree of freedom constitutes a major part of modern condensed matter physics and is very important for spintronics devices. Using the newly discovered two dimensional van der Waals ferromagnetic CrI3 as a prototypic material, we theoretically demonstrated a giant magneto band-structure (GMB) effect whereby a change of magnetization direction significantly modifies the electronic band structure. Our density functional theory calculations and model analysis reveal that rotating the magnetic moment of CrI3 from out-of-plane to in-plane causes a direct-to-indirect bandgap transition, inducing a magnetic field controlled photoluminescence. Moreover, our results show a significant change of Fermi surface with different magnetization directions, giving rise to giant anisotropic magnetoresistance. Additionally, the spin reorientation is found to modify the topological states. Given that a variety of properties are determined by band structures, our predicted GMB effect in CrI3 opens a new paradigm for spintronics applications.

cond-mat.mtrl-sci↗

arXiv2018

Reinforced Co-Training

Co-training is a popular semi-supervised learning framework to utilize a large amount of unlabeled data in addition to a small labeled set. Co-training methods exploit predicted labels on the unlabeled data and select samples based on prediction confidence to augment the training. However, the selection of samples in existing co-training methods is based on a predetermined policy, which ignores the sampling bias between the unlabeled and the labeled subsets, and fails to explore the data space. In this paper, we propose a novel method, Reinforced Co-Training, to select high-quality unlabeled samples to better co-train on. More specifically, our approach uses Q-learning to learn a data selection policy with a small labeled dataset, and then exploits this policy to train the co-training classifiers automatically. Experimental results on clickbait detection and generic text classification tasks demonstrate that our proposed method can obtain more accurate text classification results.

cs.CL↗

arXiv2018

Image Domain Dual Material Decomposition for Dual-Energy CT using Butterfly Network

Dual-energy CT (DECT) has been increasingly used in imaging applications because of its capability for material differentiation. However, material decomposition suffers from magnified noise from two CT images of independent scans, leading to severe degradation of image quality. Existing algorithms achieve suboptimal decomposition performance since they fail to accurately depict the mapping relationship between DECT and the basis material images. Inspired by the impressive potential of CNN, we developed a new Butterfly network to perform the image domain dual material decomposition due to its strong approximation ability to the mapping functions in DECT. The Butterfly network is derived from the image domain DECT decomposition model by exploring the geometric relationship between mapping functions of data model and network components. The network is designed as the double-entry double-out crossover architecture based on the decomposition formulation. It enters a pair of dual-energy images as inputs and defines the ground true decomposed images as each label. The crossover architecture, which plays an important role in material decomposition, is designed to implement the information exchange between the two material generation pathways in the network. Network components exhibit different sensitivity to basis materials in the visualization. By analyzing their sensitivity to different basis materials, we determined the roles of network components in material decomposition. This visualization evaluation reveals what the network can learn and verifies the rationality of network design. The qualitative and quantitative evaluations in material decomposition of patient data indicate that the proposed network outperforms its counterpart.

physics.med-ph↗

arXiv2018

A note on one-dimensional time fractional ODEs

In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work \cite{fllx17}. Here we use the definition of Caputo derivative proposed in \cite{liliu17frac1,liliu2017} based on a convolution group. In particular, we establish generalized comparison principles consistent with the new definition of Caputo derivatives. In addition, we establish the full asymptotic behaviors of the solutions for $D_c^γu=Au^p$. Lastly, we provide a simplified proof for the strict monotonicity and stability in initial values for the time fractional differential equations with weak assumptions.

math.CA↗

arXiv2018

Cauchy problems for Keller-Segel type time-space fractional diffusion equation

This paper investigates Cauchy problems for nonlinear fractional time-space generalized Keller-Segel equation $^c_0D_t^βρ+(-\triangle)^{\fracα{2}}ρ+\nabla\cdot(ρB(ρ))=0$, where Caputo derivative $^c_0D_t^βρ$ models memory effects in time, fractional Laplacian $(-\triangle)^{\fracα{2}}ρ$ represents Lévy diffusion and $B(ρ)=-s_{n,γ}\int_{R^n}\frac{x-y}{|x-y|^{n-γ+2}}ρ(y)dy $ is the general potential with a singular kernel which takes into account the long rang interaction. We first establish $L^r-L^q$ estimates and weighted estimates of the fundamental solutions $(P(x,t), Y(x,t))$ (or equivalently, the solution operators $(S_α^β(t), T_α^β(t))$). Then, we prove the existence and uniqueness of the mild solutions when initial data are in $L^p$ spaces, or the weighted spaces. Similar to Keller-Segel equations, if the initial data are small in critical space $L^{p_c}(\mathbb{R}^n)$ ($p_c=\frac{n}{α+γ-2}$), we construct the global existence. Furthermore, we prove the $L^1$ integrability and integral preservation when the initial data are in $L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ or $L^1(\mathbb{R}^n)\cap L^{p_c}(\mathbb{R}^n)$. Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established.

math.AP↗

arXiv2018

Large-Scale Fabrication of RF MOSFETs on Liquid-Exfoliated MoS2

For the first time, thousands of RF MOSFETs were batch-fabricated on liquid-exfoliated MoS2 below 300 °C with nearly 100% yield. The large-scale fabrication with high yield allowed the average performance instead of the best performance to be reported. The DC performance of these devices were typical, but the RF performance, enabled by buried gates and on the order of 100 MHz, was reported for the first time for liquid-exfoliated MoS2. To resolve the dilemma of thin vs. thick films, gate recess was used on 20-nm thick films to improve the gate control while keeping the contact resistance lower than that on 10-nm films. These innovations may enable thin-film transistors to operate in the microwave range.

physics.app-ph↗

arXiv2018

On the diffusion approximation of nonconvex stochastic gradient descent

We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small step size regime and the presence of omnidirectional noise, our weak approximating diffusion process suggests the following dynamics for the SGD iteration starting from a local minimizer (resp.~saddle point): it escapes in a number of iterations exponentially (resp.~almost linearly) dependent on the inverse stepsize. The results are obtained using the theory for random perturbations of dynamical systems (theory of large deviations for local minimizers and theory of exiting for unstable stationary points). In addition, we discuss the effects of batch size for the deep neural networks, and we find that small batch size is helpful for SGD algorithms to escape unstable stationary points and sharp minimizers. Our theory indicates that one should increase the batch size at later stage for the SGD to be trapped in flat minimizers for better generalization.

stat.ML↗

arXiv2018

Effect of coherence of nonthermal reservoirs on heat transport in a microscopic collision model

We investigate the heat transport between two nonthermal reservoirs based on a microscopic collision model. We consider a bipartite system consisting of two identical subsystems, and each subsystem interacts with its own local reservoir, which consists of a large collection of initially uncorrelated ancillas. Then a heat transport is formed between two reservoirs by a sequence of pairwise collisions (inter-subsystem and subsystem-local reservoir). In this paper we consider two kinds of reservoir's initial states, the thermal state, and the state with coherence whose diagonal elements are the same as that of the thermal state and the off-diaganal elements are nonzero. In this way, we define the effective temperature of the reservoir with coherence according to its diagonal elements. We find that for two reservoirs having coherence the direction of the steady current of heat is different for different phase differences between the two initial states of two reservoirs, especially the heat can transfer from the "cold reservoir" to the "hot reservoir" in the steady regime for particular phase difference. And in the limit of the effective temperature difference between the two reservoirs $ΔT\rightarrow0$, for most of the phase differences, the steady heat current increases with the increase of effective temperature until to the high effective temperature limit; while for the thermal state or particular phase difference the steady heat current decreases with the increase of temperature at high temperatures, and in this case the conductance can be obtained.

quant-ph↗

arXiv2018

Black phosphorus Q-switched and mode-locked Er:ZBLAN fiber lasers at 3.5 um

With the proposal of dual-wavelength pumping (DWP) scheme, DWP Er:ZBLAN fiber lasers at 3.5 um have become a fascinating area of research. However, limited by the absence of suitable saturable absorber, passively Q-switched and mode-locked fiber lasers have not been realized in this spectral region. Based on the layer-dependent bandgap and excellent photoelectric characteristics of black phosphorus (BP), BP is a promising candidate for saturable absorber near 3.5 um. Here, we fabricated a 3.5-um saturable absorber mirror (SAM) by transferring liquid-phase exfoliated BP flakes onto a gold-coated mirror. With the as-prepared BP SAM, we realized stable Q-switching and continuous-wave mode-locking operations in the DWP Er:ZBLAN fiber lasers at 3.5 um. To the best of our knowledge, it is the first time to achieve passively Q-switched and mode-locked pulses in 3.5 um spectral region. The research results will not only promote the development of 3.5-um pulsed fiber lasers but also open the photonic application of two-dimensional materials in this spectral region

physics.optics↗

arXiv2018

Electronic structure of the candidate 2D Dirac semimetal SrMnSb2: a combined experimental and theoretical study

SrMnSb$_2$ is suggested to be a magnetic topological semimetal. It contains square, 2D Sb planes with non-symmorphic crystal symmetries that could protect band crossings, offering the possibility of a quasi-2D, robust Dirac semi-metal in the form of a stable, bulk (3D) crystal. Here, we report a combined and comprehensive experimental and theoretical investigation of the electronic structure of SrMnSb$_2$, including the first ARPES data on this compound. SrMnSb$_2$ possesses a small Fermi surface originating from highly 2D, sharp and linearly dispersing bands (the Y-states) around the (0,$π$/a)-point in $k$-space. The ARPES Fermi surface agrees perfectly with that from bulk-sensitive Shubnikov de Haas data from the same crystals, proving the Y$-$states to be responsible for electrical conductivity in SrMnSb$_2$. DFT and tight binding (TB) methods are used to model the electronic states, and both show good agreement with the ARPES data. Despite the great promise of the latter, both theory approaches show the Y-states to be gapped above E$_F$, suggesting trivial topology. Subsequent analysis within both theory approaches shows the Berry phase to be zero, indicating the non-topological character of the transport in SrMnSb$_2$, a conclusion backed up by the analysis of the quantum oscillation data from our crystals.

cond-mat.str-el↗

arXiv2018

Semi-groups of stochastic gradient descent and online principal component analysis: properties and diffusion approximations

We study the Markov semigroups for two important algorithms from machine learning: stochastic gradient descent (SGD) and online principal component analysis (PCA). We investigate the effects of small jumps on the properties of the semi-groups. Properties including regularity preserving, $L^{\infty}$ contraction are discussed. These semigroups are the dual of the semigroups for evolution of probability, while the latter are $L^{1}$ contracting and positivity preserving. Using these properties, we show that stochastic differential equations (SDEs) in $\mathbb{R}^d$ (on the sphere $\mathbb{S}^{d-1}$) can be used to approximate SGD (online PCA) weakly. These SDEs may be used to provide some insights of the behaviors of these algorithms.

math.PR↗

arXiv2017

$p$-Euler equations and $p$-Navier-Stokes equations

We propose in this work new systems of equations which we call $p$-Euler equations and $p$-Navier-Stokes equations. $p$-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-$p$ distances, with incompressibility constraint. $p$-Euler equations have similar structures with the usual Euler equations but the `momentum' is the signed ($p-1$)-th power of the velocity. In the 2D case, the $p$-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the $p$-Laplacian of the streamfunction. By adding diffusion presented by $γ$-Laplacian of the velocity, we obtain what we call $p$-Navier-Stokes equations. If $γ=p$, the {\it a priori} energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the $p$-Navier-Stokes equations in $\mathbb{R}^d$ for $γ=p$ and $p\ge d\ge 2$ through a compactness criterion.

math.AP↗

arXiv2017

On Runge-Kutta methods for the water wave equation and its simplified nonlocal hyperbolic model

There is a growing interest in investigating numerical approximations of the water wave equation in recent years, whereas the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In this work, we focus on a nonlocal hyperbolic model, which essentially inherits the features of the water wave equation, and is simplified from the latter. For the constant coefficient case, we carry out systematical stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and general Runge-Kutta method in time. In particular, we discover the optimal time step constraints, in the form of a modified CFL condition, when certain explicit Runge-Kutta method are applied. Besides, the convergence of the semi-discrete approximation of variable coefficient case is shown, which naturally connects to the water wave equation. Extensive numerical tests have been performed to verify the stability conditions and simulations of the simplified hyperbolic model in the high frequency regime and the water wave equation are also provided.

math.NA↗

arXiv2017

SAM: Semantic Attribute Modulation for Language Modeling and Style Variation

This paper presents a Semantic Attribute Modulation (SAM) for language modeling and style variation. The semantic attribute modulation includes various document attributes, such as titles, authors, and document categories. We consider two types of attributes, (title attributes and category attributes), and a flexible attribute selection scheme by automatically scoring them via an attribute attention mechanism. The semantic attributes are embedded into the hidden semantic space as the generation inputs. With the attributes properly harnessed, our proposed SAM can generate interpretable texts with regard to the input attributes. Qualitative analysis, including word semantic analysis and attention values, shows the interpretability of SAM. On several typical text datasets, we empirically demonstrate the superiority of the Semantic Attribute Modulated language model with different combinations of document attributes. Moreover, we present a style variation for the lyric generation using SAM, which shows a strong connection between the style variation and the semantic attributes.

cs.CL↗