arXiv subjects

Tao Tang

Publications and source records attributed to Tao Tang.

At least 109 records · Page 6Linked to original sources

arXiv2021

A new discrete energy technique for multi-step backward difference formulas

The backward differentiation formula (BDF) is a useful family of implicit methods for the numerical integration of stiff differential equations. It is well noticed that the stability and convergence of the $A$-stable BDF1 and BDF2 schemes for parabolic equations can be directly established by using the standard discrete energy analysis. However, such classical analysis technique seems not directly applicable to the BDF-$\mathbf{k}$ schemes for $3\leq \mathbf{k}\leq 5$. To overcome the difficulty, a powerful analysis tool based on the Nevanlinna-Odeh multiplier technique [Numer. Funct. Anal. Optim., 3:377-423, 1981] was developed by Lubich et al. [IMA J. Numer. Anal., 33:1365-1385, 2013]. In this work, by using the so-called discrete orthogonal convolution kernels technique, we will recover the classical energy analysis so that the stability and convergence of the BDF-$\mathbf{k}$ schemes for $3\leq \mathbf{k}\leq 5$ can be established. One of the theoretical advantages of our analysis technique is that less spacial regularity requirement is needed on the initial data.

math.NA↗

arXiv2021

Field-effect at electrical contacts to two-dimensional materials

The inferior electrical contact to two-dimensional (2D) materials is a critical challenge for their application in post-silicon very large-scale integrated circuits. Electrical contacts were generally related to their resistive effect, quantified as contact resistance. With a systematic investigation, this work demonstrates a capacitive metal-insulator-semiconductor (MIS) field-effect at the electrical contacts to 2D materials: the field-effect depletes or accumulates charge carriers, redistributes the voltage potential, and give rise to abnormal current saturation and nonlinearity. On the one hand, the current saturation hinders the devices' driving ability, which can be eliminated with carefully engineered contact configurations. On the other hand, by introducing the nonlinearity to monolithic analog artificial neural network circuits, the circuits' perception ability can be significantly enhanced, as evidenced using a COVID-19 critical illness prediction model. This work provides a comprehension of the field-effect at the electrical contacts to 2D materials, which is fundamental to the design, simulation, and fabrication of electronics based on 2D material.

physics.app-ph↗

arXiv2020

An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order $α\rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $α\rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative, with the help of a class of discrete orthogonal convolution kernels. This is the first time such a \textit{discrete} transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme.

math.NA↗

arXiv2020

Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators

The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators. More precisely, for a class of discrete convolution kernels relevant to variable-step time discretizations,we show that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Our proof is based on an elementary constructing strategy using the properties of discrete orthogonal convolution kernels and complementary convolution kernels. To the best of our knowledge, this is the first general result on simple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using the unified theory, the stability for some simple non-uniform time-stepping schemes can be obtained in a straightforward way.

math.NA↗

arXiv2020

Understanding the Advisor-advisee Relationship via Scholarly Data Analysis

Advisor-advisee relationship is important in academic networks due to its universality and necessity. Despite the increasing desire to analyze the career of newcomers, however, the outcomes of different collaboration patterns between advisors and advisees remain unknown. The purpose of this paper is to find out the correlation between advisors' academic characteristics and advisees' academic performance in Computer Science. Employing both quantitative and qualitative analysis, we find that with the increase of advisors' academic age, advisees' performance experiences an initial growth, follows a sustaining stage, and finally ends up with a declining trend. We also discover the phenomenon that accomplished advisors can bring up skilled advisees. We explore the conclusion from two aspects: (1) Advisees mentored by advisors with high academic level have better academic performance than the rest; (2) Advisors with high academic level can raise their advisees' h-index ranking. This work provides new insights on promoting our understanding of the relationship between advisors' academic characteristics and advisees' performance, as well as on advisor choosing.

cs.DL↗

arXiv2020

An Augmented Regression Model for Tensors with Missing Values

Heterogeneous but complementary sources of data provide an unprecedented opportunity for developing accurate statistical models of systems. Although the existing methods have shown promising results, they are mostly applicable to situations where the system output is measured in its complete form. In reality, however, it may not be feasible to obtain the complete output measurement of a system, which results in observations that contain missing values. This paper introduces a general framework that integrates tensor regression with tensor completion and proposes an efficient optimization framework that alternates between two steps for parameter estimation. Through multiple simulations and a case study, we evaluate the performance of the proposed method. The results indicate the superiority of the proposed method in comparison to a benchmark.

stat.AP↗

arXiv2020

Stability analysis for the Implicit-Explicit discretization of the Cahn-Hilliard equation

Implicit-Explicit methods have been widely used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the effect of small dissipation coefficient, most existing theoretical analysis relies on adding additional stabilization terms, mollifying the nonlinearity or introducing auxiliary variables which implicitly either changes the structure of the problem or trades accuracy for stability in a subtle way. In this work we introduce a robust theoretical framework to analyze directly the stability of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and prove energy stability under natural time step constraints which are optimal with respect to energy scaling. These settle several questions which have been open since the work of Chen and Shen \cite{CS98}.

math.AP↗

arXiv2020

Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios $r_k:=τ_k/τ_{k-1}<3.561.$ Moreover, with a novel discrete orthogonal convolution kernels argument and some new discrete convolutional inequalities, the $L^2$ norm stability and rigorous error estimates are established, under the same step-ratios constraint that ensuring the energy stability., i.e., $0<r_k<3.561.$ This is known to be the best result in literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

math.NA↗

arXiv2020

How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations

There exists a well defined energy for classical phase-field equations under which the dissipation law is satisfied, i.e., the energy is non-increasing with respect to time. However, it is not clear how to extend the energy definition to time-fractional phase-field equations so that the corresponding dissipation law is still satisfied. In this work, we will try to settle this problem for phase-field equations with Caputo time-fractional derivative, by defining a nonlocal energy as an averaging of the classical energy with a time-dependent weight function. As the governing equation exhibits both nonlocal and nonlinear behavior, the dissipation analysis is challenging. To deal with this, we propose a new theorem on judging the positive definiteness of a symmetric function, that is derived from a special Cholesky decomposition. Then, the nonlocal energy is proved to be dissipative under a simple restriction of the weight function. Within the same framework, the time fractional derivative of classical energy for time-fractional phase-field models can be proved to be always nonpositive.

math.AP↗

arXiv2020

Revisit of Semi-Implicit Schemes for Phase-Field Equations

It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semi-implicit schemes for the Allen-Cahn equation with {\em general potential} function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang \& Yang (J. Comput. Math., 34(5):471--481, 2016) which studies the semi-implicit scheme for the Allen-Cahn equation with {\em polynomial potentials}.

math.NA↗

arXiv2020

Parallel Programming Models for Heterogeneous Many-Cores : A Survey

Heterogeneous many-cores are now an integral part of modern computing systems ranging from embedding systems to supercomputers. While heterogeneous many-core design offers the potential for energy-efficient high-performance, such potential can only be unlocked if the application programs are suitably parallel and can be made to match the underlying heterogeneous platform. In this article, we provide a comprehensive survey for parallel programming models for heterogeneous many-core architectures and review the compiling techniques of improving programmability and portability. We examine various software optimization techniques for minimizing the communicating overhead between heterogeneous computing devices. We provide a road map for a wide variety of different research areas. We conclude with a discussion on open issues in the area and potential research directions. This article provides both an accessible introduction to the fast-moving area of heterogeneous programming and a detailed bibliography of its main achievements.

cs.DC↗

arXiv2020

Optimizing Streaming Parallelism on Heterogeneous Many-Core Architectures: A Machine Learning Based Approach

This article presents an automatic approach to quickly derive a good solution for hardware resource partition and task granularity for task-based parallel applications on heterogeneous many-core architectures. Our approach employs a performance model to estimate the resulting performance of the target application under a given resource partition and task granularity configuration. The model is used as a utility to quickly search for a good configuration at runtime. Instead of hand-crafting an analytical model that requires expert insights into low-level hardware details, we employ machine learning techniques to automatically learn it. We achieve this by first learning a predictive model offline using training programs. The learnt model can then be used to predict the performance of any unseen program at runtime. We apply our approach to 39 representative parallel applications and evaluate it on two representative heterogeneous many-core platforms: a CPU-XeonPhi platform and a CPU-GPU platform. Compared to the single-stream version, our approach achieves, on average, a 1.6x and 1.1x speedup on the XeonPhi and the GPU platform, respectively. These results translate to over 93% of the performance delivered by a theoretically perfect predictor.

cs.DC↗

arXiv2020

On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation

In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=τ_k/τ_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

math.NA↗

arXiv2020

Efficient and High-quality Sparse Graph Coloring on the GPU

Graph coloring has been broadly used to discover concurrency in parallel computing. To speedup graph coloring for large-scale datasets, parallel algorithms have been proposed to leverage modern GPUs. Existing GPU implementations either have limited performance or yield unsatisfactory coloring quality (too many colors assigned). We present a work-efficient parallel graph coloring implementation on GPUs with good coloring quality. Our approach employs the speculative greedy scheme which inherently yields better quality than the method of finding maximal independent set. In order to achieve high performance on GPUs, we refine the algorithm to leverage efficient operators and alleviate conflicts. We also incorporate common optimization techniques to further improve performance. Our method is evaluated with both synthetic and real-world sparse graphs on the NVIDIA GPU. Experimental results show that our proposed implementation achieves averaged 4.1x (up to 8.9x) speedup over the serial implementation. It also outperforms the existing GPU implementation from the NVIDIA CUSPARSE library (2.2x average speedup), while yielding much better coloring quality than CUSPARSE.

cs.DC↗

arXiv2019

A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

In this work, we present a second-order nonuniform time-stepping scheme for the time-fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete maximum principle, and by using the convolution structure of consistency error, we present sharp maximum-norm error estimates which reflect the temporal regularity. As our analysis is built on nonuniform time steps, we may resolve the intrinsic initial singularity by using the graded meshes. Moreover, we propose an adaptive time-stepping strategy for large time simulations. Numerical experiments are presented to show the effectiveness of the proposed scheme. This seems to be the first second-order maximum principle preserving scheme for the time-fractional Allen-Cahn equation.

math.NA↗

arXiv2019

On energy dissipation theory and numerical stability for time-fractional phase field equations

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter $α$ reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and time-fractional molecular beam epitaxy model, while there exists a $-α/3$ power law coarsening stage.

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

Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach.

math.NA↗