In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using ...In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.展开更多
The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based...The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented. To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1 functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance.展开更多
Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This...Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This scheme takes into ac- count the periodicity of the analytic solution by using derivatives of a periodic quasi-interpolant (multiquadric trigonometric B-spline quasi-interpolant) to approximate the spatial derivatives of the equations. Thus, it overcomes the difficulties of the previous schemes based on quasi-interpolation (requiring some additional boundary conditions and yielding unwanted high-order discontinuous points at the boundaries in the spatial domain). Moreover, the scheme also overcomes the dif- ficulty of the meshless collocation methods (i.e., yielding a notorious ill-conditioned linear system of equations for large collocation points). The numerical examples that are presented at the end of the paper show that the scheme provides excellent approximations to the analytic solutions.展开更多
In this paper, we use a kind of univariate multiquadric quasi-interpolation to solve a parabolic equation with overspecified data, which has arisen in many physical phenomena. We obtain the numerical scheme by using t...In this paper, we use a kind of univariate multiquadric quasi-interpolation to solve a parabolic equation with overspecified data, which has arisen in many physical phenomena. We obtain the numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a simple forward difference to approximate the temporal derivative of the dependent variable. The advantage of the presented scheme is that the algorithm is very simple so it is very easy to implement. The results of the numerical experiment are presented and are compared with the exact solution to confirm the good accuracy of the presented scheme.展开更多
Estimating the global state of a networked system is an important problem in many application domains.The classical approach to tackling this problem is the periodic(observation)method,which is inefficient because it ...Estimating the global state of a networked system is an important problem in many application domains.The classical approach to tackling this problem is the periodic(observation)method,which is inefficient because it often observes states at a very high frequency.This inefficiency has motivated the idea of event-based method,which leverages the evolution dynamics in question and makes observations only when some rules are triggered(i.e.,only when certain conditions hold).This paper initiates the investigation of using the event-based method to estimate the equilibrium in the new application domain of cybersecurity,where equilibrium is an important metric that has no closed-form solutions.More specifically,the paper presents an event-based method for estimating cybersecurity equilibrium in the preventive and reactive cyber defense dynamics,which has been proven globally convergent.The presented study proves that the estimated equilibrium from our trigger rule i)indeed converges to the equilibrium of the dynamics and ii)is Zeno-free,which assures the usefulness of the event-based method.Numerical examples show that the event-based method can reduce 98%of the observation cost incurred by the periodic method.In order to use the event-based method in practice,this paper investigates how to bridge the gap between i)the continuous state in the dynamics model,which is dubbed probability-state because it measures the probability that a node is in the secure or compromised state,and ii)the discrete state that is often encountered in practice,dubbed sample-state because it is sampled from some nodes.This bridge may be of independent value because probability-state models have been widely used to approximate exponentially-many discrete state systems.展开更多
The aim of this survey paper is to propose a new concept "generator". In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than ba...The aim of this survey paper is to propose a new concept "generator". In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.展开更多
This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM met...This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM method for wave equations,however,by transforming the system into a first order hyperbolic system,the HUM method for 1-D first order hyperbolic systems,established by Li-Lu(2022)and Lu-Li(2022),can be applied to get the corresponding results.展开更多
In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact inter...In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.展开更多
In this paper, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced couple...In this paper, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.展开更多
In this paper,the authors define the strong (weak) exact boundary controllability and the strong (weak) exact boundary observability for first order quasilinear hyperbolic systems,and study their properties and the re...In this paper,the authors define the strong (weak) exact boundary controllability and the strong (weak) exact boundary observability for first order quasilinear hyperbolic systems,and study their properties and the relationship between them.展开更多
In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups....In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the synchronization by 3-groups, respectively.展开更多
This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce ...This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next,in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.展开更多
In this survey paper, the synchronization will be initially studied for infinite dimensional dynamical systems of partial differential equations instead of finite dimensional systems of ordinary differential equations...In this survey paper, the synchronization will be initially studied for infinite dimensional dynamical systems of partial differential equations instead of finite dimensional systems of ordinary differential equations,and will be connected with the control theory via boundary controls in a finite time interval. More precisely,various kinds of exact boundary synchronization and approximate boundary synchronization will be introduced and realized by means of fewer boundary controls for a coupled system of wave equations with Dirichlet boundary controls. Moreover, as necessary conditions for various kinds of approximate boundary synchronization, criteria of Kalman's type are obtained. Finally, some prospects will be given.展开更多
We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the ...We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.展开更多
We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here ...We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here κ > 0 does not depend on the initial data.This is achieved by proving the regularity of solutions under■ This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ > 1.展开更多
In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally ...In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary(null) controllability and the exact boundary observability for first order hyperbolic systems.展开更多
Background:COVID-19 has been impacting on the whole world critically and constantly since late December 2019.Rapidly increasing infections has raised intense worldwide attention.How to model the evolution of COVID-19 ...Background:COVID-19 has been impacting on the whole world critically and constantly since late December 2019.Rapidly increasing infections has raised intense worldwide attention.How to model the evolution of COVID-19 effectively and efficiently is of great significance for prevention and control.Methods:We propose the multi-chain Fudan-CCDC model based on the original single-chain model in[Shao et al.2020]to describe the evolution of COVID-19 in Singapore.Multi-chains can be considered as the superposition of several single chains with different characteristics.We identify the parameters of models by minimizing the penalty function.Results:The numerical simulation results exhibit the multi-chain model performs well on data fitting.Though unsteady the increments are,they could still fall within the range of±30%fluctuation from simulation results.Conclusion:The multi-chain Fudan-CCDC model provides an effective way to early detect the appearance of imported infectors and super spreaders and forecast a second outbreak.It can also explain the data from those countries where the single-chain model shows deviation from the data.展开更多
Making use of the periodic unfolding method,the authors give an elementary proof for the periodic homogenization of the elastic torsion problem of an infinite 3dimensional rod with a multiply-connected cross section a...Making use of the periodic unfolding method,the authors give an elementary proof for the periodic homogenization of the elastic torsion problem of an infinite 3dimensional rod with a multiply-connected cross section as well as for the general electroconductivity problem in the presence of many perfect conductors(arising in resistivity well-logging).Both problems fall into the general setting of equi-valued surfaces with corresponding assigned total fluxes.The unfolding method also gives a general corrector result for these problems.展开更多
The l1 norm is the tight convex relaxation for the l0 norm and has been successfully applied for recovering sparse signals.However,for problems with fewer samples than required for accurate l1 recovery,one needs to ap...The l1 norm is the tight convex relaxation for the l0 norm and has been successfully applied for recovering sparse signals.However,for problems with fewer samples than required for accurate l1 recovery,one needs to apply nonconvex penalties such as lp norm.As one method for solving lp minimization problems,iteratively reweighted l1 minimization updates the weight for each component based on the value of the same component at the previous iteration.It assigns large weights on small components in magnitude and small weights on large components in magnitude.The set of the weights is not fixed,and it makes the analysis of this method difficult.In this paper,we consider a weighted l1 penalty with the set of the weights fixed,and the weights are assigned based on the sort of all the components in magnitude.The smallest weight is assigned to the largest component in magnitude.This new penalty is called nonconvex sorted l1.Then we propose two methods for solving nonconvex sorted l1 minimization problems:iteratively reweighted l1 minimization and iterative sorted thresholding,and prove that both methods will converge to a local minimizer of the nonconvex sorted l1 minimization problems.We also show that both methods are generalizations of iterative support detection and iterative hard thresholding,respectively.The numerical experiments demonstrate the better performance of assigning weights by sort compared to assigning by value.展开更多
This article concerns the weak Galerkin mixed finite element method(WGMFEM)for second order elliptic equations on 2D domains with curved boundary.The Neumann boundary condition is considered since it becomes the essen...This article concerns the weak Galerkin mixed finite element method(WGMFEM)for second order elliptic equations on 2D domains with curved boundary.The Neumann boundary condition is considered since it becomes the essential boundary condition in this case.It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretizationwith polynomial order a>1.The purpose of this paper is two-fold.First,we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains,which exhibits an O(h^(1/2))convergence for all a≥1.It is a little surprising to see that even the lowest-order WG-MFEM(a=1)experiences a loss of accuracy.This is different from known results for the finite element method(FEM)or the mixed FEM,and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain.Second,we propose a remedy to bring the approximation rate back to optimal by employing two techniques.One is a specially designed boundary correction technique.The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes,which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements.Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all a.Numerical results further confirm this conclusion.展开更多
基金supported by the State Key Development Program for Basic Research of China (Grant No 2006CB303102)Science and Technology Commission of Shanghai Municipality,China (Grant No 09DZ2272900)
文摘In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.
基金Supported by 973-Project of China(2006cb303102)the National Science Foundation of China(11461161006,11201079)
文摘The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented. To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1 functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance.
基金supported by the Shanghai Guidance of Science and Technology,China(Grant No.12DZ2272800)the Natural Science Foundation of Education Department of Anhui Province,China(Grant No.KJ2013B203)the Foundation of Introducing Leaders of Science and Technology of Anhui University,China(Grant No.J10117700057)
文摘Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This scheme takes into ac- count the periodicity of the analytic solution by using derivatives of a periodic quasi-interpolant (multiquadric trigonometric B-spline quasi-interpolant) to approximate the spatial derivatives of the equations. Thus, it overcomes the difficulties of the previous schemes based on quasi-interpolation (requiring some additional boundary conditions and yielding unwanted high-order discontinuous points at the boundaries in the spatial domain). Moreover, the scheme also overcomes the dif- ficulty of the meshless collocation methods (i.e., yielding a notorious ill-conditioned linear system of equations for large collocation points). The numerical examples that are presented at the end of the paper show that the scheme provides excellent approximations to the analytic solutions.
基金Project supported by the National Key Basic Research Program of China (Grant No. 2006CB303102)Science and Technology Commission of Shanghai Municipality, China (Grant No. 09DZ2272900)
文摘In this paper, we use a kind of univariate multiquadric quasi-interpolation to solve a parabolic equation with overspecified data, which has arisen in many physical phenomena. We obtain the numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a simple forward difference to approximate the temporal derivative of the dependent variable. The advantage of the presented scheme is that the algorithm is very simple so it is very easy to implement. The results of the numerical experiment are presented and are compared with the exact solution to confirm the good accuracy of the presented scheme.
基金supported in part by the National Natural Sciences Foundation of China(62072111)。
文摘Estimating the global state of a networked system is an important problem in many application domains.The classical approach to tackling this problem is the periodic(observation)method,which is inefficient because it often observes states at a very high frequency.This inefficiency has motivated the idea of event-based method,which leverages the evolution dynamics in question and makes observations only when some rules are triggered(i.e.,only when certain conditions hold).This paper initiates the investigation of using the event-based method to estimate the equilibrium in the new application domain of cybersecurity,where equilibrium is an important metric that has no closed-form solutions.More specifically,the paper presents an event-based method for estimating cybersecurity equilibrium in the preventive and reactive cyber defense dynamics,which has been proven globally convergent.The presented study proves that the estimated equilibrium from our trigger rule i)indeed converges to the equilibrium of the dynamics and ii)is Zeno-free,which assures the usefulness of the event-based method.Numerical examples show that the event-based method can reduce 98%of the observation cost incurred by the periodic method.In order to use the event-based method in practice,this paper investigates how to bridge the gap between i)the continuous state in the dynamics model,which is dubbed probability-state because it measures the probability that a node is in the secure or compromised state,and ii)the discrete state that is often encountered in practice,dubbed sample-state because it is sampled from some nodes.This bridge may be of independent value because probability-state models have been widely used to approximate exponentially-many discrete state systems.
基金Supported by the 973program-2006CB303102SGST 09DZ 2272900NSFC No.11026089
文摘The aim of this survey paper is to propose a new concept "generator". In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.
基金supported by the National Natural Science Foundation of China (Nos. 11831011,11901082)the Natural Science Foundation of Jiangsu Province (No. BK20190323)the Fundamental Research Funds for the Central Universities of China
文摘This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM method for wave equations,however,by transforming the system into a first order hyperbolic system,the HUM method for 1-D first order hyperbolic systems,established by Li-Lu(2022)and Lu-Li(2022),can be applied to get the corresponding results.
基金supported by the National Natural Science Foundation of China(No.11831011)。
文摘In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.
文摘In this paper, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.
基金supported by the Basic Research Program of China(No. 2007CB814800)
文摘In this paper,the authors define the strong (weak) exact boundary controllability and the strong (weak) exact boundary observability for first order quasilinear hyperbolic systems,and study their properties and the relationship between them.
基金supported by the National Natural Science Foundation of China(No.11121101)the National Basic Research Program of China(No.2013CB834100)
文摘In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the synchronization by 3-groups, respectively.
基金supported by the National Basic Research Program of China(No.2013CB834100)the National Natural Science Foundation of China(No.11121101)
文摘This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next,in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.
基金supported by National Basic Research Program of China(Grant No.2013CB834100)National Natural Science Foundation of China(Grant No.111211101)
文摘In this survey paper, the synchronization will be initially studied for infinite dimensional dynamical systems of partial differential equations instead of finite dimensional systems of ordinary differential equations,and will be connected with the control theory via boundary controls in a finite time interval. More precisely,various kinds of exact boundary synchronization and approximate boundary synchronization will be introduced and realized by means of fewer boundary controls for a coupled system of wave equations with Dirichlet boundary controls. Moreover, as necessary conditions for various kinds of approximate boundary synchronization, criteria of Kalman's type are obtained. Finally, some prospects will be given.
基金partially supported by National Natural Science Foundation of China (Grant No.11001058)Specialized Research Fund for the Doctoral Program of Higher Educationthe Fundamental Research Funds for the Central Universities
文摘We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.
基金supported by National Natural Science Foundation of China (Grant No. 11725102)National Support Program for Young Top-Notch Talents+3 种基金SGST 09DZ2272900 from Shanghai Key Laboratory for Contemporary Applied Mathematicssupported by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants (Grant Nos. CUHK-14305315, CUHK-14300917 and CUHK-14302917)NSFC/RGC Joint Research Scheme Grant (Grant No. N-CUHK 443-14)a Focus Area Grant from the Chinese University of Hong Kong
文摘We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here κ > 0 does not depend on the initial data.This is achieved by proving the regularity of solutions under■ This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ > 1.
基金supported by the National Natural Science Foundation of China(Nos.11831011,11901082)the Natural Science Foundation of Jiangsu Province(No.BK20190323)the Fundamental Research Funds for the Central Universities of China。
文摘In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary(null) controllability and the exact boundary observability for first order hyperbolic systems.
基金the National Natural Science Foundation of China(No.11671098)partially supported by Shanghai Science and technology research program(No.19JC1420101)+1 种基金J.C.is supported in part by the National Natural Science Foundation of China(No.11971121)Y.Y.is supported by Shanghai Sailing Program(No.20YF1412400).
文摘Background:COVID-19 has been impacting on the whole world critically and constantly since late December 2019.Rapidly increasing infections has raised intense worldwide attention.How to model the evolution of COVID-19 effectively and efficiently is of great significance for prevention and control.Methods:We propose the multi-chain Fudan-CCDC model based on the original single-chain model in[Shao et al.2020]to describe the evolution of COVID-19 in Singapore.Multi-chains can be considered as the superposition of several single chains with different characteristics.We identify the parameters of models by minimizing the penalty function.Results:The numerical simulation results exhibit the multi-chain model performs well on data fitting.Though unsteady the increments are,they could still fall within the range of±30%fluctuation from simulation results.Conclusion:The multi-chain Fudan-CCDC model provides an effective way to early detect the appearance of imported infectors and super spreaders and forecast a second outbreak.It can also explain the data from those countries where the single-chain model shows deviation from the data.
基金Supported by the National Natural Science Foundation of China (No. 11121101)the National Basic Research Program of China (No. 2013CB834100)
文摘Making use of the periodic unfolding method,the authors give an elementary proof for the periodic homogenization of the elastic torsion problem of an infinite 3dimensional rod with a multiply-connected cross section as well as for the general electroconductivity problem in the presence of many perfect conductors(arising in resistivity well-logging).Both problems fall into the general setting of equi-valued surfaces with corresponding assigned total fluxes.The unfolding method also gives a general corrector result for these problems.
基金This work is partially supported by European Research Council,the National Natural Science Foundation of China(No.11201079)the Fundamental Research Funds for the Central Universities of China(Nos.20520133238 and 20520131169)the National Natural Science Foundation of United States(Nos.DMS-0748839 and DMS-1317602).
文摘The l1 norm is the tight convex relaxation for the l0 norm and has been successfully applied for recovering sparse signals.However,for problems with fewer samples than required for accurate l1 recovery,one needs to apply nonconvex penalties such as lp norm.As one method for solving lp minimization problems,iteratively reweighted l1 minimization updates the weight for each component based on the value of the same component at the previous iteration.It assigns large weights on small components in magnitude and small weights on large components in magnitude.The set of the weights is not fixed,and it makes the analysis of this method difficult.In this paper,we consider a weighted l1 penalty with the set of the weights fixed,and the weights are assigned based on the sort of all the components in magnitude.The smallest weight is assigned to the largest component in magnitude.This new penalty is called nonconvex sorted l1.Then we propose two methods for solving nonconvex sorted l1 minimization problems:iteratively reweighted l1 minimization and iterative sorted thresholding,and prove that both methods will converge to a local minimizer of the nonconvex sorted l1 minimization problems.We also show that both methods are generalizations of iterative support detection and iterative hard thresholding,respectively.The numerical experiments demonstrate the better performance of assigning weights by sort compared to assigning by value.
基金supported by Postgraduate Research&Practice Innovation Program of Jiangsu Province 1812000024768supported by the NSFC grant 12171244.Chen is supported by NSFC grant 12071090.
文摘This article concerns the weak Galerkin mixed finite element method(WGMFEM)for second order elliptic equations on 2D domains with curved boundary.The Neumann boundary condition is considered since it becomes the essential boundary condition in this case.It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretizationwith polynomial order a>1.The purpose of this paper is two-fold.First,we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains,which exhibits an O(h^(1/2))convergence for all a≥1.It is a little surprising to see that even the lowest-order WG-MFEM(a=1)experiences a loss of accuracy.This is different from known results for the finite element method(FEM)or the mixed FEM,and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain.Second,we propose a remedy to bring the approximation rate back to optimal by employing two techniques.One is a specially designed boundary correction technique.The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes,which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements.Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all a.Numerical results further confirm this conclusion.