We used discrete dipole approximation(DDA)to examine the scattering and absorption characteristics of spherical ice crystal particles.On this basis,we studied the scattering characteristics of spherical ice crystal pa...We used discrete dipole approximation(DDA)to examine the scattering and absorption characteristics of spherical ice crystal particles.On this basis,we studied the scattering characteristics of spherical ice crystal particles at different frequencies and non-spherical ice crystal particles with different shapes,aspect ratios,and spatial orientations.The results indicate that the DDA and Mie methods yield almost the same results for spherical ice crystal particles,illustrating the superior calculation accuracy of the DDA method.Compared with the millimeter wave band,the terahertz band particles have richer scattering characteristics and can detect ice crystal particles more easily.Different frequencies,shapes,aspect ratios,and spatial orientations have specific effects on the scattering and absorption characteristics o f ice crystal particles.The results provide an important theoretical basis for the design of terahertz cloud radars and related cirrus detection methods.展开更多
We show that the best L_p-approximant to continuous functions by n-convex functions is the limit of discrete n-convex approximations.The techniques of the proof are then used to show the existence of near interpolants...We show that the best L_p-approximant to continuous functions by n-convex functions is the limit of discrete n-convex approximations.The techniques of the proof are then used to show the existence of near interpolants to discrete n-convex data by continuous n-convex functions if the data points are close.展开更多
This paper investigates the stabilization issue for a class of sampled-data nonlinear Markov switching system with indistinguishable modes.In order to handle indistinguishable modes,the authors reconstruct the origina...This paper investigates the stabilization issue for a class of sampled-data nonlinear Markov switching system with indistinguishable modes.In order to handle indistinguishable modes,the authors reconstruct the original mode space by mode clustering method,forming a new merged Markov switching system.By specifying the difference between the Euler-Maruyama(EM)approximate discrete-time model of the merged system and the exact discrete-time model of the original Markov switching system,the authors prove that the sampled-data controller,designed for the merged system based on its EM approximation,can exponentially stabilize the original system in mean square sense.Finally,a numerical example is given to illustrate the effectiveness of the method.展开更多
This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the aut...This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.展开更多
Optimization problems with L^(1)-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l^(1)-regularization optimization,the resul...Optimization problems with L^(1)-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l^(1)-regularization optimization,the resulting discretized L^(1)norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem.A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L^(1)-norm.In this paper,a new discretized scheme for the L^(1)-norm is presented.Compared to the new discretized scheme for L^(1)-norm with the nodal quadrature formula,the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation.Moreover,finite element error estimates results for the primal problem with the new discretized scheme for the L^(1)-norm are provided,which confirms that this approximation scheme will not change the order of error estimates.To solve the new discretized problem,a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD)method is introduced to solve it via its dual.The proposed sGS-mABCD algorithm is illustrated at two numerical examples.Numerical results not only confirm the finite element error estimates,but also show that our proposed algorithm is efficient.展开更多
We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization...We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.展开更多
This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time ...This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time discretization is based on the backward Euler method.The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions.We derive the superconvergence properties of finite element solutions.By using the superconvergence results,we obtain recovery type a posteriori error estimates.Some numerical examples are presented to verify the theoretical results.展开更多
In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior lay...In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior layers appear in these problems.If classical discretisation methods are used, the solution of the finite differencescheme and the approximation of the diffusive flux do not converge uniformly withrespect to this parameter. Using the method of special, adapted grids, we canconstruct difference schemes that allow approximation of the solution and the nor-malised diffusive flux uniformly with respect to the small parameter.We also consider sillgularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly We study what problems ap-pear, when classical schemes are used for the approximation of the spatial deriva-tives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Diriclilet, Neumann and RDbin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions-展开更多
We use the method of discrete dipole approximation with surface interaction to construct a model in which a plurality of nanoparticles is arranged on the surface of BK7 glass. Nanoparticles are in air medium illuminat...We use the method of discrete dipole approximation with surface interaction to construct a model in which a plurality of nanoparticles is arranged on the surface of BK7 glass. Nanoparticles are in air medium illuminated by evanescent wave generated from total internal reflection. The effects of the wavelength, the polarization of the incident wave, the number of nanoparticles and the spacing of multiple nanoparticles on the field enhancement and extinction efficiency are calculated by our model. Our work could pave the way to improve the field enhancement of multiple nanoparticles systems.展开更多
Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize t...Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize the result of P.Gniadek.As an application,the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial.In addition,for the case of n≥3,the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points,such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.展开更多
In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior laye...In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior layers appear in theseproblems. If classical discretisation methods are used, the solution of the finitedifference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, edapted grids,we can construct difference schemes that allow apprcximation of the solution andthe normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions.展开更多
In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior lay...In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions展开更多
Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di- mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generaliz...Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di- mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U) ---- w(X) (where "w"is the topological weight) for each open nonempty subset U of X, then X itself i,~ homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W(X, *) ---- {(xn)=l C X : x~ = * for almost all n} is homeomorphic to a pre-Hilbert space E with E EE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.展开更多
Customary stochastic programming with recourse assumes that the probability distribution of random parameters is independent of decision variables.Recent studies demonstrated that stochastic programming models with en...Customary stochastic programming with recourse assumes that the probability distribution of random parameters is independent of decision variables.Recent studies demonstrated that stochastic programming models with endogenous uncertainty can better reflect many real-world activities and applications accompanying with decision-dependent uncertainty.In this paper,we concentrate on a class of decision-dependent two-stage stochastic programs(DTSPs)and investigate their discrete approximation.To develop the discrete approximation methods for DTSPs,we first derive the quantitative stability results for DTSPs.Based on the stability conclusion,we examine two discretization schemes when the support set of random variables is bounded,and give the rates of convergence for the optimal value and optimal solution set of the discrete approximation problem to those of the original problem.Then we extend the proposed approaches to the general situation with an unbounded support set by using the truncating technique.As an illustration of our discretization schemes,we reformulate the discretization problems under specific structures of the decision-dependent distribution.Finally,an application and numerical results are presented to demonstrate our theoretical results.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61527805 and 41776181).
文摘We used discrete dipole approximation(DDA)to examine the scattering and absorption characteristics of spherical ice crystal particles.On this basis,we studied the scattering characteristics of spherical ice crystal particles at different frequencies and non-spherical ice crystal particles with different shapes,aspect ratios,and spatial orientations.The results indicate that the DDA and Mie methods yield almost the same results for spherical ice crystal particles,illustrating the superior calculation accuracy of the DDA method.Compared with the millimeter wave band,the terahertz band particles have richer scattering characteristics and can detect ice crystal particles more easily.Different frequencies,shapes,aspect ratios,and spatial orientations have specific effects on the scattering and absorption characteristics o f ice crystal particles.The results provide an important theoretical basis for the design of terahertz cloud radars and related cirrus detection methods.
文摘We show that the best L_p-approximant to continuous functions by n-convex functions is the limit of discrete n-convex approximations.The techniques of the proof are then used to show the existence of near interpolants to discrete n-convex data by continuous n-convex functions if the data points are close.
基金supported by the National Key Research and Development Program of China under Grant Nos.2018AAA0100800 and 2018YFE0106800the National Natural Science Foundation of China under Grant Nos.61725304 and 61673361the Science and Technology Major Project of Anhui Province under Grant No.912198698036。
文摘This paper investigates the stabilization issue for a class of sampled-data nonlinear Markov switching system with indistinguishable modes.In order to handle indistinguishable modes,the authors reconstruct the original mode space by mode clustering method,forming a new merged Markov switching system.By specifying the difference between the Euler-Maruyama(EM)approximate discrete-time model of the merged system and the exact discrete-time model of the original Markov switching system,the authors prove that the sampled-data controller,designed for the merged system based on its EM approximation,can exponentially stabilize the original system in mean square sense.Finally,a numerical example is given to illustrate the effectiveness of the method.
基金supported by the National Natural Science Foundation of China under Grant Nos.11171133 and 11271156
文摘This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.
文摘Optimization problems with L^(1)-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l^(1)-regularization optimization,the resulting discretized L^(1)norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem.A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L^(1)-norm.In this paper,a new discretized scheme for the L^(1)-norm is presented.Compared to the new discretized scheme for L^(1)-norm with the nodal quadrature formula,the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation.Moreover,finite element error estimates results for the primal problem with the new discretized scheme for the L^(1)-norm are provided,which confirms that this approximation scheme will not change the order of error estimates.To solve the new discretized problem,a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD)method is introduced to solve it via its dual.The proposed sGS-mABCD algorithm is illustrated at two numerical examples.Numerical results not only confirm the finite element error estimates,but also show that our proposed algorithm is efficient.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11271145), the Foundation for Talent Introduction of Guangdong Provincial University, the Specialized Research Fund for the Doctoral Program of Higher Education (20114407110009), and the Project of Department of Education of Guangdong Province (2012KJCX0036).
文摘We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.
基金supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China(10971074)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(20114407110009)Hunan Provinical Innovation Foundation for Postgraduate(lx2009 B120)。
文摘This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time discretization is based on the backward Euler method.The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions.We derive the superconvergence properties of finite element solutions.By using the superconvergence results,we obtain recovery type a posteriori error estimates.Some numerical examples are presented to verify the theoretical results.
文摘In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior layers appear in these problems.If classical discretisation methods are used, the solution of the finite differencescheme and the approximation of the diffusive flux do not converge uniformly withrespect to this parameter. Using the method of special, adapted grids, we canconstruct difference schemes that allow approximation of the solution and the nor-malised diffusive flux uniformly with respect to the small parameter.We also consider sillgularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly We study what problems ap-pear, when classical schemes are used for the approximation of the spatial deriva-tives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Diriclilet, Neumann and RDbin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions-
基金supported by the Zhejiang Provincial Natural Science Foundation of China(No.LGF20C050001)the National Nature Science Foundation of China(No.61805213)。
文摘We use the method of discrete dipole approximation with surface interaction to construct a model in which a plurality of nanoparticles is arranged on the surface of BK7 glass. Nanoparticles are in air medium illuminated by evanescent wave generated from total internal reflection. The effects of the wavelength, the polarization of the incident wave, the number of nanoparticles and the spacing of multiple nanoparticles on the field enhancement and extinction efficiency are calculated by our model. Our work could pave the way to improve the field enhancement of multiple nanoparticles systems.
基金supported by the National Natural Science Foundation of China under Grant Nos.11901402 and 11671169。
文摘Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize the result of P.Gniadek.As an application,the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial.In addition,for the case of n≥3,the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points,such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.
文摘In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior layers appear in theseproblems. If classical discretisation methods are used, the solution of the finitedifference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, edapted grids,we can construct difference schemes that allow apprcximation of the solution andthe normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions.
文摘In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions
文摘Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di- mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U) ---- w(X) (where "w"is the topological weight) for each open nonempty subset U of X, then X itself i,~ homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W(X, *) ---- {(xn)=l C X : x~ = * for almost all n} is homeomorphic to a pre-Hilbert space E with E EE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.
文摘Customary stochastic programming with recourse assumes that the probability distribution of random parameters is independent of decision variables.Recent studies demonstrated that stochastic programming models with endogenous uncertainty can better reflect many real-world activities and applications accompanying with decision-dependent uncertainty.In this paper,we concentrate on a class of decision-dependent two-stage stochastic programs(DTSPs)and investigate their discrete approximation.To develop the discrete approximation methods for DTSPs,we first derive the quantitative stability results for DTSPs.Based on the stability conclusion,we examine two discretization schemes when the support set of random variables is bounded,and give the rates of convergence for the optimal value and optimal solution set of the discrete approximation problem to those of the original problem.Then we extend the proposed approaches to the general situation with an unbounded support set by using the truncating technique.As an illustration of our discretization schemes,we reformulate the discretization problems under specific structures of the decision-dependent distribution.Finally,an application and numerical results are presented to demonstrate our theoretical results.