We study the long-time asymptotic behaviour of viscosity solutions u(x,t)of the Hamilton-Jacobi equation u_(t)(x,t)+H(x,u(x,t),Du(x,t))=0 in T^(n)×(0,∞)with a PDE approach,where H=H(x,u,p)is coercive in p,non-de...We study the long-time asymptotic behaviour of viscosity solutions u(x,t)of the Hamilton-Jacobi equation u_(t)(x,t)+H(x,u(x,t),Du(x,t))=0 in T^(n)×(0,∞)with a PDE approach,where H=H(x,u,p)is coercive in p,non-decreasing in u and strictly convex in(u,p),and establish the uniform convergence of u(x,t)to an asymptotic solution u_(∞)(x)as t→∞.Moreover,u_(∞) is a viscosity solution of Hamilton-Jacobi equation H(x,u(x),Du(x))=0.展开更多
We present a parallel numerical method of simulating the interaction of atoms with a strong laser field by solving the time-depending Schr?dinger equation(TDSE) in spherical coordinates. This method is realized by com...We present a parallel numerical method of simulating the interaction of atoms with a strong laser field by solving the time-depending Schr?dinger equation(TDSE) in spherical coordinates. This method is realized by combining constructing block diagonal matrices through using the real space product formula(RSPF) with splitting out diagonal sub-matrices for short iterative Lanczos(SIL) propagator. The numerical implementation of the solver guarantees efficient parallel computing for the simulation of real physical problems such as high harmonic generation(HHG) in these interaction systems.展开更多
In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for U<sub>t&l...In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for U<sub>t</sub>. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.展开更多
Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational...Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Utilizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.展开更多
基金the National Natural Science Foundation of China(11971344)Jiangsu Graduate Science Innovation Project(KYCX20-2746)。
文摘We study the long-time asymptotic behaviour of viscosity solutions u(x,t)of the Hamilton-Jacobi equation u_(t)(x,t)+H(x,u(x,t),Du(x,t))=0 in T^(n)×(0,∞)with a PDE approach,where H=H(x,u,p)is coercive in p,non-decreasing in u and strictly convex in(u,p),and establish the uniform convergence of u(x,t)to an asymptotic solution u_(∞)(x)as t→∞.Moreover,u_(∞) is a viscosity solution of Hamilton-Jacobi equation H(x,u(x),Du(x))=0.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11534004,11627807,11774131,and 11774130)the Scientific and Technological Project of Jilin Provincial Education Department in the Thirteenth Five-Year Plan,China(Grant No.JJKH20170538KJ)
文摘We present a parallel numerical method of simulating the interaction of atoms with a strong laser field by solving the time-depending Schr?dinger equation(TDSE) in spherical coordinates. This method is realized by combining constructing block diagonal matrices through using the real space product formula(RSPF) with splitting out diagonal sub-matrices for short iterative Lanczos(SIL) propagator. The numerical implementation of the solver guarantees efficient parallel computing for the simulation of real physical problems such as high harmonic generation(HHG) in these interaction systems.
文摘In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for U<sub>t</sub>. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.
基金supported by the National Key R&D Program of China under Grant No.2021ZD0110400.
文摘Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Utilizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.
