Phase space approach for optimizing grid representations: The mapped Fourier method

Fattal, Eyal; Baer, Roi; Kosloff, Ronnie

doi:10.1103/physreve.53.1217

Cited by 109 publications

(101 citation statements)

References 17 publications

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“…t = 2, l = 0) and the spaces V K,T with T > s 2 the resulting approximation order is dependent on T and dependent on the number of particles N whereas for T ≤ s 2 the resulting order is independent of T and N . Here, for T ∈ (0, s 2 ] the dimension of V K,T according to (24) is independent of N . If we restrict the class of functions for example to H 1,1 mix (i.e.…”

Section: Optimized Sparse Grid Spacesmentioning

confidence: 99%

Sparse grids for the Schrödinger equation

Griebel¹,

Hamaekers²

2007

ESAIM: M2AN

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Abstract. We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system. Mathematics Subject Classification. 35J10, 65N25, 65N30, 65T40, 65Z05.

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Section: Optimized Sparse Grid Spacesmentioning

confidence: 99%

Sparse grids for the Schrödinger equation

Griebel¹,

Hamaekers²

2007

ESAIM: M2AN

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“…In fact, pseudospectral global grid representation approaches are difficult to use in multiscale problems. This is why much of work has been devoted recently to the development of the mapping procedures in order to enhance sampling efficiency in the regions of the rapid variation of the wave function [8,11,12,13,14,15]. Though mapping procedure, based on the variable change x = f (ξ) is very efficient in 1D, it is far from being universal.…”

Section: Introductionmentioning

confidence: 99%

“…Development of the pseudospectral global grid representation approaches yielded a very efficient way to tackle this problem. The discrete variable representation (DVR) [5] and the Fourier grid Hamiltonian method (FGH) [6,7] have been widely used in time-dependent molecular dynamics [1,2,8], S-matrix [9,10] or eigenvalue [11] calculations. The FGH method based on the Fast Fourier Transform (FFT) algorithm is especially popular.…”

Section: Introductionmentioning

confidence: 99%

“…In the standard FGH method the wave function is represented on a grid of equidistant point in coordinate and momentum space. It is well suited when the the local de Broglie wavelength does not change much over the physical region spanned by the wave function of the system [11]. There are, however, lot of examples where the system under consideration spans the physical regions with very different properties.…”

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confidence: 99%

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The wave packet propagation using wavelets

Borisov

Shabanov

2002

Chemical Physics Letters

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It is demonstrated that the wavelets can be used to considerably speed up simulations of the wave packet propagation in multiscale systems. Extremely high efficiency is obtained in the representation of both bound and continuum states. The new method is compared with the fast Fourier algorithm. Depending on ratios of typical scales of a quantum system in question, the wavelet method appears to be faster by a few orders of magnitude.1. Owing to the fast development of the computational tools the direct solution of the time-dependent Schroedinger equation has become one of the basic approaches to study the evolution of quantum systems. Thus, the wave packet propagation (WPP) method is successfully applied to time dependent and time-independent problems in gas-surface interactions, molecular and atomic physics, and quantum chemistry [1,2,3,4]. One of the main issues of the numerical approaches to the time-dependent Schroedinger equation is the representation of the wave function |Ψ(t) of the system. Development of the pseudospectral global grid representation approaches yielded a very efficient way to tackle this problem. The discrete variable representation (DVR) [5] and the Fourier grid Hamiltonian method (FGH) [6,7] have been widely used in time-dependent molecular dynamics [1,2,8], S-matrix [9,10] or eigenvalue [11] calculations. The FGH method based on the Fast Fourier Transform (FFT) algorithm is especially popular. This is because for the mesh of N points the evaluation of the kinetic energy operator requires only NlogN operations and can be easily implemented numerically. In the standard FGH method the wave function is represented on a grid of equidistant point in coordinate and momentum space. It is well suited when the the local de Broglie wavelength does not change much over the physical region spanned by the wave function of the system [11]. There are, however, lot of examples where the system under consideration spans the physical regions with very different properties. Consider, for example, the scattering of a slow particle on a narrow and deep potential well. Despite the short wave lengths occur only in the well, in the FGH method they would determine the lattice mesh over entire physical space leading to high computational cost. In fact, pseudospectral global grid representation approaches are difficult to use in multiscale problems. This is why much of work has been devoted recently to the development of the mapping procedures in order to enhance sampling efficiency in the regions of the rapid variation of the wave function [8,11,12,13,14,15]. Though mapping procedure, based on the variable change x = f (ξ) is very efficient in 1D, it is far from being universal. One obvious drawback 1 is that it is difficult to implement in higher dimensions. In the case of several variables, there is no simple procedure to define the mapping functions f so that the lattice would be fine only in some designated regions of space. Next, the topology of the new coordinate surfaces can be different from that ...

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“…First, we test a novel time-domain algorithm for the Maxwell's theory of linear, passive (dispersive and absorbing) media. The algorithm is based on (i) the Hamiltonian formalism for evolution differential equations [1], on (ii) the time leapfrog scheme [2], and on (iii) the Fourier pseudospectral method [3] in combination with a change of variables that enhances the spatial grid resolution in designated domains (in the vicinity of medium interfaces) and, thereby, prevents the loss of accuracy due to the aliasing problem of the Fourier transform, while keeping the total spatial grid size fixed [4], [5]. Boundary conditions at medium interfaces are not fixed in the algorithm, but rather medium parameters are allowed to have spatial discontinuities so that the correct boundary conditions are enforced dynamically [6], similarly to the wave packet method for quantum mechanical systems with discontinuous potentials.…”

Section: Introductionmentioning

confidence: 99%

Applications of the wave packet method to resonant transmission and reflection gratings

Borisov

Shabanov

2004

Journal of Computational Physics

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Scattering of femtosecond laser pulses on resonant transmission and reflection gratings made of dispersive (Drude metals) and dielectric materials is studied by a time-domain numerical algorithm for Maxwell's theory of linear passive (dispersive and absorbing) media. The algorithm is based on the Hamiltonian formalism in the framework of which Maxwell's equations for passive media are shown to be equivalent to the first-order equation, ∂Ψ/∂t = HΨ, where H is a linear differential operator (Hamiltonian) acting on a multi-dimensional vector Ψ built of the electromagnetic inductions and auxiliary matter fields describing the medium response. The initial value problem is then solved by means of a modified time leapfrog method in combination with the Fourier pseudospectral method applied on a non-uniform grid that is constructed by a change of variables and designed to enhance the sampling efficiency near medium interfaces. The algorithm is shown to be highly accurate at relatively low computational costs. An excellent agreement with previous theoretical and experimental studies of the gratings is demonstrated by numerical simulations using our algorithm. In addition, our algorithm allows one to see real time dynamics of long leaving resonant excitations of electromagnetic fields in the gratings in the entire frequency range of the initial wide band wave packet as well as formation of the reflected and transmitted wave fronts.

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Phase space approach for optimizing grid representations: The mapped Fourier method

Cited by 109 publications

References 17 publications

Sparse grids for the Schrödinger equation

Sparse grids for the Schrödinger equation

The wave packet propagation using wavelets

Applications of the wave packet method to resonant transmission and reflection gratings

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