A fast solution method for time dependent multidimensional Schrödinger equations

Lanzara, Flavia; Maz′ya, Vladimir; Schmidt, G.

doi:10.1080/00036811.2017.1359571

Cited by 6 publications

(3 citation statements)

References 15 publications

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“…In the next tables we consider w(x) = (x 2 − 1) 2 e x for x ∈ [−1, 1]; w(x) = 0 otherwise, the extension w(x) = w(x) (other extensions can also be considered, see e.g. [10]), D = 3 and the parameters in the quadrature rule τ = 10 −6 , a = 6, b = 4 in (4.16).…”

Section: Separated Representation and Numerical Resultsmentioning

confidence: 99%

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

Lanzara,

Maz'ya,

Schmidt

2018

Preprint

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We propose a fast method for high order approximation of potentials of the Helmholtz type operator ∆ + κ 2 over hyper-rectangles in R n . By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of onedimensional integrals with separable integrands. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Numerical tests show that these formulas are accurate and provide approximations of order 6 up to dimension 100 and κ 2 = 100.

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Section: Separated Representation and Numerical Resultsmentioning

confidence: 99%

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

Lanzara,

Maz'ya,

Schmidt

2018

Preprint

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“…High dimensionality, sometimes involving millions or billions of dimensions, is usually an outcome of mathematical modeling of complex systems. For example, high-dimensional spaces in quantum physics result from solving the multiparticle Schrödinger equation [7,31], where every particle adds to the dimension of the problem. Analogously in the study of dynamical systems [15,17,81], the evolution of an ensemble of degrees of freedom is studied in the system's phase space.…”

Section: Introductionmentioning

confidence: 99%

GNN-Assisted Phase Space Integration with Application to Atomistics

Saxena¹,

Bastek²,

Miguel³

et al. 2023

Preprint

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Overcoming the time scale limitations of atomistics can be achieved by switching from the statespace representation of Molecular Dynamics (MD) to a statistical-mechanics-based representation in phase space, where approximations such as maximum-entropy or Gaussian phase packets (GPP) evolve the atomistic ensemble in a time-coarsened fashion. In practice, this requires the computation of expensive high-dimensional integrals over all of phase space of an atomistic ensemble. This, in turn, is commonly accomplished efficiently by low-order numerical quadrature. We show that numerical quadrature in this context, unfortunately, comes with a set of inherent problems, which corrupt the accuracy of simulations-especially when dealing with crystal lattices with imperfections. As a remedy, we demonstrate that Graph Neural Networks, trained on Monte-Carlo data, can serve as a replacement for commonly used numerical quadrature rules, overcoming their deficiencies and significantly improving the accuracy. This is showcased by three benchmarks: the thermal expansion of copper, the martensitic phase transition of iron, and the energy of grain boundaries. We illustrate the benefits of the proposed technique over classically used third-and fifth-order Gaussian quadrature, we highlight the impact on time-coarsened atomistic predictions, and we discuss the computational efficiency. The latter is of general importance when performing frequent evaluation of phase space or other high-dimensional integrals, which is why the proposed framework promises applications beyond the scope of atomistics.

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“…The procedure has been applied in [16,17] to advection-diffusion potentials and in [18] to parabolic problems of second order. In [19] our approach has been extended to the computation of the Schrödinger potential, where standard cubature methods are very expansive due to the fast oscillations of the kernel. Here we consider the problem of constructing fast cubature formulas for higher order problems.…”

Section: Introductionmentioning

confidence: 99%

Fast cubature of high dimensional biharmonic potential based on approximate approximations

Lanzara

Maz′ya

Schmidt³

2019

Ann Univ Ferrara

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We derive new formulas for the high dimensional biharmonic potential acting on Gaussians or Gaussians times special polynomials. These formulas can be used to construct accurate cubature formulas of an arbitrary high order which are fast and effective also in very high dimensions. Numerical tests show that the formulas are accurate and provide the predicted approximation rate O(h 8 ) up to the dimension 10 7 .

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A fast solution method for time dependent multidimensional Schrödinger equations

Cited by 6 publications

References 15 publications

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

GNN-Assisted Phase Space Integration with Application to Atomistics

Fast cubature of high dimensional biharmonic potential based on approximate approximations

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