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Antoine J. Cerfon - One of the best experts on this subject based on the ideXlab platform.
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a Spectral Transform method for singular sturm liouville problems with applications to energy diffusion in plasma physics
Siam Journal on Applied Mathematics, 2015Co-Authors: Jon Wilkening, Antoine J. CerfonAbstract:We develop a Spectrally accurate numerical method to compute solutions of a model PDE used in plasma physics to describe diffusion in velocity space due to Fokker--Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the Spectral Transform associated with a singular Sturm--Liouville operator. We present a new algorithm for computing the Spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh--Weyl $m$-function from the complex upper half-plane to the real axis. The eigenfunctions and density function are rescaled, and a new formula for the limiting value of the $m$-function is derived to avoid amplification of roundoff errors when the solution is reconstructed. The complexity of the algorithm is also analyzed, showing that the cost of computing the Spectral density function at a point grows less rapidly than any fractional inverse power of the desired acc...
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A Spectral Transform Method for Continuum Kinetic Diffusion Equations in Velocity Space
arXiv: Classical Analysis and ODEs, 2013Co-Authors: Jon Wilkening, Antoine J. CerfonAbstract:We develop a Spectrally accurate numerical algorithm to compute solutions of a model partial differential equation used in plasma physics to describe diffusion in velocity space due to Fokker-Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and non-normalizable eigenfunctions via the Spectral Transform associated with a singular Sturm-Liouville operator. We present a new algorithm for computing the Spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh-Weyl m-function from the complex upper half-plane to the real axis. The eigenfunctions and density function are rescaled and a new formula for the limiting value of the m-function is derived to avoid amplification of roundoff errors when the solution is reconstructed. We highlight key properties of the partial differential equation and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.
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a Spectral Transform method for singular sturm liouville problems with applications to energy diffusion in plasma physics
arXiv: Classical Analysis and ODEs, 2013Co-Authors: Jon Wilkening, Antoine J. CerfonAbstract:We develop a Spectrally accurate numerical method to compute solutions of a model partial differential equation used in plasma physics to describe diffusion in velocity space due to Fokker-Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and non-normalizable eigenfunctions via the Spectral Transform associated with a singular Sturm-Liouville operator. We present a new algorithm for computing the Spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh-Weyl $m$-function from the complex upper half-plane to the real axis. The eigenfunctions and density function are rescaled and a new formula for the limiting value of the $m$-function is derived to avoid amplification of roundoff errors when the solution is reconstructed. The complexity of the algorithm is also analyzed, showing that the cost of computing the Spectral density function at a point grows less rapidly than any fractional inverse power of the desired accuracy. A WKB analysis is used to prove that the Spectral density function is real analytic. Using this new algorithm, we highlight key properties of the partial differential equation and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.
Shixing Yan - One of the best experts on this subject based on the ideXlab platform.
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Ternary Walsh Spectral Transform decision diagrams
2007 6th International Conference on Information Communications & Signal Processing, 2007Co-Authors: Bogdan J. Falkowski, Shixing YanAbstract:The ternary Walsh Transform operated over Galois field (3) (GF(3)) provides a simpler arithmetic and requires less storage space for the calculations than standard Walsh Transform for Boolean functions. The Spectral Transform decision diagrams for ternary Walsh Transform are developed which can provide efficient calculations of the Spectral coefficients.
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ISMVL - Arithmetic-Haar Spectral Transform Decision Diagrams
36th International Symposium on Multiple-Valued Logic (ISMVL'06), 2006Co-Authors: Bogdan J. Falkowski, Shixing YanAbstract:The generalization of multi-polarity arithmetic-Haar Transform in the form of layered Kronecker matrices and its corresponding representations is proposed. As the new generalized arithmetic-Haar Transform has a structure similar to that of the Haar and arithmetic Transform matrices, similar Spectral Transform decision diagrams are held in the expanded Transform as well. The new kind of Spectral Transform decision diagram has multiple-valued terminals corresponding to the multi-polarity arithmetic-Haar Transform to represent the arithmetic-Haar spectra of discrete functions.
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ISIT - Arithmetic-Walsh Spectral Transform Decision Diagrams
2006 IEEE International Symposium on Information Theory, 2006Co-Authors: Bogdan J. Falkowski, Shixing YanAbstract:The generalization of multi-polarity arithmetic-Walsh Transform in the form of layered Kronecker matrices and its corresponding representations is proposed. As the new hybrid arithmetic-Walsh Transform has a structure similar to that of the Walsh and arithmetic Transform matrices, similar Spectral Transform decision diagrams are held in the expanded Transform as well. The new kind of Spectral Transform decision diagram has terminals corresponding to the multi-polarity arithmetic-Walsh Transform to represent the arithmetic-Walsh spectra of discrete functions.
P. Miller - One of the best experts on this subject based on the ideXlab platform.
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A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit
Communications on Pure and Applied Mathematics, 2019Co-Authors: O. Assainova, Christian Klein, K. Mclaughlin, P. MillerAbstract:The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrodinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering Transform, we consider the direct Spectral Transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct Spectral Transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the Spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct Spectral Transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k not equal 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as e down arrow 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.
Huagen Yu - One of the best experts on this subject based on the ideXlab platform.
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neural network iterative diagonalization method to solve eigenvalue problems in quantum mechanics
Physical Chemistry Chemical Physics, 2015Co-Authors: Huagen YuAbstract:We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided Spectral Transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value via a linear Transform diagonalization approach. Since the algorithm uses the Spectral Transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO2 and CH4 as examples.
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a Spectral Transform minimum residual filter diagonalization method for interior eigenvalues of physical systems
Journal of Chemical Physics, 1999Co-Authors: Huagen Yu, Gunnar NymanAbstract:A Spectral Transform technique is introduced into the minimum residual (MINRES) filter diagonalization (FD) algorithm for the computation of eigenvalues of large Hermitian matrices. It is a low storage method, i.e., only four real vectors are required to calculate all bound states of the system. In the MINRES FD step, the finite Krylov subspace is built up by a Lanczos iteration using a Spectral Transform operator which is expanded in a series of Chebyshev polynomials. A guided Spectral Transform method is suggested to achieve high efficiency of this new algorithm. As an example, all even parity bound states of NO2 have been calculated on the adiabatic ground state potential energy surface of NO2 by a single propagation using a hyperbolic tangent function guided filter operator. The results show that the method is accurate and highly efficient. A statistical analysis of the spectrum is also given.
J.b. Drake - One of the best experts on this subject based on the ideXlab platform.
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Parallelizing the Spectral Transform method, part II
Concurrency: Practice and Experience, 1992Co-Authors: D.w. Walker, P.h. Worley, J.b. DrakeAbstract:The Spectral Transform method is a widely used numerical technique for solving partial differential equations on the sphere in global climate modeling. This paper describes the parallelization and performance of the Spectral method for solving the non-linear shallow water equations on the surface of a sphere using a 128-node Intel iPSC/860 hypercube. Solving the shallow water equations represents a computational kernel of more complex climate models. This work is part of a research program to develop climate models that are capable of much longer simulations at a significantly finer resolution than current models. Such models are important in understanding the effects of the increasing atmospheric concentrations of greenhouse gases, and the computational requirements are so large that massively parallel multiprocessors will be necessary to run climate model simulations in a reasonable amount of time. The Spectral method involves the Transformation of data between the physical, Fourier and Spectral domains. Each of these domains is two-dimensional. The Spectral method performs Fourier Transforms in the longitude direction followed by summation in the latitude direction to evaluate the discrete Spectral Transform. A simple way of parallelizing the Spectral code is to decompose the physical problem domain in just the latitude direction. This allows an optimized sequential FFT algorithm to be used in the longitude direction. However, this approach limits the number of processors that can be brought to bear on the problem. Decomposing the problem over both directions allows the parallelism inherent in the problem to be exploited more effectively-the grain size is reduced, so that more processors can be used. Results are presented that show that decomposing over both directions does result in a more rapid solution of the problem. The results show that, for a given problem and number of processors, the optimum decomposition has approximately equal numbers of processors in each direction. Load imbalance also has an impact on the performance of the method. The importance of minimizing communication latency and overlapping communication with calculation is stressed. General methods for doing this, that may be applied to many other problems, are discussed.
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Parallelizing the Spectral Transform method---Part 2
1991Co-Authors: D.w. Walker, P.h. Worley, J.b. DrakeAbstract:This paper describes the parallelization and performance of the Spectral method for solving the shallow water equations on the surface of a sphere using a 128-node Intel iPSC/860 hypercube. The shallow water equations form a computational kernel of more complex climate models. This work is part of a research program to develop climate models that are capable of much longer simulations at a significantly finer resolution than current models. Such models are important in understanding the effects of the increasing atmospheric concentrations of greenhouse gases, and the computational requirements are so large that massively parallel multiprocessors will be necessary to run climate models simulations in a reasonable amount of time. The Spectral method involves the Transformation of data between the physical, Fourier, and Spectral domains. Each of these domains is two-dimensional. The Spectral method performs Fourier Transforms in the longitude direction followed by summation in the latitude direction to evaluate the discrete Spectral Transform. A simple way of parallelizing the Spectral code is to decompose the physical problem domain in just the latitude direction. This allows an optimized sequential FFT algorithm to be used in the longitude direction. However, this approach limits the number of processors that can be broughtmore » to bear on the problem. Decomposing the problem over both directions allows the parallelism inherent in the problem to be exploited more effectively -- the grain size is reduced and more processors can be used. Results are presented that show that decomposing over both directions does result in a more rapid solution of the problem. The importance of minimizing communication latency and overlapping communication with calculation is stressed. General methods for doing this, that may be applied to many other problems, are discussed.« less
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Parallelizing the Spectral Transform method, Part 1
1991Co-Authors: P.h. Worley, J.b. DrakeAbstract:The Spectral Transform method is a standard numerical technique used to solve partial differential equations on the sphere in global climate modeling. In particular, it is used in CCM1 and CCM2, the Community Climate Models developed at the National Center for Atmospheric Research. This paper describes initial experiences in parallelizing a program that uses the Spectral Transform method to solve the nonlinear shallow water equations on the sphere, showing that an efficient implementation is possible on the Intel iPSC/860. The use of PICL, a portable instrumented communication library, and ParaGraph, a performance visualization tool, in tuning the implementation is also described. The Legendre Transform and the Fourier Transform comprise the computational kernel of the Spectral Transform method. This paper is a case study of parallelizing the Legendre Transform. For many problem sizes and numbers of processors, the Spectral Transform method can be parallelized efficiently by parallelizing only the Legendre Transform. A subsequent paper will discuss parallelizing the Fourier Transform as well. 42 refs., 5 figs., 8 tabs.
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Parallelizing the Spectral Transform Method
The Sixth Distributed Memory Computing Conference 1991. Proceedings, 1Co-Authors: P.h. Worley, D.w. Walker, J.b. DrakeAbstract:The Spectral Transform method is the standard numerical technique for solving partial differential equations on the sphere in global climate modeling. This paper describes the parallelization and performance of the Spectral Transform method for solving the nonlinear shallow water equations on the surface of a sphere using a 128-node Intel iPSC/860 hypercube. This work is part of a research program to develop climate models that are capable of much longer and more numerous simulations at a significantly finer resolution than are currently available. Such models are important in understanding the effects of the increasing atmospheric concentrations of greenhouse gases, and the computational requirements are such that massively parallel multiprocessors will be necessary to run such simulations in a reasonable amount of time. 10 refs., 6 figs., 1 tab.