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Wang, Yu Guang - One of the best experts on this subject based on the ideXlab platform.
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FaVeST: Fast Vector Spherical Harmonic Transforms
2021Co-Authors: Quoc Thong Le Gia, Li Ming, Wang, Yu GuangAbstract:Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have broad applications in geophysics, quantum mechanics and astrophysics. In the representation of a Tangent Vector field, one needs to evaluate the expansion and the Fourier coefficients of Vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for Vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of Vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated Tangent fields illustrate the accuracy, efficiency and stability of FaVeST.Comment: 23 pages, 6 figures, 3 table
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FaVeST: Fast Vector Spherical Harmonic Transforms
2019Co-Authors: Quoc Thong Le Gia, Li Ming, Wang, Yu GuangAbstract:Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have wide applications in geophysics, quantum mechanics and astrophysics. In the representation of a Tangent Vector field, one needs to evaluate the expansion and the Fourier coefficients of Vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for Vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of Vector spherical harmonics with degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated Tangent fields illustrate the accuracy and efficiency of FaVeST.Comment: 20 pages, 4 figures, 2 table
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Fast Tensor Needlet Transforms for Tangent Vector Fields on the Sphere
2019Co-Authors: Li Ming, Broadbridge Philip, Olenko Andriy, Wang, Yu GuangAbstract:This paper constructs a semi-discrete tight frame of tensor needlets associated with a quadrature rule for Tangent Vector fields on the unit sphere $\mathbb{S}^2$ of $\mathbb{R}^3$ --- tensor needlets. The proposed tight tensor needlets provide a multiscale representation of any square integrable Tangent Vector field on $\mathbb{S}^2$, which leads to a multiresolution analysis (MRA) for the field. From the MRA, we develop fast algorithms for tensor needlet transforms, including the decomposition and reconstruction of the needlet coefficients between levels, via a set of filter banks and scalar FFTs. The fast tensor needlet transforms have near linear computational cost proportional to $N\log \sqrt{N}$ for $N$ evaluation points or coefficients. Numerical examples for the simulated and real data demonstrate the efficiency of the proposed algorithm.Comment: 29 pages, 5 figures, 2 table
Quoc Thong Le Gia - One of the best experts on this subject based on the ideXlab platform.
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FaVeST: Fast Vector Spherical Harmonic Transforms
2021Co-Authors: Quoc Thong Le Gia, Li Ming, Wang, Yu GuangAbstract:Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have broad applications in geophysics, quantum mechanics and astrophysics. In the representation of a Tangent Vector field, one needs to evaluate the expansion and the Fourier coefficients of Vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for Vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of Vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated Tangent fields illustrate the accuracy, efficiency and stability of FaVeST.Comment: 23 pages, 6 figures, 3 table
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FaVeST: Fast Vector Spherical Harmonic Transforms.
arXiv: Numerical Analysis, 2019Co-Authors: Quoc Thong Le Gia, Yu Guang WangAbstract:Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have wide applications in geophysics, quantum mechanics and astrophysics. In the representation of a Tangent Vector field, one needs to evaluate the expansion and the Fourier coefficients of Vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for Vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of Vector spherical harmonics with degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated Tangent fields illustrate the accuracy and efficiency of FaVeST.
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FaVeST: Fast Vector Spherical Harmonic Transforms
2019Co-Authors: Quoc Thong Le Gia, Li Ming, Wang, Yu GuangAbstract:Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have wide applications in geophysics, quantum mechanics and astrophysics. In the representation of a Tangent Vector field, one needs to evaluate the expansion and the Fourier coefficients of Vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for Vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of Vector spherical harmonics with degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated Tangent fields illustrate the accuracy and efficiency of FaVeST.Comment: 20 pages, 4 figures, 2 table
Yann Ollivier - One of the best experts on this subject based on the ideXlab platform.
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Ricci curvature of Markov chains on metric spaces
2020Co-Authors: Yann OllivierAbstract:Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a Tangent Vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein-Uhlenbeck process. Moreover this generalization is consistent with the Bakry-Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is shown to imply a spectral gap, a Lévy-Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples
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ricci curvature of markov chains on metric spaces
Journal of Functional Analysis, 2009Co-Authors: Yann OllivierAbstract:Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a Tangent Vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein–Uhlenbeck process. Moreover this generalization is consistent with the Bakry–Emery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is shown to imply a spectral gap, a Levy–Gromov–like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.
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ricci curvature of markov chains on metric spaces
arXiv: Probability, 2007Co-Authors: Yann OllivierAbstract:We define the Ricci curvature of Markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability measures equipped with a Wasserstein transportation distance. For Brownian motion on a Riemannian manifold this gives back the value of Ricci curvature of a Tangent Vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein--Uhlenbeck process. Moreover this generalization is consistent with the Bakry--Emery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is easily shown to imply a spectral gap, a Levy--Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. These bounds are sharp in several interesting examples.
Eduardo Garciario - One of the best experts on this subject based on the ideXlab platform.
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relating the curvature tensor and the complex jacobi operator of an almost hermitian manifold
Advances in Geometry, 2008Co-Authors: M Brozosvazquez, Eduardo GarciarioAbstract:Let J be a unitary almost complex structure on a Riemannian manifold (M, g). If x is a unit Tangent Vector, let π := Span{x, Jx} be the associated complex line in the Tangent bundle of M . The complex Jacobi operator and the complex curvature operators are defined, respectively, by J (π) := J (x) + J (Jx) and R(π) := R(x, Jx). We show that if (M, g) is Hermitian or if (M,g) is nearly Kahler, then either the complex Jacobi operator or the complex curvature operator completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show this result fails for general almost Hermitian manifolds.
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relating the curvature tensor and the complex jacobi operator of an almost hermitian manifold
arXiv: Differential Geometry, 2006Co-Authors: M Brozosvazquez, Eduardo GarciarioAbstract:Let J be a unitary almost complex structure on a Riemannian manifold (M,g). If x is a unit Tangent Vector, let P be the associated complex line spanned by x and by Jx. We show that if (M,g) is Hermitian or if (M,g) is nearly Kaehler, then either the complex Jacobi operator (JC(P)y=R(y,x)x+R(y,Jx)Jx) or the complex curvature operator (RC(P)y=R(x,Jx)y) completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show this result fails for general almost Hermitian manifold.
Li Ming - One of the best experts on this subject based on the ideXlab platform.
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FaVeST: Fast Vector Spherical Harmonic Transforms
2021Co-Authors: Quoc Thong Le Gia, Li Ming, Wang, Yu GuangAbstract:Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have broad applications in geophysics, quantum mechanics and astrophysics. In the representation of a Tangent Vector field, one needs to evaluate the expansion and the Fourier coefficients of Vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for Vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of Vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated Tangent fields illustrate the accuracy, efficiency and stability of FaVeST.Comment: 23 pages, 6 figures, 3 table
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FaVeST: Fast Vector Spherical Harmonic Transforms
2019Co-Authors: Quoc Thong Le Gia, Li Ming, Wang, Yu GuangAbstract:Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have wide applications in geophysics, quantum mechanics and astrophysics. In the representation of a Tangent Vector field, one needs to evaluate the expansion and the Fourier coefficients of Vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for Vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of Vector spherical harmonics with degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated Tangent fields illustrate the accuracy and efficiency of FaVeST.Comment: 20 pages, 4 figures, 2 table
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Fast Tensor Needlet Transforms for Tangent Vector Fields on the Sphere
2019Co-Authors: Li Ming, Broadbridge Philip, Olenko Andriy, Wang, Yu GuangAbstract:This paper constructs a semi-discrete tight frame of tensor needlets associated with a quadrature rule for Tangent Vector fields on the unit sphere $\mathbb{S}^2$ of $\mathbb{R}^3$ --- tensor needlets. The proposed tight tensor needlets provide a multiscale representation of any square integrable Tangent Vector field on $\mathbb{S}^2$, which leads to a multiresolution analysis (MRA) for the field. From the MRA, we develop fast algorithms for tensor needlet transforms, including the decomposition and reconstruction of the needlet coefficients between levels, via a set of filter banks and scalar FFTs. The fast tensor needlet transforms have near linear computational cost proportional to $N\log \sqrt{N}$ for $N$ evaluation points or coefficients. Numerical examples for the simulated and real data demonstrate the efficiency of the proposed algorithm.Comment: 29 pages, 5 figures, 2 table
