Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

An, Congpei; Chen, Xiaojun; Sloan, Ian H.; Womersley, Robert S.

doi:10.1137/100795140

Cited by 40 publications

(68 citation statements)

References 25 publications

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“…Different point systems can be generated from the notion of so-called spherical designs which possess good qualities for quadrature as well as interpolation, as the growth of the Lebesgue constant for these systems also seems to be close to linear with N . An et al [2] find I N S ≈ 0.8025(N + 1)…”

Section: Convergence Propertiesmentioning

confidence: 97%

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Sparse Discrete Ordinates Method in Radiative Transfer

Grella¹,

Schwab²

2011

Comput. Methods Appl. Math.

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-The stationary monochromatic radiative transfer equation (RTE) is a partial differential transport equation stated on a five-dimensional phase space, the Cartesian product of physical and angular domain. We solve the RTE with a Galerkin FEM in physical space and collocation in angle, corresponding to a discrete ordinates method (DOM). To reduce the complexity of the problem and to avoid the "curse of dimension", we adapt the sparse grid combination technique to the solution space of the RTE and show that we obtain a sparse DOM which uses essentially only as many degrees of freedom as required for a purely spatial transport problem. For smooth solutions, the convergence rates deteriorate only by a logarithmic factor. We compare the sparse DOM to the standard full DOM and a sparse tensor product approach developed earlier with Galerkin FEM in physical space and a spectral method in angle. Numerical experiments confirm our findings.2010 Mathematical subject classification: 35Q79; 65N12; 65N30; 65N35.

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Section: Convergence Propertiesmentioning

confidence: 97%

“…In the radiative transfer problem, which is stated in five dimensions for (d, d S ) = (3,2), this makes the solution of any problems larger than example size in practice computationally very challenging.…”

Section: Radiative Transfermentioning

confidence: 99%

Sparse Discrete Ordinates Method in Radiative Transfer

Grella¹,

Schwab²

2011

Comput. Methods Appl. Math.

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“…Corollary 8. Let D be 3 × 3 symmetric matrix with Gaussian density (1). As µ → ∞ with λ > −2µ/3, we have four asymptotic regimes depending on the symmetries of the mean matrixD.…”

Section: Spectral Groupingmentioning

confidence: 99%

“…If this hypothesis is accepted, we assume that we are in the asymptotic regime (1) and construct a conditional sphericity test by using the conditional distribution of VR(γ (n) ) given a (n) κ 1 (γ (n) ) 2 = t , which converges in distribution to the law of…”

Section: Testing the Sphericity Hypothesismentioning

confidence: 99%

Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments

Gasbarra

Pajevic

Basser

2017

SIAM J. Imaging Sci.

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Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and unique information, but at the cost of requiring a more complex statistical analysis. In this work we derive the distributions of eigenvalues and eigenvectors in the special but important case of m×m symmetric random matrices, D, observed with isotropic matrix-variate Gaussian noise. The properties of these distributions depend strongly on the symmetries of the mean tensor/matrix, D̄. When D̄ has repeated eigenvalues, the eigenvalues of D are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same D̄ eigenspaces. We apply these results to diffusion tensor imaging (DTI), with m = 3, addressing an important problem of detecting the symmetries of the diffusion tensor, and seeking an experimental design that could potentially yield an isotropic Gaussian distribution. In the 3-dimensional case, when the mean tensor is spherically symmetric and the noise is Gaussian and isotropic, the asymptotic distribution of the first three eigenvalue central moment statistics is simple and can be used to test for isotropy. In order to apply such tests, we use quadrature rules of order t ≥ 4 with constant weights on the unit sphere to design a DTI-experiment with the property that isotropy of the underlying true tensor implies isotropy of the Fisher information. We also explain the potential implications of the methods using simulated DTI data with a Rician noise model.

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“…Also for a given t, a spherical t-design is not unique. Our numerical results use the one near the extremal system [1,7,8]. Let t = 5 and N = (t+1) 2 = 36.…”

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confidence: 99%

Minimizing the Condition Number of a Gram Matrix

Chen¹,

Womersley²,

Ye³

2011

SIAM J. Optim.

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Abstract. The condition number of a Gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the least squares polynomial approximation. Given a polynomial basis, we consider the problem of finding a set of points and/or weights which minimizes the condition number of the Gram matrix. The objective function f in the minimization problem is nonconvex and nonsmooth. We present an expression of the Clarke generalized gradient of f and show that f is Clarke regular and strongly semismooth. Moreover, we develop a globally convergent smoothing method to solve the minimization problem by using the exponential smoothing function. To illustrate applications of minimizing the condition number, we report numerical results for the Gram matrix defined by the weighted Vandermonde-like matrix for least squares approximation on an interval and for the Gram matrix defined by an orthonormal set of real spherical harmonics for least squares approximation on the sphere.

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Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

Cited by 40 publications

References 25 publications

Sparse Discrete Ordinates Method in Radiative Transfer

Sparse Discrete Ordinates Method in Radiative Transfer

Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments

Minimizing the Condition Number of a Gram Matrix

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