Discrete spherical harmonic functions for texture representation and analysis

A basis of discrete harmonic functions for efficient representation and analysis of crystallographic texture is presented. Discrete harmonics are a numerical representation of the harmonics on the sphere. A finite element formulation is utilized to calculate these orthonormal basis functions, which provides several advantageous features for quantitative texture analysis. These include high-precision numerical integration, a simple implementation of the non-negativity constraint and computational efficiency. Simple examples of pole figure and texture interpolation and of Fourier filtering using these basis sets are presented.

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Fast Domain Decomposition Type Solver for Stiffness Matrices of Reference p-Elements

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2013-0003 ◽

2013 ◽

Vol 13 (2) ◽

pp. 161-183 ◽

Cited By ~ 4

Author(s):

Vadim Korneev

Keyword(s):

Finite Difference ◽

Domain Decomposition ◽

Elliptic Equations ◽

Harmonic Functions ◽

Schur Complement ◽

P Elements ◽

Stiffness Matrices ◽

The Family ◽

Discrete Harmonic Functions

Abstract. A key component of domain decomposition solvers for hp discretizations of elliptic equations is the solver for internal stiffness matrices of p-elements. We consider an algorithm which belongs to the family of secondary domain decomposition solvers, based on the finite-difference like preconditioning of p-elements, and was outlined by the author earlier. We remove the uncertainty in the choice of the coarse (decomposition) grid solver and suggest the new interface Schur complement preconditioner. The latter essentially uses the boundary norm for discrete harmonic functions induced by orthotropic discretizations on slim rectangles, which was derived recently. We prove that the algorithm has linear arithmetical complexity.

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Discrete harmonic functions

Graphs and Geometry - Colloquium Publications ◽

10.1090/coll/065/04 ◽

2019 ◽

pp. 39-56

Keyword(s):

Harmonic Functions ◽

Discrete Harmonic Functions

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Discrete harmonic functions on an orthant in $\mathbb{Z}^d$

Electronic Communications in Probability ◽

10.1214/ecp.v20-4249 ◽

2015 ◽

Vol 20 (0) ◽

Author(s):

Mustapha Sami ◽

Aymen Bouaziz ◽

Mohamed Sifi

Keyword(s):

Harmonic Functions ◽

Discrete Harmonic Functions

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On the positivity of discrete harmonic functions and the discrete Harnack inequality for piecewise linear finite elements

Mathematics of Computation ◽

10.1090/mcom/3117 ◽

2016 ◽

Vol 86 (305) ◽

pp. 1127-1145 ◽

Cited By ~ 1

Author(s):

D. Leykekhman ◽

M. Pruitt

Keyword(s):

Finite Elements ◽

Harnack Inequality ◽

Harmonic Functions ◽

Piecewise Linear ◽

Discrete Harmonic Functions

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Perspectives on Basis Sets Beautiful: Seasonal Plantings of Diffuse Basis Functions

Journal of Chemical Theory and Computation ◽

10.1021/ct200106a ◽

2011 ◽

Vol 7 (10) ◽

pp. 3027-3034 ◽

Cited By ~ 274

Author(s):

Ewa Papajak ◽

Jingjing Zheng ◽

Xuefei Xu ◽

Hannah R. Leverentz ◽

Donald G. Truhlar

Keyword(s):

Basis Functions ◽

Basis Sets

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Gaussian Basis Functions for Use in Molecular Calculations. I. Contraction of (9s5p) Atomic Basis Sets for the First‐Row Atoms

The Journal of Chemical Physics ◽

10.1063/1.1674408 ◽

1970 ◽

Vol 53 (7) ◽

pp. 2823-2833 ◽

Cited By ~ 4110

Author(s):

Thom H. Dunning

Keyword(s):

Basis Functions ◽

Basis Sets ◽

Gaussian Basis Functions ◽

Molecular Calculations

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Accurate atomic Gaussian basis functions for first-row atoms. Part 1. Contracted basis sets derived from 9s5p primitives

Journal of Molecular Structure THEOCHEM ◽

10.1016/0166-1280(96)04431-4 ◽

1996 ◽

Vol 362 (3) ◽

pp. 283-296 ◽

Cited By ~ 6

Author(s):

Alexander V. Mitin ◽

Gerhard Hirsch ◽

Robert J. Buenker

Keyword(s):

Basis Functions ◽

Basis Sets ◽

Gaussian Basis Functions

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Convergence and applications of reproducing kernels for classes of discrete harmonic functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700043975 ◽

1974 ◽

Vol 11 (3) ◽

pp. 339-358

Author(s):

C. Wayne Mastin

Keyword(s):

Harmonic Functions ◽

Reproducing Kernels ◽

Biharmonic Operator ◽

Convergence Properties ◽

Interpolation Problems ◽

Discrete Analogs ◽

Discrete Harmonic Functions

This paper gives convergence properties and applications of the discrete analogs of reproducing kernels for various families of harmonic functions. From these results information is obtained on the solution of interpolation problems, the convergence of the discrete Neumann's function, and the solution to problems involving the discrete biharmonic operator.

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Discrete harmonic functions in Lipschitz domains

Electronic Communications in Probability ◽

10.1214/19-ecp259 ◽

2019 ◽

Vol 24 (0) ◽

Author(s):

Sami Mustapha ◽

Mohamed Sifi

Keyword(s):

Harmonic Functions ◽

Lipschitz Domains ◽

Discrete Harmonic Functions

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Stability estimates for discrete harmonic functions on product domains

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm121204025g ◽

2013 ◽

Vol 7 (1) ◽

pp. 143-160 ◽

Cited By ~ 3

Author(s):

Maru Guadie

Keyword(s):

Fourier Series ◽

Dirichlet Problem ◽

Harmonic Measure ◽

Harmonic Functions ◽

Discrete Analog ◽

Stability Estimates ◽

Product Domains ◽

Discrete Harmonic Functions

We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm?n-Lindel?f theorem and use Fourier series to obtain a discrete analog of the three-line theorem for the gradients of harmonic functions in a strip. Then we derive estimates for the discrete harmonic measure and use elementary spectral inequalities to obtain stability estimates for Dirichlet problem in cylinder domains.

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