Questions tagged [basis-set]
A linearly independent set of elements of a vector space such that any element in that vector space can be expressed as a linear combination of the elements of the basis.
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Lowest order Raviart Thomas elements
I have questions regarding the implementation of Lowest order Raviart Thomas elements on quadrilaterals, some papers use the basis functions in this form:
for example the right edge: $N = \left[\dfrac{...
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Measuring the extent to which two sets of vectors span the same space
I have a set of measurements $y_i$, $1 \leq i \leq N$, and I want to model these measurements with a linear model. I have two possible models I can use,
$$
y \approx A c
$$
and
$$
y \approx B d
$$
...
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FEM Basis for functions of two variables in $\mathbb{R}^2$ (Applied to Linear Full Stokes Equations)
I'm trying to do FEM for a very basic version of the linear full Stokes equations in two dimensions. Say we are working in the grid $[0,1]\times[0,1]$ in the $xy-$plane.
To solve an FEM problem for a ...
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How to perform FFT from plane-wave basis function coefficients to real space?
I have a 3D grid in real space of grid spacing $L$ and say 21 grid points in each direction, containing e.g. a charge distribution. This is stored as a numpy array of shape ...
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Are there radial basis functions $\phi_n(x)$ which can be factored as $f_n(x) x^{\alpha}$ for $0<\alpha<1$ and $n>0$?
I am looking for a set of radial basis functions $\phi_n(x)$ on $\mathbb{R}^+=[0;\infty[$ which satisfy some othogonality condition
$$ \int x \phi_n(x) \phi_{n'}^*(x) dx =\delta_{nn'}$$
and "...
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Generate polynomial basis through a sequence of SVD
I need help to understand how to use the result given by an algorithm for constructing an orthonormal polynomial basis over $L^{2}(X)$, where $X\subset\mathbb{R}^2$, with respect to the inner product $...
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Quick way to find a common basis of eigenvectors between 2 matrices : valid or not?
Following the advise of @Federico Polonion a previous post, one suggested, to find a basis of common eigen vectors between 2 matrices, to simply do :
Generate 2 ...
user14831
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FEM shape functions on triangular elements: transition from 2D to 3D
I'm writing a code for solving PDEs through the finite element method. In particular, I'm facing with 3D problems, in which I don't know how to calculate shape functions derivatives on the boundaries (...
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Forming Basis Functions from 6-31G Basis Set for Carbon Atom
I am a computer science grad and I am working to write an electronic structure calculation program and I am stuck at forming basis functions using 6-31G Basis set for atoms having higher atomic ...
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Definition of Lagrange nodes in Gmsh
When gmsh uses higher-order tetrahedral elements, there is an underlying Lagrange basis used to specify the map from reference space to the element. I'm trying to load a gmsh mesh of 3rd degree ...
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Multi-matrix orthogonal basis problem
Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
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Orthonormal basis for hexahedron
Orthogonal polynomials are often preferred as basis functions. Recently I learned selecting orthonormal basis further simplifies the mass matrix from diagonal to simply the identity matrix when used ...
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Finding Shape Functions for a Triangle in 3D coordinate space
I have a Triangular Plane T given by three points in 3D space $$P_1 = (x_1,y_1,z_1)\\ P_2 = (x_2,y_2,z_2) \\ P_3 = (x_3,y_3,z_3)$$
I want to find the Shape Functions on this plane as $N_1(x,y,z),N_2(...
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Computing size of N-Dimensional Polynomial Basis and Efficient Representation of Basis
A problem I have had on my mind recently has been a compact way to compute the size of an $N$-Dimensional Polynomial basis of some order $p$, where a linear basis is $p=1$. I have attempted searching ...
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Choice of basis in FEM
Briefly, what are the different types of basis used in FEM and why is the nodal basis so popular and advantageous in the finite element context?
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Using SVD to biorthogonalize left and right eigenvectors?
I have a set of left and right eigenvectors from an nonsymmetric eigenproblem, and I'd like to biorthogonalize them.
I tried Gram-Schmidt, but this fails for most cases.
I then read that the SVD is ...
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Basis for spatially localized function [closed]
I'm solving an ODE, where currently the dependent variables are the (time dependent) spatial Fourier coefficients. It turns out that the phenomena I'm interested in describing is spatially localized, ...
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How should I choose the knot sequence when using B-splines as a basis for solving a PDE?
I'm looking to solve the Schrödinger equation with a basis made of a tensor product of basis splines. A number of papers describe calculations made with a program designed this way, but they never ...
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Size reduction of matrices in dispersion curve calculation
I have an energy dispersion curve obtained from the eigenvalues of
$$E(k) = \text{eig}(T e^{ik} + T^H e^{-ik} + H_0),$$
where $H_0$ and $T$ are $N\times N$ square matrices, $T^H$ is the Hermitian ...
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Does energy decrease with basis set size in density functional theory?
Based on the variational principle, one might expect that the ground state energy of a density functional theory (DFT) calculation will decrease as the basis set size increases. (As I understand it, ...
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Fitting one set of points to another by a rigid motion
I'm not really sure how to explain this problem clearly, so please bear with me.
I have a basis of 3 orthonormal unit vectors and a position, a standard 4x4 transform matrix in computer graphics.
...
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Trying to generate a wave function basis set
For a little project I'm working on, I am trying to generate a wavefunction basis set I can use in Quantum Monte Carlo (DMC to be specific). Preferably, it would be a linear combination of Slater ...
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Orthonormalized Bernstein polynomials using Gram-Schmidt
I was wondering, before trying to do that myself, has anyone attempted to do orthonormalization of Bernstein polynomials using Gram-Schmidt?
I discussed this with several people and have been told ...
- 1,413
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Gram-Schmidt method to identify linearly dependent vectors
A method to orthogonalize a set of vectors (vectors of unit length that are mutually orthogonal) is the Gram-Schmidt process:
http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
Note that the ...
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Efficient computation of the extension of a linear basis to completion when the basis is almost complete (ideally using LAPACK routines)
I have a $p \times n$ matrix $B$ (where $n < p$) with orthonormal columns and would like to find a numerically efficient way to extend this matrix to get a complete $p$-dimensional orthonormal ...
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Conjugate Gradient with Hierarchical Basis Functions: How can the hierarchical base be decomposed?
I'm trying to implement a Conjugate Gradient solver using Hierarchical Basis Functions, following this paper.
In section 3 the paper says that the hierarchical basis matrix $S$ can be decomposed into ...
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How can I compute a basis for a matrix Lie algebra given a finite set of generators?
Given an arbitrary set of (numerical) square complex matrices $\mathcal{A}=\{A_1,A_2,\cdots,A_m\}$, I am interested in computing the real matrix Lie algebra generated by $\mathcal{A}$, call it $\...
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Which libraries have good implementations of Basis splines?
I'm looking to use the finite element method with B-splines as my function basis. Which C/C++ libraries have good B-spline support?
Specifically, I'm looking for an implementation of a stable ...
- 3,355
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How to do FEM in sector elements?
Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there
are a lot of ...
- 1,319
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Polynomials that are orthogonal over curves in the complex plane
Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other ...
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Differences between Gaussian and Slater functions on the quality of the results?
Given two computational programs, one using a Gaussian basis, and the other using Slater basis, what are the practical differences, advantages and disadvantages for each choice ?
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What does it mean for a basis set to be "correlation consistent" ?
Some basis sets are said to be "correlation consistent". What does it mean in practice ?
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How to choose a basis set for ab-initio evaluations ?
How do I pick a basis set for an ab-initio Hartree-Fock evaluation ? In other words, what are the important characteristics of a basis set so that a proper choice can be made ?
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What is counterpoise correction?
What is counterpoise correction exactly ? Can you explain when it is needed and why ?
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