# Tag Info

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Inge Söderkvist (2009) has a nice write-up of solving the Rigid Body Movement Problem by singular value decomposition (SVD). Suppose we are given 3D points $\{x_1,\ldots,x_n\}$ that after perturbation take positions $\{y_1,\ldots,y_n\}$ respectively. We seek a rigid "motion", i.e. a rotation $R$ and translation $d$ combined, applied to points $x_i$ that ...

10

People use all sorts of bases in practice. For example, people use orthonormal bases in DG methods to ensure that the mass matrix in time stepping schemes is diagonal. People also use hierarchical bases when doing $p$ adaptivity because it makes the construction of constraints at faces where different polynomial degrees come together trivial. For higher ...

6

This is actually quite simple. Let's say you have your (two-dimensional) reference triangle $\hat K=\left\{(\xi,\eta)\in {\mathbb R}^2: 0\le \xi \le 1, 0\le \eta\le 1, \xi \le (1-\eta)\right\}$. Then you know how to define the three basis functions on this triangle. They are: $$\hat N_1(\xi,\eta) = 1-\xi-\eta,\\ \hat N_2(\xi,\eta) = \xi,\\ \hat N_3(\xi,\eta)... 6 The nodal Lagrange bases are nice because they interpolate the functions at the knots:$$\phi_i(x_j) = \delta_{ij}$$This means that you can read and plot solutions by looking only at the coefficients u_j in the representation:$$u_h(x) = \sum_j u_j\phi_j(x)$$That's much nicer than having to evaluate the sum at every point you care about knowing u_h. 5 While it doesn't answer your explict question, I offer the following important remark that might answer the intentions behind the question. The anomalies of the Gram-Scmidt process in case of dependence are avoided if one computes the orthogonalization instead by the standard Householder QR factorization (or in the sparse case the Givens variant). The ... 5 If v_2 can be expressed as a linear combination of \{ v_3, v_4, v_5 \}, but not of a subset of these vectors, then, e.g., v_5 is in the linear span of \{ v_2, v_3, v_4 \}. So your algorithm would likely map v_5 onto 0. Typically, if you apply the Gram-Schmidt proecess to the sequence of v_i, 1 \leq i \leq L, a vector v_k is mapped onto 0 ... 4 There are a variety of different bases in FEM, but most involve basis functions which are associated with topological entities, like vertices, edges, faces, and element interiors. This makes it possible to enforce various types of continuity by ensuring that degrees of freedom for such functions match at shared vertices/edges/faces. These basis functions ... 4 In Engineering nodal bases are a good starting point for solid mechanics problems because the principle of virtual work for the discretised system \mathbf{K} \mathbf{u} = \mathbf{f} reads$$ \delta \mathbf{u}^T \mathbf{K} \mathbf{u} = \delta \mathbf{u}^T \mathbf{f} $$showing that \mathbf{K} \mathbf{u} are indeed (equivalent) nodal internal forces, and \... 4 I think you've got slightly the wrong end of the stick from the documentation. As with a lot of other software in the area, GMSH started out with low order, hard coded numberings. These are the ones with the ASCII art representations, which only give first and second order numberings for tetrahedra (hence there aren't any face nodes in the 4 node or 10 node "... 3 On quad/hex you can use tensor product polynomials. For example in 2d, you map the cell to a reference cell [-1,1] \times [-1,1] and if N is the degree, you would use$$ P_i(\xi) P_j(\eta), \qquad 0 \le i,j \le N $$as the basis functions, where \xi,\eta are coordinates on the reference cell and P_i is the Legendre polynomial of degree i. These are ... 3 I recommend reading Thom Dunning Jr. “Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.” J. Chem. Phys. 90, 1007 (1989) by for an answer to this question. The abstract says the following: This leads to the concept of correlation consistent basis sets, i.e., sets which include all functions in ... 3 The paper  gives an explicit construction of the Bernstein form of a set of orthogonal polynomials on simplices based on Legendre polynomials.  Farouki, R.T., Goodman, T.N.T and Sauer, T: Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains, Computer Aided Geometric Design 20 (2003), 209-230, DOI: 10.... 2 I think Rob Kirby told me once that he had written something on using Bernstein polynomials for FEM. Take a look at his web site at Texas Tech (or now at Baylor). 2 In the standard form, KS-DFT is solved variationally, which means that additional degrees of freedom in the basis set must lead to a lower (or equal) energy. This is a very basic property of the variational method and the math is almost certainly explained on Wikipedia or equivalent. I am assuming the same functional is used. Each DFT functional provides ... 2 For the case of faceted triangle geometry, the derivatives you're looking for (\frac{d\phi}{dx}, \frac{d\phi}{dy}, \frac{d\phi}{dz}) can actually be found without resorting to calculus (chain rule / jacobian), you can deduce them from purely geometrical considerations. These derivatives are the cartesian (x,y,z) components of the vector function \... 1 From @Wolfgang Bangerth's answer, there exists a mapping function (x,y,z)=\Phi(\xi,\eta) which can be expressed as a function of the basis shape functions. Using a similar notation as @Wolfgang Bangerth, the reference or base triangle is defined as:$$ \left\{(\xi,\eta)\in {\mathbb R}^2: 0\le \xi \le 1, 0\le \eta\le 1, \xi \le (1-\eta)\right\}  The ...

1

Ideally, one would like to have an orthonormal basis with respect to the $L_2$ or energy inner product, or both (e.g., the eigenmodes of the operator at hand), because many matrices would be diagonal and thus very easy to invert. But that's hard to get (solving eigenproblems is a nonlinear procedure). Lagrange bases is provides sparse matrices, which in ...

1

I guess, the final result is suffering from instability (relative) of Gram-Schmidt, even in modified/stabilized forms. In this case, instead of going from the start: modifying the orthogonalization process, we can start with the final result instead. So, after solving a non-Hermitian eigenvalue problem (with a reliable algorithm), we obtained $X_L$ and $X_R$...

1

Density functional theory optimizes the total energy with respect to the (possibly constrained) electronic density. The usual numerical procedure for this optimization is embodied by the KS method. The KS procedure will give you the optimal total energy and the ground-state (gs) density for which this optimal energy is achieved. This optimization procedure ...

1

Not using pseudo-potentials is a great way to drive up the computational cost of a QMC enormously while not adding to the accuracy much at all. There's a very recent paper about PPs for QMC: http://jcp.aip.org/resource/1/jcpsa6/v139/i1/p014101_s1

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Also take a look at John Burkhardt's libraries at http://people.sc.fsu.edu/~jburkardt/ His code is LGPL licensed, as opposed to the GPL used by the Gnu Scientific Library.

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Here are some references that might be useful: http://pubs.acs.org/doi/abs/10.1021/cr00074a002 http://www.sciencedirect.com/science/article/pii/0167797785900036 http://books.google.com/books?id=Gt4pnp-UFhUC&lpg=PA853&vq=helgaker&pg=PA725#v=snippet&q=helgaker&f=false (if you see nothing, click next then prev). See http://www....

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