7

One part of the fundamental theorem of linear algebra is that the kernel/nullspace of $\mathbf A$ is orthogonal to the range of $\mathbf A^T$. By applying the $\mathbf Q \mathbf R$ decomposition to $\mathbf A^T$, you can generate the orthogonal projector $\mathbf P = \mathbf I - \mathbf Q \mathbf Q^T$. The vector $\mathbf P \mathbf x$ is what you're looking ...


6

The solutions for the equation are in $$\psi \in \mathbb{C}^{3M}\times\mathbb{R}^+ \enspace .$$ If the number of electrons is small enough you can just use any traditional method. Like a domain discretization method (Finite Difference, Finite Element, Boundary Element), or a pseudospectral method. Since solving this equation is not more difficult than ...


5

Take a look at active subspaces, e.g., Active Subspace Methods in Theory and Practice: http://epubs.siam.org/doi/abs/10.1137/130916138 And a PDF here: http://inside.mines.edu/~pconstan/docs/constantine-asm.pdf I have a SIAM book (Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies) coming out in March. Suppose $f$ maps $\mathbb{...


5

This feature seems to be available in CGAL


4

I think you can do this using convex hull software (e.g. QHull) via the lifting algorithm. At least, the documentation of matlab's "delaunayn" command seems to indicate as much.


4

This depends on how many dimensions you have an how many points. Also on how the points are structured. If the number of dimensions are high and the points are randomly chosen, you essentially have a Monte Carlo integration procedure. If the points are randomly chosen from a distribution, then you have a kind of importance sampling. If the number of points ...


4

Suppose that you know the orthogonal polynomial basis in a single dimension $(x)$ of each degree $i$ up to some desired order $K$. That is, we know $$p_0(x),p_1(x),...p_i(x),...,p_K(x)$$ To extend this into a two dimensions $(x,y)$, we need only consider the product between 1D polynomials in (x) and (y) and collect only the products whose total degree is ...


4

I don't have time to get all the details down, but maybe this answer can give some helpful intuition. Basically, something that will work is taking the first $N$ binary binary of your number in $[0,1]$ and assigning them as the most significant digits for each of the $N$ dimensions (so, the first decimal digit is the most significant digit in the first ...


3

You want to solve for 3 to 10 particle systems (3D per particle)? As far as I am aware, mean field theories do not work especially well for so few particles, but it seems there has been DFT work on diatomic molecules. Is this a system where Born-Oppenheimer is valid? If so, I might be inclined to expand the electronic wavefunction using a linear combination ...


2

This is really too big a question to ask in a single post, as there are at least two steps that need to be addressed: discretization and eigensolves. The first step is that you need to discretize the partial differential equation that underlies your problem. There are of course many ways of doing this, the finite element and finite differences method being ...


2

Let's consider the integral in 2D. Note that the domain where $f(x_1,x_2) = x_1 + x_2 - K$ is positive lies to the right of the dashed line $x_1+x_2=K$ in the sketch, so to account for the positive part function the domain of integration is restricted to the area to the right of the dashed line (shaded area in the sketch). The integral splits in 1D ...


2

Edit: Unfortunately I can not comment on your question. Perhaps someone can like my answer to get a reputation of 50. That's no joke! Anyway, you may use a Gauss Laguerre quadrature or Gauss Hermite quadrature to calculate it. The quadrature rules are designed for integration kernels of following form: Laguerre quadrature: $$\int_{0}^{\infty} e^{-x} f(x)\,dx ...


2

You may check my answer on Cross Validated. I didn't want to copy it here. Basically, you can use fast, randomized SVD to compute PCA basis and coefficients.


2

The short answer is that you can't do this -- it's outside our computational power today. To explain why, think of just building the box itself, where you have one degree of freedom on each vertex. In 100 dimensions, there are $2^{100}\approx 10^{10}$ vertices. That's not far from the size of the biggest finite element computations, and you'll need on the ...


1

If the index for a bin is some $d$-tuple of integers $\{k_1, \ldots, k_d\}$, one approach that might work is to come up with a hash function $h$ for bin indices and use an unordered_map from the C++ standard template library. It's hard to beat a well-tuned hash table implementation for serial performance. You will of course need a good hash function. The ...


1

Given for each element the indices of its vertices, you can efficiently retrieve face adjacencies by sorting the faces lexicographically, then the pairs of elements that share a face are contiguous in the sorted facet list. The boundary elements are those that have a facet that appears only once in the list. The more detailed procedure is given in my answer ...


1

The main difference is in the dimensionality, that is reflected in the matrices. In 1D, e.g, using (FEM) Finite Element Method or (FDM) Finite Difference Methods the Stiffness (Hamiltonian) matrix is tridiagonal. In 2D, the Stiffness matrix is pentadiagonal for the FDM and octadiagonal for the FEM with linear or bilinear elements, using structured meshes. ...


1

If you have $M$ atoms, your wave function depends on $3M$ variables. If you wanted to discretize this function on a uniform mesh with $N$ nodes in each of these directions (or with $N$ one-dimensional shape functions), you'd need a total of $N^{3M}$ unknowns -- far too many for any interesting number of electrons $M$. To give just one example, if you'd just ...


1

Since your dimension is only 3 for the moment, I guess a naive approach would be to create separate trees for each subspace. I would recommend you to use Flann (http://www.cs.ubc.ca/research/flann/) as it is very well implemented and reported to be quite fast for the datasets of your concern (http://www.cs.ubc.ca/research/flann/uploads/FLANN/flann_visapp09....


1

As @NickAlger alludes, the incremental delaunay approach can scale exponentially with the dimension of the space, even if the final tesselation has few facets. Even if some computable solutions exist for special cases, it's unlikely that any practical algorithms exist for general tesselations, which seems to be what you're looking for.


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