smh
  • Member for 3 years, 9 months
  • Last seen more than 1 year ago
3 answers
11 votes
512 views
Numerical evaluation of highly oscillatory integral
12 votes

Use Plancherel's theorem to evaluate this integral. The basic idea is that for two functions $f,g$, $$ I=\int_{-\infty}^{\infty} f(x) g^*(x)dx = \int_{-\infty}^{\infty} F(k) G^*(k) dk $$ where $F,G$...

View answer
2 answers
3 votes
132 views
DFT of $g(\omega) \exp(i C \omega^2)$. How to do it ,if uniform sampling requires too much memory?
7 votes

Expanding upon my comment above, if you only need a few digits of accuracy you can probably use the method of stationary phase. We can follow the procedure on Wikipedia. We can write the transform as ...

View answer
1 answers
2 votes
94 views
Computing Series of $ke^{-(x - h)^2}$
6 votes

Depending on how large $n$ can get and how many evaluation points $x$ you wish to use, this summation problem is well-suited to the use of fast multipole methods (FMMs); for instance, see the black-...

View answer
1 answers
1 votes
127 views
Interpolating the gradient of a cylindrically symmetric potential field that's 'supposed to' obey the Laplace equation?
4 votes

Provided that your geometry conforms to the cylindrical coordinate system, the separation of variables solution should look something like $$ \Phi(\mathbf{r}) = \sum_{n,m} a_{nm} \left\lbrace \begin{...

View answer
2 answers
3 votes
69 views
What is a good way to select a small subset (say 50) of items from a large pool of items (say 5 million) while minimizing an objective function?
Accepted answer
4 votes

Simple way: Compute the Euclidean distance between each item and the target feature specification. Sort by distance. Take the first 50 elements. Fast way for arbitrary target features: Since your ...

View answer
4 answers
4 votes
507 views
Find representatives of vector-space in set of vectors?
Accepted answer
2 votes

You can think of each vector as a point in your linear space. As such, we can use a simple quadtree/octree-like algorithm to map your points into boxes, with "nearby" vectors assigned to the same or ...

View answer
1 answers
18 votes
697 views
Why are Octrees used for Multipole space decomposition?
2 votes

The comments above give some very good reasons for using octrees (i.e., recursively halving the computational cube in each dimension as opposed to a more general orthogonal bisection). Symmetry and ...

View answer
1 answers
1 votes
86 views
Cyclic Deconvolution
1 votes

Since convolution can be written as a matrix-vector product $Ax=b$ of a circulant or Toeplitz matrix $A$ acting on a vector $x$, you can invert or pseudoinvert via SVD $A$ to obtain $x=A^{-1}b$. That ...

View answer
2 answers
1 votes
569 views
Vandermonde matrix DG Hestaven
1 votes

You're using the orthonormalized version of the Legendre polynomials, while Hesthaven is not. The polynomials in your matrix are normalized by a factor of $\sqrt{\frac{2n+1}{2}}$, i.e. $\sqrt{\frac{2(...

View answer
1 answers
0 votes
65 views
Clarification regarding 3D FMM translation operators
Accepted answer
0 votes

First, in an FMM implementation, you are always working in some coordinate frame relative to the center of some box, hence the $P-Q$ vector defining the primed coordinate system above. Make sure this ...

View answer
1 answers
1 votes
60 views
Need for translation theorem in Fast Multipole Method
Accepted answer
0 votes

The FMM has been exhaustively studied, so if people aren't doing something that seems obvious, then there's likely a fairly obvious reason they aren't doing it. This isn't to say there's nothing new ...

View answer