Most Popular
1500 questions
18
votes
4 answers
Why can't Householder reflections diagonalize a matrix?
When computing the QR factorization in practice, one uses Householder reflections to zero out the lower portion of a matrix. I know that for computing eigenvalues of symmetric matrices, the best you can do with Householder reflections is getting it…
Victor Liu
- 4,480
- 18
- 28
18
votes
1 answer
Why are Octrees used for Multipole space decomposition?
In most (all?) implementations of the Fast Multipole Method (FMM), octrees are used to decompose the relevant domain. Theoretically, octrees provide a simple volumetric bound, which is useful for proving the O(n) runtime of a FMM. Beyond this…
Ben Thompson
- 476
- 3
- 8
18
votes
5 answers
20% performance penalty for a nice software design
I'm writing a small library for sparse matrix computations as a way to teach myself to make the best use of object-oriented programming. I've worked really hard on having a nice object model, where the parts (sparse matrices and the graphs which…
Daniel Shapero
- 10,263
- 1
- 28
- 59
18
votes
3 answers
Efficient computation of the matrix square root inverse
A common problem in statistics is computing the square root inverse of a symmetric positive definite matrix. What would be the most efficient way of computing this?
I came across some literature (which I haven't read yet), and some incidental R code…
tchakravarty
- 345
- 3
- 8
18
votes
1 answer
How can wavelets be applied to PDE?
I would like to learn how wavelet methods can be applied to PDE, but unfortunately I do not know a good resource to learn about this topic.
It seems that many introductions to wavelets focus on interpolation theory, e.g., assembling a signal by a…
shuhalo
- 3,660
- 1
- 20
- 31
17
votes
5 answers
Databases of results for numerical codes
In the numerical methods literature, many research papers consist of a description of a new algorithmic variation, followed by a few test problems comparing the new method with one or two existing methods. This makes it difficult to determine
How…
David Ketcheson
- 16,522
- 4
- 54
- 105
17
votes
1 answer
Can a Krylov subspace method be used as a smoother for multigrid?
As far as I am aware, multigrid solvers use iterative smoothers such as Jacobi, Gauss-Seidel, and SOR to dampen the error at various frequencies. Could a Krylov subspace method (like conjugate gradient, GMRES, etc.) be utilized instead? I don't…
Paul
- 12,045
- 7
- 56
- 129
17
votes
8 answers
Parsing protein structure data in C
My background is in genomics, but I have recently been working with problems related to protein structure. I wrote a few relevant programs in C, building my own PDB file parser from scratch in the process. I didn't worry about making a really robust…
Daniel Standage
- 663
- 5
- 17
17
votes
1 answer
How do you debug numerical code, what could be source of this oscillatory error?
Quiet a lot of insight can be gained form experience, I was just wondering if anybody has seen something similar to this before. The plot shows the initial condition (green) for the advection-diffusion equation, then the solution at iteration 200…
boyfarrell
- 5,409
- 3
- 35
- 67
17
votes
1 answer
How should boundary conditions be applied when using finite-volume method?
Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh,
I would like to apply a Robin type boundary condition to the l.h.s. of the domain ($x=x_L)$, such that,
$$
\sigma_L = \left( d u_x…
boyfarrell
- 5,409
- 3
- 35
- 67
17
votes
1 answer
Convergence rate of FFT Poisson solver
What is the theoretical convergence rate for an FFT Poison solver?
I am solving a Poisson equation:
$$\nabla^2 V_H(x, y, z) = -4\pi n(x, y, z)$$
with
$$n(x, y, z) = {3\over\pi} ((x-1)^2 + (y-1)^2 + (z-1)^2 - 1)$$
on the domain $[0, 2] \times [0, 2]…
Ondřej Čertík
- 2,930
- 18
- 40
17
votes
3 answers
Desktop software with HPC resources for back end number crunching
Our workgroup produces a desktop application that simulates building energy performance. It is a .NET application and when the user is running a lot of simulations, they can be quite time consuming. The simulations are totally parallelizable, and we…
Tangurena
- 576
- 6
- 18
17
votes
5 answers
Are there operator splitting approaches for multiphysics PDEs that achieve high order convergence?
Given an evolution PDE
$$u_t = Au + Bu$$
where $A,B$ are (possibly nonlinear) differential operators that don't commute, a common numerical approach is to alternate between solving
$$u_t = Au$$
and
$$u_t = Bu.$$
The simplest implementation of this…
David Ketcheson
- 16,522
- 4
- 54
- 105
17
votes
2 answers
Stopping criteria for iterative linear solvers applied to nearly singular systems
Consider $Ax=b$ with $A$ nearly singular which means there is an eigenvalue $\lambda_0$ of $A$ that is very small. The usual stop criterion of an iterative method is based on the residual $r_n:=b-Ax_n$ and regards the iterations can stop when…
Hui Zhang
- 1,319
- 7
- 16
17
votes
3 answers
Confusion about Armijo rule
I have this confusion about Armijo rule used in line search. I was reading back tracking line search but didn't get what this Armijo rule is all about. Can anyone elaborate what Armijo rule is? The wikipedia doesn't seem to explain well. Thanks
user34790
- 473
- 1
- 3
- 8