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1500 questions
9
votes
1 answer
Forcing an ODE solver to preserve the norm
I have an ODE of the form
$$ \frac{dy}{dt} = -i H y \enspace .$$
where $y$ is a complex vector and $H$ is a time dependent Hermitian matrix.
The norm of the solution $y(t)$ at any point in time should be 1, but due to accumulation of small…
Krastanov
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- 4
9
votes
3 answers
Construction of $C^1$/$H^2$-conforming finite element basis for triangular or tetrahedral mesh
In the paper Hierarchical Conforming Finite Element Methods for the Biharmonic Equation, P. Oswald claimed Clough-Tocher type elements has $C^1$-continuity while being a cubic polynomial on each triangle. He didn't give a set of explicit basis…
Shuhao Cao
- 2,552
- 17
- 30
9
votes
3 answers
What is the fastest opensource implementation of Bessel functions computation?
I'm looking for an open-source (to use and learn from) software which computes Bessel functions of integer order of real argument to double precision the fastest among all such implementations. Currently I've tried Boost.Math and GSL. From these GSL…
Ruslan
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9
votes
2 answers
How does weak convergence feel, numerically?
Consider, you have a problem in an infinite dimensional Hilbert or Banach space (think of a PDE or an optimization problem in such a space) and you have an algorithm that converges weakly to a solution. If you discretize the problem and apply the…
Dirk
- 1,738
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9
votes
2 answers
Estimate Norm of a black-box functional
Let $V$ be a finite-dimensional vector space with norm $\|\cdot\|$ and let $F : V \rightarrow \mathbb R$ be a bounded linear functional. It is only given as black-box.
I would like to estimate the norm of $F$ (from above and below). As $F$ is a…
shuhalo
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- 1
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- 31
9
votes
6 answers
Super C++ optimization of matrix multiplication with Armadillo
I'm using Armadillo to do very intensive matrix multiplications with side lengths $2^n$, where $n$ can be up to 20 or even more. I'm using Armadillo with OpenBLAS for matrix multiplication, which seems to be doing a very good job in parallel cores,…
The Quantum Physicist
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9
votes
2 answers
Is there a generalization of the Sylvester Inertia Law for the symmetric generalized eigenvalue problem?
I know that in order to solve symmetric eigenvalue problem $Ax = \lambda x$, we can use the Sylvester Inertia Law, that is the number of eigenvalues of $A$ less than $a$ equals the number of negative entries of $D$ where diagonal matrix $D$ comes…
Willowbrook
- 601
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- 10
9
votes
2 answers
Does the matrix condition number affect accuracy of iterative linear solvers?
I have a rather specific question regarding the condition number. I run FEM simulations which have multiple length scales to them which results in a huge disparity between the largest entries and the smallest entries in my matrix. The condition…
CraigJ
- 113
- 6
9
votes
1 answer
Optimal use of Strang splitting (for reaction diffusion equation)
I made a strange observation while computing the solution to a simple 1D reaction diffusion equation:
$\frac{\partial}{\partial t}a=\frac{\partial^2}{\partial x^2}a-ab$
$\frac{\partial}{\partial t}b=-ab$
$\frac{\partial}{\partial t}c = a$
The…
Thomas Klimpel
- 2,141
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9
votes
1 answer
Guidelines for nested preconditioners
Consider the situation where you want to solve a linear system using a preconditioned Krylov method, but applying the preconditioner itself involves solving an auxiliary system, which is done with another preconditioned Krylov method.
On one…
Nick Alger
- 3,143
- 15
- 25
9
votes
3 answers
Numerics: How do I renormalize the following ODE
This question is more about how to tackle a problem numerically.
In a small project I wanted to simulate the coorbital motion of Janus and Epimetheus. This is basically a three body problem. I choose Saturn to be fixed at the origin, let $r_1$ and…
bios
9
votes
4 answers
Generating Symmetric Positive Definite Matrices using indices
I was trying to run test cases for CG and I need to generate:
symmetric positive definite matrices
of size > 10,000
FULL DENSE
Using only matrix indices and if necessary 1 vector
(Like $A(i,j) = \dfrac{x(i) - x(j)}{(i+j)}$)
With condition number…
Inquest
- 3,394
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9
votes
0 answers
Fast algorithms to solve Markov Decision Processes
In my master thesis I used an Algorithm called Approximative Dynamic Programming [1] to solve equations of the form
$$
\max_{\pi}\mathbb{E}^{\pi}\left\{\sum_{t=0}^{T}\gamma^tC_t^{\pi}(S_t,A_t^{\pi}(S_t))\right\}.
$$
It uses Monte-Carlo sampling…
Reza
- 191
- 2
9
votes
1 answer
Intuition behind Alternating Direction Method of Multipliers
I've been reading a lot of papers on ADMM lately, and also tried to solve several problems using it, in all of which it was very effective. In contrast to other optimization methods, I can't get a good intuition as to how and why this method is so…
olamundo
- 599
- 4
- 14
9
votes
2 answers
Evaluating oscillatory integrals with many independent periods and no closed forms
Most methods for oscillatory integrals I know about deal with integrals of the form
$$ \int f(x)e^{i\omega x}\,dx $$
where $\omega$ is large.
If I have an integral of the form
$$ \int f(x)g_1(x)\cdots g_n(x)\,dx, $$
where $g_k$ are oscillatory…
Kirill
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