Questions tagged [convergence]
Questions related to whether the sequence of iterates generated by an iterative method has one or more limit points, and if those limit points have the correct properties.
208
questions
-1
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1
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60
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optimal gradient algorithm to determine best $α_k$
Let's consider an optimal-step gradient algorithm and assume that:
$g(α) := f(X_k - α∇f(X_k)) = 2α^2-4α+17$, how can we determine the optimal $α_k$?
Here is my simple solution:
$g(α) = 2α^2-4α+17$
$g'(...
- 15
0
votes
1
answer
42
views
step-fixed algorithm first iterates
let us have the fixed-step gradient algorithm, with $p = 2$ and we assume that for $X = (x, y)$,
$∇ f(X) = \begin{pmatrix}
x -1\\
y -2
\end{pmatrix}$
Let me assume we intialize with $X_0 = (0,0)$ what ...
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0
votes
1
answer
59
views
step-fixed algorithm to minimize f, which step to ensure convergence?
If we want to apply the fixed-step gradient algorithm to the minimization of $f(x) = \frac{1}{2}(Ax, x)$ where $A$ is a symmetric 2x2 matrix with eigenvalues $\lambda_1 > \lambda_2 > 0$, for ...
- 15
0
votes
1
answer
38
views
Estimating the rate of convergence of Projected Gradient Descent on constrained polynomial objectives
I am estimating the order of convergence of Projected Gradient Descent (GD) on quadratic polynomials with random coefficients independently drawn from Uniform(-1,1) and bounded by a unit hypercube. I'...
- 23
1
vote
1
answer
127
views
How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE
Setting
I am solving for $u(x,y,t)$ the wave equation $\partial_{tt} u - \partial_{xx} u = f$ on $(x,y)\in\Omega=[0,1]\times \mathbb{R}$ by splitting it into an equivalent first order system:
$$\...
- 13
0
votes
0
answers
110
views
Is there a fast matrix-free inverse power iteration?
Problem: I want to solve the eigenvalue problem
$$x=Ax$$
to the eigenvalue $1$ for a large matrix (roughly $N^3\times N^3$ and $N$ ranges from 10 to 100) where $A$ is stochastic (i.e. all entries are ...
0
votes
0
answers
51
views
Calculating a 2D Ewald sum for a multipolar expansion
I am attempting to calculate the potential of a particle at the center of an infinite two-dimensional lattice as per the following reference:
Reference: Lambin, PH & Senet, P. Ewald Summation of ...
- 43
3
votes
1
answer
186
views
Stability of Euler forward method
I am trying to solve a linear system of ODEs of the form:
$$ \frac{du}{dt} = A u, \quad u(0)=k$$
where $A$ is a 2x2 matrix and $u(t)$ is a 2x1 column vector. I want to solve this numerically, using ...
- 31
3
votes
1
answer
150
views
Role of rotation's pivot point in optimization?
In this paper, the authors describe how to use locally rigid transformations (sampled on nodes in space) to deform mesh vertices. In the paper, rotations are relative to the pivot point, which ...
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1
vote
2
answers
79
views
Why is the definition of convergence different for root finding algorithms as compared to sequences?
The definition of convergence for root finding algorithms is given in a few sources as: A sequence ${x^k}$ generated by a numerical method is said to converge to the root $\alpha$ with order $p\geq 1$ ...
- 11
2
votes
1
answer
297
views
Why do we use modified pressure in incompressible multiphase solvers with gravity?
The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ...
0
votes
0
answers
66
views
Faster convergence for minimizing least squares of forward modelling problems
This specific question was raised from optimizing parameters of column experiments in the hydrogeological context. I want to optimize a parameter of interest (in this case $D$), based on experimental ...
- 23
3
votes
1
answer
244
views
Convergence-test for ODE approximates wrong limit
I am trying to numerically solve a differential equation but I am having trouble getting the convergence test to run properly. The problem is as follows:
Consider an ODE
$$y'(t) \enspace = \enspace f(...
- 185
6
votes
1
answer
318
views
Why FEM for incompressible materials is ill-posed?
I am an engineer who is trying to get a deeper understanding of FEM. I have been using the Zienkiewicz texts as my bible. It touches on the issue of incompressibility but I need a more intuitive way ...
- 201
0
votes
0
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77
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Convergence rate for not smooth solution with classical $P^1$ Lagrangian FEM
I'm using classical $P^1$ finite elements to solve $- \Delta u = f$ with Dirichlet BC in a 2D domain $\Omega$.
I know from theory that the solution is not in $H^1$ for my particular choice of $f$, so ...
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