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.

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1answer
52 views

The definition of asymptotic convergence?

What is the difference between convergence and asymptotic convergence? Why say the convergence is asymptotic?
2
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2answers
105 views

How many generations does it typically take for a differential evolution method to reach a global optimum?

For differential evolution methods in optimization, how many generations does it typically take to reach a global optimum? How do we know if the values are never going to converge?
0
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0answers
12 views

Convergence criterion for overset grids

If there are two overset grids, how do you decide whether convergence is reached or not? What I did was, after interpolating from one grid to another, I check the rms of conservative variables and if ...
3
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1answer
93 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + ...
2
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2answers
107 views

What are some reasons that Conjugate Gradient iteration does not converge?

I would greatly appreciate it if you could share some reasons the Conjugate Gradient iteration for Ax = b does not converge? My matrix A is symmetric positive definite. Thank you so much! Edit with ...
2
votes
1answer
131 views

How to test convergence of an algorithm for constrained optimization

I am applying an iterative method (projected newton) to an optimization problem. Theoretically, the method should converge linearly. I would greatly appreciate it if you could share how should I test ...
0
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0answers
15 views

convergence interval of an infinite series without the general term

I am trying to find the convergence of an infinite series of which I do not have the nth term. Instead of applying the ratio test for the nth term, I divided the first two terms, then the next two and ...
7
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1answer
128 views

Finite element convergence rates for mixed problems

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh. ...
1
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0answers
49 views

Stationary phase approximation for an integral with infinity saddle points?

I need a hand with the numerical evaluation, in Mathematica, for this integral: $$f(t)=\int_{-\infty}^\infty Exp\{it(\omega_H-\omega_l-\omega_k) - \sum _{j\neq(l,k)} S_j [1-e^{-it\omega_j}]\}\, dt$$ ...
1
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0answers
106 views

BFGS Fails to Converge

The model I'm working on is a multinomial logit choice model. It's a very specific dataset so other existing MNLogit libraries don't fit with my data. So basically, it's a very complex function ...
3
votes
0answers
29 views

Solving ODE boundary problem with additional conditions

I need to solve two point bondary problem for ODE. The solution of the problem itself is very easy. The real issue is that when solving arising non linear system of equations it always converges to ...
1
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1answer
101 views

Problem with convergence of Jacobi iterative algorithm

I'm dealing with Jacobi iterative method for solving sparse system of linear equations. For small matrices it works well and gives right answers even if matrix is not strictly diagonal dominant, ...
4
votes
0answers
105 views

Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
8
votes
0answers
587 views

Comparing Jacobi and Gauss-Seidel methods for nonlinear iterations

It is well known that for certain linear systems Jacobi and Gauss-Seidel iterative methods have the same convergence behavior, e.g. Stein-Rosenberg Theorem. I am wondering if similar results exist for ...
4
votes
1answer
123 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
1
vote
1answer
948 views

Numerically determining convergence order of Euler's method

I need to numerically determine the convergence order of Euler's method for various step-sizes. I am unsure how to go about doing this. Here is the question: Problem statement: $\frac{dy}{dt}=\alpha ...
11
votes
3answers
204 views

Computing slightly oscillatory series to high precision?

Suppose I have the following interesting function: $$ f(x) = \sum_{k\geq1} \frac{\cos k x}{k^2(2-\cos kx)}. $$ It has some unpleasant properties, like its derivative not being continous at rational ...
3
votes
1answer
126 views

Excluding roots from a system of nonlinear equations

I have a system of nonlinear equations of which I know it has a single root I am interested in, and has a continuum of roots I am not interested in. I am currently using Newton with line-searching in ...
0
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0answers
63 views

Increase convergence of non-linear equations resulting from ODEs

I am trying to solve a set of couple ODEs: $V_l(r) - r W_l(r) - f1(r) W_l' = 0\tag 1$ $r^2 h''_l(r) + f2 r h_l'(r) + f3 h_l(r) - f4 U_l(r) = 0 \tag 2$ $\kappa (U_l + h_l) + V_{l+1} + W_{l+1} = ...
2
votes
1answer
70 views

Does Lanczos have trouble with large matrix elements?

I have a large, yet very sparse, matrix that I'd like to diagonalize. Both my own Lanczos implementation and the ARPACK that's built in with scipy fail to converge properly, though. I know that my ...
3
votes
1answer
96 views

What is the origin of the preasymptotic convergence behaviour in FEM?

When you have fine-scale features (e.g. boundary layers) in the solution, its FEM approximation on coarse meshes converge at strange apparent rates. Looking at Cea's lemma, is this behaviour because ...
1
vote
3answers
106 views

Guess the final term of a converging series [closed]

I have a non-linear equation that converges, and reaches suitable accuracy after around 20 steps, however each step is very expensive to calculate. The series are never quite the same, but they are ...
3
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0answers
62 views

analytic or numeric integral of diverging function

I'm trying to carry out the following integral numerically $$\int_{r_\mathrm{in}}^{r_\mathrm{out}} \Sigma\left(r'\right) \frac{r'}{r} \left( \frac{1}{r-r'}\, E(L) + \frac{1}{r+r'}\, K(L) \right) ...
2
votes
1answer
135 views

Error norm in finite difference calculation

I've used an explicit finite difference scheme to model the 1D time dependent temperature distribution in a friction weld. I want to now verify the consistency and convergence of my algorithm. I have ...
6
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3answers
337 views

Significance of p-convergence studies

Consider a method (e.g., FEM) with variable approximation order $p$. Now, we know that the optimal order of convergence is given by $$e = C h^{p+1},$$ where $h$ denotes the mesh size and a constant ...
4
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0answers
130 views

Understanding the meaning of Computational Order of Convergence

I am a postgraduate student with interest in numerical methods for solving nonlinear systems of equations. I have read some papers that discussed about 'computational order of convergence' for some ...
7
votes
2answers
315 views

Convergence/stagnation of BiCGStab(l)

I am solving 3D time-harmonic Maxwell FDFD problems (which result in huge sparse linear systems) using BiCGStab(l). I have tried out a bunch of different methods and for my specific use case, it seems ...
2
votes
2answers
74 views

Normalizing error when data passes through zero

Given a time-dependent measure of error, $e(t)$, and a time-dependent function to normalize it with, $f(t)$, one would report the normalized error as $$E(t) = \left|\frac{e(t)}{f(t)}\right|.$$ ...
2
votes
2answers
143 views

Does energy decrease with basis set size in density functional theory?

Based on the variational principle, one might expect that the ground state energy of a density functional theory (DFT) calculation will decrease as the basis set size increases. (As I understand it, ...
1
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0answers
47 views

Hatree-Fock, reasons for convergence/ non-convergence

I'm new here so please forgive me if I lack proper stack exchange etiquette. So, I was wondering if anyone here could provide insight on a problem that I am running into with with a Hartree-Fock ...
4
votes
0answers
65 views

Non-convergance when calculating temperature/heat flows through a section of rock

I am attempting to calculate temperature of section of rock in the earth as a function of vertical position in the rock and time. Along with it I am calculating the heat flow through the rock as a ...
0
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0answers
44 views

Question about Logarithmic convergence

I examine the following recursion $X_{n+1}=\frac{t_n}{t_{n+1}}X_n+\frac{Y_n}{t_{n+1}}$ where $X_n,Y_n$ are positive finite random variables and $t_n$ the time. I have shown that $\lim_{n \to \infty} ...
14
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1answer
563 views

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 + ...
2
votes
1answer
188 views

Is it possible to ensure global convergence of a fixed point iteration?

Suppose I have a fixed point iteration of the form $$x_{n+1}=f(x_n).$$ Suppose further that after some initial testing, I found that it does not converge to an a priori known fixed point $x^*$. I ...
2
votes
1answer
249 views

Convergence of GMRES

From what I understand the GMRES method is (using Arnoldi Iterations/Modified Gram-Schmidt): The first vector of the Krylov subspace span of A is the normalized vector $\frac{\vec b - A\vec x_0} {|| ...
2
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0answers
332 views

Newton Iteration method convergence

I wrote a Python code which solves a second degree nonlinear differential equation using the Newton iteration method. The code converges to a 2-cycle within 50 or so iterations. The cycle only ...
1
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0answers
962 views

Stationary solution converge but time dependent doesn't

I've coupled a COMSOL model for fluid dynamics with a very simple pde that model the transport of humidity in air. When I solve it for the stationary case, the solution converge easily, but when I ...
2
votes
4answers
212 views

What is the meaning of “preasymptotic” and “superconvergent”?

Precisely the title of the question. I have encountered these terms in two areas: conjugate gradient method, and adaptive finite elements.
1
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3answers
322 views

necessary and sufficient tests to show order of convergence for the numerical method

I would like to know what are the necessary and sufficient tests one has to perform in order to show the convergence of the algorithm. I have not found a good reference to state for that as I am ...
4
votes
1answer
139 views

Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde

Suppose I have the parabolic system $$u_t=\nabla\cdot(k(x)\nabla u)+f,\quad (x,t)\in\Omega\times I$$ with dirichlet boundary conditions $$u=g, \quad x\in\partial\Omega$$ and initial condition ...
9
votes
2answers
880 views

Strategies for Newton's Method when the Jacobian at the solution is singular

I'm trying to solve the following system of equations for the variables $P,x_1$ and $x_2$ (all else are constants): $$\frac{A(1-P)}{2}-k_1x_1=0 \\ \frac{AP}{2}-k_2x_2=0 \\ ...
10
votes
3answers
1k views

Understanding the “rate of convergence” for iterative methods

According to Wikipedia the rate of convergence is expressed as a specific ratio of vector norms. I'm trying to understand the difference between "linear" and "quadratic" rates, at different points of ...
7
votes
1answer
155 views

Demonstrating that the time step size is small enough in a code with automatic step size selection

I recently inherited a large body of legacy code that solves a very stiff, transient problem. I would like to demonstrate that the spatial and temporal step sizes are small enough that the ...
2
votes
1answer
135 views

Root Convergence rate of Iterative Scheme

I have an iterative sequence for optimizing an EM (Expectation Maximization) algorithm based loss function $L(X)$ with $t$ being the iteration number as: $X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is ...
5
votes
2answers
1k views

When to stop Gauss-Seidel-iterations?

I want to have an estimation, that my solution has an error, let's say less than 1e-8. Usually, I stop the Gauss-Seidel algorithm, when the residual is "small enough" and this is already the problem. ...
7
votes
2answers
268 views

Basin of attraction for Newton's method

Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution. What is "sufficiently close"? Is there literature ...
2
votes
2answers
306 views

Finite volume solution of electrostatics using magnetic vector potential

I would like to solve for the electric potential and magnetic vector potential using the finite volume method (collocated grid). My equations are: $\nabla\cdot(\sigma\nabla\phi)=0$ $\nabla \cdot ...
3
votes
1answer
745 views

Gauss-Seidel iterations node spacing

I am working on an assignment where I am determining the temperature distribution of a chip on a substrate. When I decrease the nodal spacing the results change drastically. The smaller the nodal ...
5
votes
2answers
205 views

Proving convergence of 5 point scheme for the Poisson equation

So, we are solving the Biharmonic equation ($\Delta^2 u = f$) on a rectangle by solving the Poisson equation ($\nabla^2 u = f$) two times. We have nice boundary conditions, $u = 0$ and $\Delta u = 0$ ...
5
votes
1answer
444 views

Which iterative linear solvers converge for positive semidefinite matrices?

I want to know which of the classic linear solvers (e.g Gauss-Seidel, Jacobi, SOR) are guaranteed to converge for the problem $Ax=b$ where $A$ is positive semi definite and of course $b \in im(A)$ ...