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.
0
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0answers
15 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 ...
3
votes
0answers
43 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
39 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} ...
13
votes
1answer
240 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
$$n(x, y, z) = {3\over\pi} ((x-1)^2 + (y-1)^2 + ...
2
votes
1answer
68 views
Is it possible to ensure global convergence of a fixed point iteration?
Suppose I have a fixed point iteration of the form
$$x_{n+1}=f(x_n).$$
Suppose further that after some initial testing, I found that it does not converge to an a priori known fixed point $x^*$. I ...
1
vote
1answer
95 views
Convergence of GMRES
From what I understand the GMRES method is (using Arnoldi Iterations/Modified Gram-Schmidt):
The first vector of the Krylov subspace span of A is the normalized vector $\frac{\vec b - A\vec x_0} {|| ...
2
votes
0answers
151 views
Newton Iteration method convergence
I wrote a Python code which solves a second degree nonlinear differential equation using the Newton iteration method. The code converges to a 2-cycle within 50 or so iterations. The cycle only ...
0
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0answers
59 views
Stationary solution converge but time dependent doesn't
I've coupled a COMSOL model for fluid dynamics with a very simple pde that model the transport of humidity in air.
When I solve it for the stationary case, the solution converge easily, but when I ...
3
votes
4answers
103 views
What is the meaning of “preasymptotic” and “superconvergent”?
Precisely the title of the question.
I have encountered these terms in two areas: conjugate gradient method, and adaptive finite elements.
2
votes
3answers
208 views
necessary and sufficient tests to show order of convergence for the numerical method
I would like to know what are the necessary and sufficient tests one has to perform in order to show the convergence of the algorithm. I have not found a good reference to state for that as I am ...
4
votes
1answer
66 views
Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde
Suppose I have the parabolic system $$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$$ and initial condition
...
9
votes
2answers
238 views
Strategies for Newton's Method when the Jacobian at the solution is singular
I'm trying to solve the following system of equations for the variables $P,x_1$ and $x_2$ (all else are constants):
$$\frac{A(1-P)}{2}-k_1x_1=0 \\ \frac{AP}{2}-k_2x_2=0 \\ ...
10
votes
3answers
307 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
96 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
103 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
370 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. ...
5
votes
2answers
154 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
218 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:
$\nabla\cdot(\sigma\nabla\phi)=0$
$\nabla \cdot ...
3
votes
1answer
581 views
Gauss-Seidel iterations node spacing
I am working on an assignment where I am determining the temperature distribution of a chip on a substrate. When I decrease the nodal spacing the results change drastically. The smaller the nodal ...
5
votes
2answers
133 views
Proving convergence of 5 point scheme for the Poisson equation
So, we are solving the Biharmonic equation ($\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
274 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 $b \in im(A)$
...
8
votes
1answer
153 views
Why do we have to rerun the CFD solver for higher Reynolds number?
I started to learn OpenFOAM from the Cavity tutorial which is provided at the web-site. When experimenting with different Reynolds numbers, in section "2.1.8.2 Running the code", tutorial says to ...
9
votes
1answer
206 views
How to establish that an iterative method for large linear systems is convergent in practice?
In computational science we often encounter large linear systems which we are required to solve by some (efficient) means, e.g. by either direct or iterative methods. If we focus on the latter, how ...
7
votes
3answers
297 views
Why does iteratively solving the Hartree-Fock equations result in convergence?
In the Hartree-Fock self-consistent field method of solving the time-independent electronic Schroedinger equation, we seek to minimize the ground state energy, $E_{0}$, of a system of electrons in an ...
4
votes
2answers
246 views
Sufficient conditions to ensure convergence of the conjugate gradient method
I know that a conjugate gradient method is guaranteed to converge to the solution of a linear system if the matrix is positive definite. I'm working with a family of matrices that have the following ...
13
votes
3answers
466 views
What is the principle behind the convergence of Krylov subspace methods for solving linear systems of equations?
As I understand it, there are two major categories of iterative methods for solving linear systems of equations:
Stationary Methods (Jacobi, Gauss-Seidel, SOR, Multigrid)
Krylov Subspace methods ...
4
votes
0answers
106 views
Richardson extrapolation for strong rate of convergence of SDE
Is it possible to apply Richardson extrapolation with Euler-Maruyama scheme to improve strong rate of convergence of stochastic differential equations?
4
votes
1answer
145 views
Discretization of Classical Density Functional Theory (CDFT) problems
I formulate questions about ion densities in biological and materials problems using Classical Density Functional Theory, as in this paper which is also on the arXiv. The discretization in that paper ...
13
votes
2answers
463 views
How to determine if a numerical solution to a PDE is converging to a continuum solution?
The Lax equivalence theorem states that consistency and stability of a numerical scheme for a linear initial value problem is a necessary and sufficient condition for convergence. But for nonlinear ...
