Questions tagged [iterative-method]
A method which produces a sequence of numerical approximations which converges (provided technical conditions are satisfied) to the solution of a problem, generally through repeated applications of some procedure. Examples include Newton's method for root finding, and Jacobi iteration for matrix-vector solves.
312 questions
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Solving linear systems with a clustered spectrum except for 1 eigenvalue
I have a fairly general question, which I don't think depend a lot on my particular application.
Consider a linear system $Ax = b$ that I would like to solve with preconditioned MinRes. Call the ...
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Euler method and energy drift for a simple harmonic oscillator
I was numerically solving ordinary differential equations and I encountered some very interesting properties.
The forward Euler method for a simple harmonic oscillator diverges, it has energy drift ...
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Multigrid with Jacobi iteration seems to converge wrong
I've been working on a simple multigrid solver that uses Jacobi iterations to solve the Poisson equation as a little exercise. What I'm finding, however is that my solver doesn't seem to converge, or ...
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Generalizable Preconditioners for Solving Linear Equations with Positive Definite Matrices. $Ax=b$
For large positive definite matrices $A$, conjugate gradients is the method of choice for solving linear systems
$$Ax=b.$$
Convergence of conjugate gradients heavily relies on having a good ...
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Why does MOM fail for large number of subdivision for finite dipole?
In Constantine A. Balanis' book about antennas, he introduced the method of moments for current distribution over a finite dipole.
I found that the method of moments works very bad for a half-...
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How to Integrate in Energy Domain for a Tight-Binding System?
I am working on evaluating the integral:
$
I(\vec{r}) = \int f(k) e^{i\vec{k} \cdot \vec{r}} d\vec{k},
$
for a system with a tight-binding dispersion relation given by:
$
\epsilon_k = -2t \left[\cos(...
1
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0
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ILU Preconditiner Implementation in Python
I have a question regarding the computational complexity of the ILU preconditioner in Python. I am trying to implement an ILU(0) preconditioner using the following code:
ILUfact = sla.spilu(...
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How does the Arnoldi iterations algorithm deal with repeated eigenvalues?
The simplest possible matrix I can think of to use an arnoldi method is the identity matrix.
In this case the Krylov sequence is just $\{v, v, v, v, \cdots\}$ for any $v$.
Thus the span of the krylov ...
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Diagonal and Upper-Triangular pre-conditioning for Jacobi
I am interested in analyzing convergence of the Jacobi method to solve the linear system $Ax=b$,
$$\begin{pmatrix}
2 & 4 \\
1 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
=
\...
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1
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Efficient Solver for Solving a Large Linear System Sequentially of a Positive Definite Matrix
In my case, I am solving $AX=B$ repeatedly, but the solution usually doesn't change much. So it'd probably be faster than me when I start from the previous solution and iterative, rather than solving ...
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Preconditioner Implementation with matrix-free methods (sparse iterative solvers)
How can I define preconditioners (SPILU, SPAI, etc.) for sparse iterative methods (TFQMR, GMRES, CGS, etc.) for the matrix-free left-hand side? I defined $Ax=b$ using matrix-free $A$ (with ...
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Computational efficiency of Galerkin projection in AMG
I have been using recently AMG as preconditioner for CG with several meshes for simple elliptic problems discretised with linear elements on "complicated" three dimensional geometries and I ...
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0
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Iterative PDE solver for 1D Burgers equation
I am looking for an Iterative Numerical PDE solver for 1D Burgers equation. I need to have access to the intermediate solutions of the Numerical Solver. By iterative methods, I mean techniques which ...
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Is there a fast matrix-free inverse power iteration?
Problem: I want to solve the eigenvalue problem
$$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 ...
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Flexible Conjugate Residual
If we want to use variable preconditioning in Conjugate Gradient, we can replace the Fletcher–Reeves by the Polak–Ribière formula (https://en.wikipedia.org/wiki/Conjugate_gradient_method#...
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