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# 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.

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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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### Solving for expectation using iteration in a implicit function

For a implicit function $V(k,l)$, taking $l$ as given and $k$ to be the only variable, $k$ is sampling from an unknown distribution and $\mathbb{E}k = \bar{K}$. Using Taylor expansion on $V(k,l)$ ...
88 views

### 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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1 vote
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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 ...
40 views

### 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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### Eigenvectors of Laplacian

I am studying introduction to Multigrid methods. In all tutorials, authors write that eigenvectors of Laplacian (1D, finite difference) are given as $w_k(x_i) = \sin(k \pi x_i),$ where $x_i$ is a ...
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1 vote
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### Memory issues with iterative solvers

Was trying to implement a poisson 2d solver using Conjugate Gradient Method, so from 10x10 grid the matrix becomes 100x100 (since we have 100 nodes to find the values at), 100x100 grid goes to ...
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### Solving Kepler's Equation with Newton-Raphson Method

Note (2022/03/07): This question is solved. Unfortunately, I'm not able to accept the correct answer by Lutz Lehmann, because I screwed up my registration and the account which posted this question is ...
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### Simulate circular mold spread using cellular automata - square emerges instead

I am trying to simulate the spread of mold in a petri dish using a cellular automata based approach. Thanks to the answer in my other question Stochastic cellular automata - algorithm limited by 1 ...
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### Stochastic cellular automata - algorithm limited by 1 cell per timestep

Context Let's say I am trying to model the spread of mold in a petri dish, using a stochastic cellular automata approach. The petri dish can be thought of as a grid of 1mm x 1mm squares, each called ...
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203 views

### Best search algorithm for optimal weight factor in SOR method

I had written an algorithm that searches for the optimal weight parameter to be implemented in the successive-over relaxation (SOR) method which worked cleanly by vectorizing the interval and for ...
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### How to obtain smallest eigenvalues with Arnoldi iteration

I understand that the Arnoldi iteration produces a basis which tends to include in its span the eigenvectors corresponding to eigenvalues of large magnitude (hence the analogy between the last vector ...
Are there iterative methods for the solution of nonsymmetric linear systems $Ax=b$ that can take (theoretical or practical) advantage from knowing that $A+A^T$ is positive definite? These matrices are ...