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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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 $V(k,l)$ ...
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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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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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Iterative solver for high order DG methods (3D Laplace problem)
I have a 3D Laplace problem on quite a complicated geometry where I am using Discontinuous Galerkin method. My mesh is composed by hexas, hence I am employing classical tensor product basis functions $...
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How conservation of momentum is ensured in (Projected) Gauss-Seidel constrain solver
I'm developing molecular dynamics where my time-step is limited by stiffness of the bonds. I trying to get inspiration from game-engines, where they solve similar problem (hard bond constrains). These ...
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Approximately, at any given time, what proportion of the world's total HPC resources are dedicated towards inverting matrices?
I had heard in a lecture, perhaps 15 years ago, that the vast majority of the world's HPC resources were dedicated to solving linear systems by iterative methods. I seem to remember it was 90%. I can'...
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Implementing matrix term version of Gauss-seidel
I am trying to implement the below description from Ch. 11 of Heath's "Scientific Computing An Introductory Survey" of the Gauss-Seidel iterative method for solving a system of linear ...
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Stochastic power iteration for generalized eigenvalue problems?
Suppose $\mathbf{x}$ is a random variable in $n$ dimensions, and $u$ is a vector. How can I estimate the following quantity in an online fashion?
$$f(x)=\max_{\|u\|=1} \frac{ E\left[\langle u\cdot x\...
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How can I find the current for a nonlinear electrical circuit using the Shockley equation, in Octave?
For this electrical circuit:
The voltage $ V_D $ can be found by solving a nonlinear equation:
$$ \frac{V_{DD}}{R} - \frac{V_D}{R} - I_se^{V_D/V_T} = 0 $$
In this example, let $R=1000$, $V_T = 0.025$,...
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Iteration counts of AMG solver changes in parallel
I am solving the linear elasticity equation within a FEM library with a complex 3D geometry. The resulting linear system is solved with CG, preconditioned by AMG (Algebraic Multigrid). The computed ...
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Algorithm to solve system of nonlinear equations
I would like some tips in figuring out a good algorithm to find the solution of the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some ...
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Equilibrium position finding with DSM
I've coded a framework that can be used to simulate the dynamic behavior of a system discretized by particles (nodes) that are connected by spring-damper elements. However, I want to compare it to a ...
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Professional orbital simulation software methods, accuracy, runtime
I'm working on an orbital body simulation as a personal project. I understand that it's not going to be as good as professional software, but I'm wondering what contemporary,
general purpose, ...
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