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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45 views

When will the Orthomin/CG iteration fails

I know that the the Conjugate Gradient iteration fails when $0\in \mathcal {W}(A^{H})$, which means there's a complex vector $x+iy$ such that $(x+iy)^{T}A^{H}(x+iy)=0$. I wonder how to derive a real ...
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How to derive the simplified Newton iteration in the TR-BDF2 ODE integration scheme

The Problem The TR-BDF2 explained in this paper [1], is quite a popular numerical scheme used to integrate $\dot{y} = f(t,y)$, consistent of the following two stages: \begin{align} y_{n+\gamma} &...
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How can I color my Mandelbrot set like this?

I have a background image of a fractal on my phone that I would like in a higher resolution with super sampling, and decided to write my own program for it. I've got down rendering a Mandelbrot set, ...
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Calculating Error for Poisson Equation using Successive Over-Relaxation technique, Python

I am trying to solve the Poisson Equation $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 32(x(x-1) + y(y-1))$ for a 61x61 grid using Python3 with boundary conditions being $T=...
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120 views

When is it easy to invert a sparse matrix?

(Crossposted on cstheory.SE) When is it easy to invert a sparse matrix? Specifically, I'm wondering about the cases in which matrix inversion has similar cost to sparse matrix multiplication, hence ...
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58 views

Is there any function to calculate condition number of sparse matrix in Eigen libraray?

The function JacobiSVD and BDCSVD can calcuate condtion number of a dense matrix via singular values. However I need to know condition number of a sparese matrix due to slow computation speed using ...
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On the reordering of sparse matrices

I have been reading on different techniques used to reorder sparse matrices to achieve better performance, the most popular being the Cuthill-McKee or Reverse Cuthill-McKee algorithm. Most of those ...
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53 views

2-norm of solution update suddenly becomes zero after a few iterations

I am trying to solve the Poisson equation in 2D for heterostructure devices. I have linearized the equation and discretized it using FDM. I am using BiCGStab to iteratively solve for the solution as ...
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1answer
44 views

Effecient method for iterating over sparse dataset

Apologies if this isn't the appropriate forum for this question. I have a set of elements that I need to iterate over as part of a modeling workflow. The elements exists over a set of dimensions (i, ...
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57 views

1-D Conduction Steady state heat transfer using FD

I have tried to solve 1-D Conduction Steady state heat transfer problems in Matlab (see below). Here is the 1-D model: T''[x] == 0, T[0] == 100, T'[100] == 0 ...
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The error in SOR algorithm suddenly falls to zero when it reaches 1e-7 range

I am solving the Poisson equation for heterojunction using Fortran90. I use the SOR algorithm to arrive at the potential profile. I see the weird behavior where the error (the difference between the $...
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Is it possible to predict solution oscillation before solving the system by looking at coefficient matrix?

Question When it is about solving a system of equations, is it possible to predict that whether high-frequency noise (e.g. checker-boarding) is likely to appear in the converged solution by looking at ...
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Project to nearest point on convex polyhedron

I have a point $y \in \mathbb{R}^d$ and a convex polyhedron $\mathcal{P}$ given as the intersection of half-spaces: $$\mathcal{P} = \{x \in \mathbb{R}^d \mid a_1 \cdot x \le b_1, \dots, a_n \cdot x \...
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Preconditioning the $[1 \quad-2 \quad 1]$ Finite Difference matrix

Let $A$ be the well known tridiagonal matrix coming from the 1D Finite difference discretization of the Laplacian, with stencil $\frac{[1 \quad-2 \quad 1]}{h^2}$. The system $Ax = b$ is very large, so ...
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Is there an iterative solver for dense matrices with possible zero diagonal entries?

Is there an iterative solver that can handle potentially zero entries on the central diagonal? I am implementing a polynomial fitting algorithm (up to $10^{th}$-order) and my matrix is a "...
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1answer
56 views

2D heat equation in both steady state and Transient state using iterative solvers

While solving a 2D heat equation in both steady-state and Transient state using iterative solvers like Jacobi, Gauss seidel, SOR. Should the answers, I mean the converged results of Temperature ...
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Without positive definiteness, does an iterative solver work?

Question Does lacking positive definiteness of the matrix of coefficients in a system of equations, make using iterative solvers impractical? Description Using the finite volume method, I have ...
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fixed point iteration on DD method

I have to solve the the problem $u_t+\Delta^2u=f(u)$, where $f(u)$ is non-linear, using domain-decomposition method. My approach is first using fixed point iteration on mixed form i.e to say $u^{k+1}...
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Why does smoothed aggregation multigrid method used as preconditioner in conjugate gradient slows down the solution time?

I'm solving a system of linear equations obtained from the FEM discretization of a simple linear elasticity problem on a cube with zero displacements at one plane and a load on the opposite one. The ...
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2answers
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Solution of symmeric/non-symmetric linear system

I would like to understand what happens in the following: I have a really simple Poisson problem, in 1D, with $u_0 = u_N = 0$. I assembled the stiffness matrix and the right-hand side, and I applied ...
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Bifurcation points on homotopy path by numerical continuation?

I am trying to implement an algorithm that finds (possibly) all solutions of a system of nonlinear polynomial equations $$F(X) = 0$$ I thought about using the (convex) homotopy $$H(X,T) = TF(X) + (1-T)...
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Calculating coarse grid matrix in geometric multigird

The coarse grid matrix is calculated via RAP where R,P are the restriction and interpolation matrix,respectively.By checking a typical MG algorithm I want to ask how to calculate efficiently coarse ...
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1answer
38 views

Fixed-point iteration when image and domain are not the same

I have a function $f(x)$ defined on a domain $D$, but such that the image $f(D)$ may contain extra regions not included in its domain. I am interested in solving the fixed-point equation $x=f(x)$. If ...
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MInimizing cost function using iterative search for a minimum method

I want to estimated the parameters $\ \hat{\theta} $ of a model using an iterative search for the minimum of a cost function. The cost function is defined as follows: $$ V_N(\hat{\theta}) = \frac{1}{...
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How to implement the Hessenberg QR Algorithm?

For context, I'm creating a linear algebra library from scratch for learning purposes in C. Right now I'm working on calculating eigenvalues but my implementation of the QR Algorithm is diverging. ...
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Solving a nonlinear problem with a very small components with finite element method

In solving nonlinear hyperelastic solid mechanics problems, to converge to the correct solution we need to do step-by-step loading which makes the deformation at each step very small (for my ...
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What online optimisation algorithm can be used for a noisy cost function?

I am trying to optimise a function, but the function can be noisy and give varying results for the same parameters. Furthermore, it needs to be online, as the data from each new iteration happens ...
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1answer
95 views

Best way to check if SOR solution has converged for 2d matrix

I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10. I can obtain the fully converged solution for ...
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How to solve for f(A)x=b without GMRES?

How to solve for $f(A)x=b$? For GMRES, an answer is given in this book chapter: http://link.springer.com/chapter/10.1007%2F978-3-642-58333-9_2. Ungated version: https://www.researchgate.net/profile/...
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Residual of Poisson equation with periodic boundaries

I am trying to write a multigrid solver for Poisson's equation, $-\Delta u=f$, on the unit square, $\Omega=(0,1)^2$ with periodic boundaries. My primary source has been Multigrid by Trottenberg, ...
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How the number of pre-smoothing and post-smoothing steps affect the asymtotic convergence rate of geometrical Multigrid?

Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately?
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A name for a numerical phenomena when using numerical methods

I have a nonlinear solver for equation $g= c_1f(x_1,y_1)+c_2f(x_2,y_2)$. Note that $c_1$ is much bigger than $c_2$. So after using Levenberg–Marquardt algorithm, I could only get $x_1$, $y_1$ and $...
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Trying to understand splitting-based iterative method for 2D Laplace problem

I am trying to understand the theory behind a splitting based iterative method which uses the incomplete Cholesky factorization. Before giving the specific details, let me first give the problem ...
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How to reproduce the numerical examples in Prof. Saad's Book about Krylov subspace methods?

After reading Prof. Saad' Book, "Iterative methods for Sparse Linear Systems, 2nd version", I want to do the numerical examples about the Krylov subspace methods not only to reproduce the results in ...
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Why MATLAB chooses the Householder in its built-in function gmres.m?

Recently, I have studied how to construct an orthonormal basis for Krylov subspace to solve $Ax=b$, where $A\in \mathbb{R}^{n\times n}$ is nonsingular. As we know, there are usually 4 ways to ...
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How to understand the choice of Krylov subspace orthonormal basis?

This semester, I study the Krylov subspace iterative methods (about Ax=b) using the book H. A. Van der Vorst. Iterative Krylov Methods for Large Linear Systems, volume 13. Cambridge University Press, ...
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203 views

What's wrong with the **PCG and MINRES** in matlab?

Last week, I have learned the details of the robust iterative methods of PCG, MINRES, GMRES, which will converges to the exact solution $x^*$ of nonsingular system within $N$ steps for $A\in \mathbb{R}...
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When should I write a matrix-vector function to handle the sparse matrix vector multiplication?

This semster, I have been studying the iterative methods for large sparse matrix system. But I have some questions. For large sparse matrix, we must use an economic storage to store them. The most ...
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Does the k-th approximate solution of a stationary iteration belong to the k-th Krylov subspace?

For an stationary iteration method solving $Ax=b$ as follows: $$ Mx_k = Nx_{k-1}+b, $$ I have known that when $M = I$, i.e., the Richardson iteration, the k-th solution $x_k = x_{k-1}+r_{k-1}$ is in ...
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What is the standard, extrapolation, and modified version of Richardson iteration method?

I have been studying the iterative methods recently. For classical iterative methods solving $Ax=b$, I have seen that the most simplest iteration method is the so-called "Richardson iteration". But I ...
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Why Krylov subspace iterative methods are faster than classical iteration?

This semester, I have been studying the most popular iterative methods, i.e., Krylov subspace iteration methods. For a large sparse system linear $$ Ax=b, $$ where $A$ is nonsingular, I know that ...
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Why do not we choose the error solution norm as an iterative method's criterion?

For solving linear system $$ Ax=b, $$ using iterative mehods, we often use the terminate criterion as follows: $$ \frac{\|r_k\|}{\|r_0\|}=\frac{\|b-Ax_k\|}{\|b-Ax_0\|}<eps. $$where $x_0$ is the ...
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Why does the initial guess for linear system usually choose by zero vector?

For solving linear system $$ Ax=b, $$ using iterative mehods, we often use the terminate criterion as follows: $$ \frac{\|r_k\|}{\|r_0\|}=\frac{\|b-Ax_k\|}{\|b-Ax_0\|}<eps. $$where $x_0$ is the ...
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68 views

How to implement Geometric Multigrid in non-rectangular grids?

It is quite easy to implement multigrid on a rectangular grid but what about an non-rectangular?How to coarse a non-rectangular grid and apply multigrid(assume an easy non-rectangular grid capital ...
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Subspaces for Iterative methods

In the original paper of Conjugate Gradients, the authors mention that if we pick the canonical basis $\{e_1,e_2,\ldots,e_n\}$, to obtain A-orthonormal vectors, we end up with the Gaussian elimination ...
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Are there any other better methods than block diagnoal and block upper triangular precondtioner for saddle point problems?

For stokes problems, $$ -\Delta \vec{u} + \nabla p =\vec{f},\qquad \nabla . \vec{u} = 0; $$ with appropriate boundary conditions which guarantee there is a unique solution. Using FDM or FEM, ...
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What is the **contraction factor and convergence factor** of a iteration method?

For any iteration method from A=M-N, e.g., $$ Mx_{k+1}=Nx_{k}+b,\quad k=0,1,... $$ we know that it converges iff $\rho(M^{-1}N)<1$. And when it converges, there exists a concept called asymtotic ...
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1answer
173 views

How suitable is multigrid method for time-dependent PDEs?

For elliptic PDEs (Poisson-type), the multigrid method is very sufficient, but how about time-dependent problems (i.e parabolic or hyperbolic PDEs)? Is it efficient to solve such problems using a ...
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Can the standard multigrid performance be used for time-dependent PDEs?

Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ...
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Can a direct method like Thomas be used in a multigrid method as a smoother?

As far as I know, multigrid uses stationary iterative methods as smoothers (i.e GS), but can we use a direct method also? For example, in case we have a tridiagonal system (for example 1D heat ...

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