Questions tagged [gmres]
Refers to the (G)eneral (M)inimal (RES)idual algorithm, which is a popular Krylov subspace method for solving linear systems.
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Struggling to understand the Householder Arnoldi Algorithm
I have spent almost three frustrating weeks trying to understand the Householder GMRES algorithm. I understand the basic GMRES algorithm the maths of Householder transformations used in QR ...
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GMRES implementation does not converge for singular Hermitian problems
I've just implemented the GMRES algorithm based on chapter 4 of Fundamentals of Numerical Mathematics for Physicists and Engineers using the problems in Numerical Analysis by Timothy Sauer for ...
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Matrix Free alternatives in dealii
I am implementing a Fast Multipole Method (FMM) in deal ii.
I do not want to store a dense matrix, but lower rank matrices and to use matrix free methods. By now, I store the elements of the low-rank ...
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Preconditioned GMRES for nearly diagonalizable systems
I have been working with a matrix $A$ and preconditioner $P\approx A^{-1}$ that I've then applied GMRES to the (left) preconditioned linear system
\begin{equation}
P^{-1}Ax=P^{-1}b
\end{equation}
$P^{-...
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Comparing block versus non-block Krylov methods for handling multiple right-hand-sides
Suppose I wish to solve a linear system $AX=B$ iteratively where $A$ is an $m\times m$ matrix and $X,B$ are $m \times s $ matrices (not single vectors). Instead of solving $s$ independent systems I'm ...
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When do not use preconditioners for sparse linear system of equations?
I'm implementing a solver of Finite Element Method, and to solve the linear system of equations I'm using gmres from MKL of Intel. Exists the option with and without a preconditioning. In what case it ...
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Help with debugging block GMRES
I have written block version of GMRES by referring [1] and MATLAB implementation of gmres. I need to write it for complex matrices. My block implementation when run on single RHS is giving correct ...
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Doubt regarding GMRES(m) and preconditioned GMRES
I have the two following algorithms for GMRES(m) and left preconditioned GMRES.
GMRES(m)
Left preconditioning
I would like to know if anyone could explain why steps 10 through 12 are not used in the ...
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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 ...
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relres in gmres MATLAB
I think the relres in MATLABis the form that relres = norm(M(b-Ax))/norm(M\b),when it smaller than tol then stop the iteration.
I want to know how to change relres to norm((b-Ax))/norm(b). Or use ...
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Solving PDEs: What is the best way to deal with non-banded/dense jacobians?
I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
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Why minimizing with respect to A-norm?
Assume solving the linear system $A \textbf x = \textbf b$, with an $A$ so large that nothing but iterative methods may be employed. Assuming $A$ induces a norm, I realized that it is often desired to ...
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How GMRES method finds smallest singular value and the corresponding singular vectors of a matrix?
https://stackoverflow.com
Krylov solvers for iterative computation of the smallest singular value and the corrensponding singular vectors of a matrix
Edit:
This is a follow-up question to How to ...
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Library to solve dense linear system with GMRES
I have a fortran 90 code and I want to solve a dense linear system with GMRES. I would prefer the restarted GMRES with preconditioning. Is there some library that you know of that I could use? Now I ...
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Arnoldi Decomposition Algorithm
I try to get into GMRES via Arnoldi-Decomposition. For my understanding, I Implemented the Arnoldi-Decomposition in python.
...
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How to implement flexible gmres in matlab?
About the flexible GMRES (fgmres), we know that it is a variant of right preconditioned gmres. And the robust command gmres in matlab as follows:
...
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How to compute the computational cost and storage of the Full Orthogonalization Method?
About the analysis of Full Orthogonalization Method (FOM) in Prof. Saad's book, wrote as follows:
Algorithm 6.4 (FOM):
\begin{array}{l}
r_0=b-Ax_0,\beta=\|r_0\|_2,v_1 = r_0/\beta\\
Define \quad H_m ...
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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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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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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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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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How does gmres method iteration behave for this non-diagonalizable matrix?
Recently, I have been studied my lessons about gmres iteration, probably the most popular iteration method for general large sparse linear system of equations Ax=b. And the convergence is obtained ...
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Number of GMRES iterations increase when stepping forward in time, using the Newton method
I am solving a system of nonlinear time-dependent equations using the Newton method in a finite-element-setting, i.e. first I create the jacobian matrix for the current time, and afterwards I try to ...
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How to pass matrices to parallel workers quickly in matlab?
I am trying to solve many different linear systems in parallel in matlab. The problem is, each linear system has entirely different parts and are fairly large, so passing the information to each of ...
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How to set an initial guess for the iterative solver in Comsol?
How to set the initial guess for the iterative solver GMRES or FGMRES for linear problems (Helmholtz equation of RF module) in Comsol?
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How to construct an effective preconditioner for this particular problem
A quick introduction to my problem
I am currently developing a method for simulation of water waves in three dimensions based on potential flow theory. The computational bottleneck of the method is ...
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GMRES vs Newton-GMRES for Solving nonlinear PDE's
Often when numerically solving nonlinear PDE's using method of lines approach with an implicit integrator a system of nonlinear equations have to be solved.
To be more specific, let's say we have ...
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What are the major differences between GMRES and FOM?
I am reading Professor Saad's "Iterative Methods for Sparse Linear Systems" (2nd edition).
The basic algorithm for FOM is given on page 166 and the basic algorithm for GMRES is given on page 172.
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Operation count for GMRES
One can use GMRES as it is, but there is also a version of GMRES called k-step restarted GMRES, which is used for large matrices, where $k$ is some fixed number of steps after which we take a new $x_0$...
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Preconditioned residual converges, but true residual doesn't
I'm using Albany w/ Trilinos to solve an elasticity problem with thermal expansion mismatch. I'm using block GMRES with MueLu preconditioning. It works for problem size of several million dofs, but ...
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Why do we need orthonormal basis of Krylov subspaces for GMRES?
The GMRES method is for solving the linear system $Ax=b$.
Given an initial guess $x_0$ and the corresponding residual $r_0:=b-Ax_0$, we have the Krylov subspace
$$\mathcal{K}_m:=\mathop{span}\{r_0,...
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Preconditioner for the GMRES method in the Uzawa algorithm
I'm trying to solve
\begin{equation}\left\{
\begin{split}
\frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\frac1\rho\nabla p&=f\;\;\;\text{in }\Lambda\\
u&=0\;\;\;\text{on }\partial\...
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Preconditioner for dense matrix "with diagonal predominance"
For a CFD panel-based potential method, I'm trying to reduce the time to solve the linear system. The matrix has the larger values on the diagonal, since the influence of a panel on itself is maximum, ...
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GMRES : incomplete Krylov-subspace
At each iteration $i$ of the GMRES method, is calculated a single new orthonormal vector of the existing Krylov subspace. If the norm of that vector is 0 (or close to 0), then the subspace is "...
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Preconditioning technique for large sparse non-hermitian matrix
I am attempting to solve a computational acoustics problem that involves solving an underlying sparse matrix. The size of the problem varies with grid size (3D) and fill-in's obviously make direct ...
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A numerical GMRES example
I'm having trouble understanding how GMRES works. I've read the part in Saad's book and a few others but still I am confused. Can someone provide me a numerical example to understand it better? Or if ...
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What factors are relevant when deciding between GMRES and schur complement solves?
Suppose I have a linear system
$$
\left\lbrack
\begin{array}{cc}
M_1& S\\
S^{\mathrm{T}}& M_2
\end{array}
\right\rbrack \left\lbrack \begin{array}{c} X\\ Y\end{array} \right\rbrack=
\...
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MATLAB: code for restarted gmres
I have a question about Matlab and restarted gmres. I would like to use gmres.m provided here. This code seems to be popular for the scientific computation newcomer....
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OpenMP threaded nonlinear solver for complex numbers
Problem:
I have translated Jacobian-Free Newton-Krylov solver written by
C. T. Kelley to Fortran and now want to parallelize it on a shared-memory system with OpenMP. In addition, I want to ...
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High quality flexible GMRES (FGMRES) implementation
What are the best FGMRES implementations in various languages/frameworks? In particular, are there any good quality Matlab implementations?
I am referring to the variation of GMRES where a changing ...
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GMRES: Making the matrix square without solving for boundaries
How do we define the matrix for GMRES, if we do not want to solve the boundary elements but only the interior ones.
I am using pentagonal elements so in a row there are 6 elements (cell itself + 5 ...
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Preconditioning a Krylov method with another Krylov method
In methods like gmres or bicgstab it could be attractive to use another Krylov method as a preconditioner. After all they are easy to implement in a matrix-free way and in a parallel environment. For ...
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Not enough memory for GMRES
After realizing that Gauss-Seidel is extremely slow for my simulation, i wanted to try GMRES and luckily found the C++ code here without diving into the theory. The size of the matrix in my case is <...
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GMRES implementations
I am looking for simple GMRES implementations. Unfortunately, I have a few requirements:
should be C or Fortran, Fortran preferred
needs a license that permits inclusion in an MIT/BSD licensed ...
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Problems where Conjugate gradient works much better than GMRES
I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD (symmetric-positive-definite) because it requires less ...
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Parameter selection difficulty in GMRES
we are using a first order implicit finite volume code for simulation of incompressible flows. At its core, the code utilizes a (non-preconditioned) GMRES method for solving linear systems given in ...
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GMRES Matlab 'tol' parameter
I need to use GMRES solver in MATLAB, and I need to play around with the codes parameters and I had a very simple question about its usage.
The documentation of the solver here mentions a parameter <...
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Best choice of solver for a large sparse symmetric (but not positive definite) system
I am presently working on solving very large symmetric (but not positive definite) systems, generated by some certain algorithms. These matrices have a nice block sparsity which can be used for ...
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Cusp Library performance worse than PETSC (GMRES 200 iterations) Why?
I wanted to compare the speeds of the GMRES implementations in the CUSP and the PETSc libraries.
The matrix (A) used for testing was a 3d Laplacian matrix obtained by using the 7 point stencil on a ...
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