Questions tagged [krylov-method]
Referring to Krylov Subspaces and the methods of solutions to linear systems of equations which exploit these spaces.
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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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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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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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Implementation of Lanczos method that returns tridiagonal matrix
The Lanczos method can be used to obtain extremal eigenpairs of sparse symmetric or hermitian matrices. I know there are several implementations of the Lanczos method (as well as Arnoldi, Davidson, ...
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Why does GMRES converge much slower for large Dirichlet boundary conditions?
I'm trying to numerically solve a simple Laplace equation in 2D, with a nonlinear source term:
$\nabla^2 u = u^2$
with boundary conditions as $u=0$ everywhere except for $y=1$ where $u=u_0$. I'm ...
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Solving nonlinear PDE with finite difference based on Newton-Krylov
I am now working on solving MHD equations with finite difference method, which include nonlinear equations:
$$
\frac{\partial\rho}{\partial t}+\nabla\cdot\left[\left(\rho_0+\rho\right){v}\right]-\...
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241 views
What are the major differences between GMRES and FOM?
I am reading Professor Saad's "Iterative Methods for Sparse Linear Systems" (2nd edition).
The basic algorithm for FOM is given on page 166 and the basic algorithm for GMRES is given on page 172.
...
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How to find a good preconditioner to the system $(A^T A + \lambda I) x = A^T b$?
The system in the title has a damper factor $\lambda > 0$ and the matrix $A$ is sparse and rectangular, with a structure I can exploit to solve matrix vector products very fast. My current solver, ...
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Questions about iterative projection methods in Saad book
I am reading Chapter 5 of Saad's iterative methods book, and I don't understand section 5.2.1 about the two propositions of optimality results.
In the statements of the propositions, what does it mean ...
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216 views
Is reduced stiffness matrix positive definite too?
The stiffness matrix $K$ in a finite element analysis is a symmetric positive definite matrix. When we introduce essential boundary conditions, we remove rows and columns associated with prescribed ...
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Name for vectors in a Krylov space but not the preceding one
It seems to me that a useful concept to define when studying Krylov subspace methods is the idea of a vector that belongs to a Krylov subspace $\mathcal{K}_{n+1}(A,b)$ but not to the preceding one $\...
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Why do we need orthonormal basis of Krylov subspaces for GMRES?
The GMRES method is for solving the linear system $Ax=b$.
Given an initial guess $x_0$ and the corresponding residual $r_0:=b-Ax_0$, we have the Krylov subspace
$$\mathcal{K}_m:=\mathop{span}\{r_0,...
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For which problems Krylov subspace methods are preferred over multigrid methods?
As multigrid methods are known to have grid independent convergence rates with $O(N)$ computational cost, then why would one be interested in using Krylov subspace methods at all, for which ...
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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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Iterative single variable solutions in large linear systems
I have a system where $A$ is a large $n\times n$ marix with fast MVMs. It may have many nonzero entries (albeit in a structured way so as to allow fast MVMs), and is not necessarily diagonally ...
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Generalized eigenvalue with null space
Define $S\in\mathbb{R}^{n\times n}$ as
$$S:=H+Q^\top V^{-1} Q.$$
$H,V$ are positive semidefinite. Here, $H$, $Q$, and $V$ are large, dense matrices but they are structured: I can write code for ...
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PetSc vs Sundials for serial numerical computations?
I am currently working on a physics problem that turns into a non-linear boundary value problem. I need an efficient numerical solver that I could run on my laptop with i5 dual core CPU. I am ...
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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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role of initial guess for iterative linear solver
Suppose we use a preconditioned iterative solver for a linear system. If the initial state for the solver can be chosen very close to the exact solution - does this reduce requirements for the ...
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Why does conjugate gradient work with this nonsymmetric preconditioner?
In this previous thread the following multiplicative way to combine symmetric preconditioners $P_1$ and $P_2$ for the symmetric system $Ax=b$ was suggested:
\begin{align}
P_\text{combo}^{-1} :=& ...
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Conjugate gradient: the 1-norm of the residual
I am trying to solve $Ax=b$ using the conjugate gradient method. However, it is important to me to obtain a bound not only on the usual residual $||b-Ax_k||_2$ but also on the quantity $||b-Ax_k||_1$. ...
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Does Conjugate Residual really have convergence properties similar to that of Conjugate Gradient?
I have coded up a toy implementation of Conjugate Residual and have been testing it. Both wikipedia and the Saad claim that Conjugate Residual and Conjugate Gradient have similar convergence behavior....
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preconditioned Uzawa method with Petsc
I am trying to improve the resolution of a Stokes problem (P2/P1 on unstructured mesh) defined by the matrix $M$:
$M=
\begin{pmatrix}
A_u & 0 & B_u \\
0 & A_v & B_v\\
B_u^T & B_v^...
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240 views
Appropriate iterative linear solver for an eigenvalue problem
I'm trying to solve a generalized eigenvalue problem
$$Ax = \lambda Bx, \quad A = A^\top > 0,\; B = B^\top > 0$$
with $\lambda \approx \sigma$ using Rayleigh Quotient Iteration (RQI) (RQI is ...
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1answer
185 views
Choosing preconditioner for unsymmetric pressure-velocity coupled system
I'm working with pressure-velocity coupled systems. It means that instead of solving 4 different linear systems in segregated approach (1 for pressure and 3 for Ux, Uy, Uz), we can solve only one ...
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1answer
209 views
Trust-region Newton: implementation issue with Conjugate Gradient calculations
UPDATE: The problem turned out to be the step (refer penultimate paragraph below) where I was factoring out a small value from the vectors of the numerator and denominator and then computed dot ...
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831 views
How is Krylov-accelerated Multigrid (using MG as a preconditioner) motivated?
Multigrid (MG) may be used to solve a linear system $Ax=b$ by constructing an initial guess $x_0$ and repeating the following for $i=0,1..$ until convergence:
Compute the residual $r_i = b-Ax_i$
...
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Search Direction in Conjugate Gradient
Could you help me with a Conjugate Gradient question? In using CG to solve Ax=b, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous direction ...
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146 views
Guidelines for nested preconditioners
Consider the situation where you want to solve a linear system using a preconditioned Krylov method, but applying the preconditioner itself involves solving an auxiliary system, which is done with ...
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Algorithm to calculate the exponential of an Hessenberg matrix
I am interested in computing the solution of a lage system of ODEs using a krylov method as in [1]. Such method involve functions related to the exponential (the so-called $\varphi$-functions). It ...
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preconditioning a krylov method with another krylov method
In method 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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Wanted: sequences of linear systems for recycling Krylov solver analysis
In the solution of sequences of linear systems
$$A_ix_i=b_i\quad\text{for}\quad i=1,2,\dots$$
with Krylov subspace methods, data can be recycled from already solved linear systems in order to speed up ...
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Is there any way to do “double preconditioning”
Question:
Suppose that you have two different (factored) preconditioners for a symmetric positive definite matrix $A$:
$$A \approx B^TB$$
and
$$A \approx C^TC,$$
where the inverses of the factors $B, ...
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Convergence/stagnation of BiCGStab(l)
I am solving 3D time-harmonic Maxwell FDFD problems (which result in huge sparse linear systems) using BiCGStab(l). I have tried out a bunch of different methods and for my specific use case, it seems ...
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Are there any libraries out there that implement block Krylov subspace methods?
Question
Are there libraries out there that implement block Krylov subspace methods? (I was not able to find any from a simple Google search.)
Background
Right now, I am working with a code that ...
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Convergence of adaptive finite elements with inexact solves
I'm working on some adaptive discontinuous Galerkin codes for time harmonic wave propagation, currently just Helmholtz, but will be branching out once I have a working prototype in this case.
There ...
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Which preconditioners (and solver) in PETSc for indefinite symmetric systems should I use?
My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow):
\begin{pmatrix}A & B^T \\ B & C\end{pmatrix}
where $C = 0$ is the typical case (I have even ...
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2answers
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Are there any specialized methods available for solving structurally symmetric sparse linear systems?
When solving $Ax=b$, prior knowledge about $A$'s structure can help in designing an efficient solver which exploits this information (e.g conjugate gradient method is to be used when $A$ is ...
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3answers
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PRIMA gives an unstable result?
I am working with Modified Nodal Admittance representation of circuits. I am doing Model Order Reduction using PRIMA on MATLAB. I am considering these circuits as Descriptor State-Space systems.
I ...
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4answers
843 views
Calculating determinant while solving $Ax=b$ using CG
I am solving $Ax=b$ for a huge sparse positive definite matrix $A$ using the conjugate gradient (CG) method. It is possible to compute the determinant of $A$ using the information produced during the ...
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2answers
267 views
Krylov subspace iterative methods in floating point arithmetic
Is there any work that considers Krylov subspace iterative methods in floating point arithmetic? I'm especially interested in how rounding errors influence the convergence and the accuracy of the ...
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1answer
991 views
Why is pinning a point to remove a null space bad?
A Poisson equation with all Neumann boundary conditions has a single constant dimensional null space. When solving via a Krylov method, the null space can be removed either by subtracting the mean of ...
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568 views
What is the current state of polynomial preconditioners?
I wonder what has happened to polynomial preconditioners. I am interested in them, because they appear to be comparatively elegant from a mathematical perspective, but as far as I have read in surveys ...
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How do low rank modifications affect Krylov method convergence?
Say I have a linear system $A x = b$, which converges quickly using a suitable Krylov method (such as CG or GMRES) for all $b$. If $B$ is a matrix with low rank $r$, will the same Krylov method on ...
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Identifying the name/provenance of a technique to find the nullspace vectors of a matrix by random sampling and the conjugate residual method
I have got a large sparse matrix $A \in \mathbb R^{n \times n}$ and I want to find non-trivial elements in the kernel/nullspace of this matrix. How can this be done? I would like to learn more about a ...
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Solving $(G^TA^{-1}G)x = b$ without inverting $A$
I have matrices $A$ and $G$. $A$ is sparse and is $n\times n$ with $n$ very large (can be on the order of several million.) $G$ is an $n\times m$ tall matrix with $m$ rather small ($1 \lt m \lt 1000$) ...
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1answer
942 views
A Comparison between GMRES, QMR and LU for Dense Matrices
As I see it, there are 3 ways to solve unstructured dense system of equations: GMRES, QMR and LU.
Has anyone done a comparison for these three? As far as I know, LU is the preferred choice and it is ...
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1answer
430 views
How to establish that an iterative method for large linear systems is convergent in practice?
In computational science we often encounter large linear systems which we are required to solve by some (efficient) means, e.g. by either direct or iterative methods. If we focus on the latter, how ...
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Understanding OpenCL performance
I'm using ViennaCL's interface to Eigen as a way to leverage OpenCL. Specifically, I'm using the ::viennacl::linalg::bicgstab_tag with an Eigen sparse matrix. ...
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What is the principle behind the convergence of Krylov subspace methods for solving linear systems of equations?
As I understand it, there are two major categories of iterative methods for solving linear systems of equations:
Stationary Methods (Jacobi, Gauss-Seidel, SOR, Multigrid)
Krylov Subspace methods (...
