# 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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### Why is my iterative linear solver not converging?

What can go wrong when using preconditoned Krylov methods from KSP (PETSc's linear solver package) to solve a sparse linear system such as those obtained by discretizing and linearizing partial ...
3k views

### 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 (...
277 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. ...
459 views

### 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 ...
2k views

### Can a Krylov subspace method be used as a smoother for multigrid?

As far as I am aware, multigrid solvers use iterative smoothers such as Jacobi, Gauss-Seidel, and SOR to dampen the error at various frequencies. Could a Krylov subspace method (like conjugate ...
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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 ...
997 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 ...
275 views

### 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} :=& ...
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 ...
261 views

### 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 ...
414 views

### 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, ...