Questions tagged [krylov-method]
Referring to Krylov Subspaces and the methods of solutions to linear systems of equations which exploit these spaces.
84 questions
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Solving linear systems with a clustered spectrum except for 1 eigenvalue
I have a fairly general question, which I don't think depend a lot on my particular application.
Consider a linear system $Ax = b$ that I would like to solve with preconditioned MinRes. Call the ...
- 575
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Preconditioner for "Generalized stokes" problem
Let $\mu>0$. For the Stokes problem,
\begin{align}
-\mu \Delta u + \nabla p &= f
\\
\nabla \cdot u & = 0
\\
u&= 0 \text{ on } \partial \Omega,
\end{align}
after discretization (say, ...
- 191
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Improving convergence rate of krylov schur iterations?
I am trying to implement krylov schur iterations. I am noticing that although my implementation converges, it does so really, really slowly. For a 40x40 matrix it is taking hundreds of iterations to ...
- 473
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How does the Arnoldi iterations algorithm deal with repeated eigenvalues?
The simplest possible matrix I can think of to use an arnoldi method is the identity matrix.
In this case the Krylov sequence is just $\{v, v, v, v, \cdots\}$ for any $v$.
Thus the span of the krylov ...
- 473
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Iterative solvers for problems in solid and structural mechanics
I am looking for comprehensive literature (papers, books, reports etc..) on iterative solvers for solid and structural mechanics problems to understand the best iterative solvers and preconditioners ...
- 984
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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^{-...
- 189
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Why multigrid is inefficient？
I am trying to solve the Stokes equation containing viscosity nonlinearity using the open source finite element software underworld2 with nested PETSc. The resolution is 2000*200. The solution results ...
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How is the Alternating Schwarz Method used as a Preconditioner to a Krylov Method?
I am reading "Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations" (Smith 1996), and I am confused as to how the below Alternating Schwarz algorithm ...
- 269
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1
answer
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Ways to fix block Lanczos tridiagonalisation numerical instability for matrix with degenerate, closely spaced eigenvalues?
I want to run a block Lanczos block-tridiagonalization on a hermitian, sparse matrix (of relatively small size $\sim 10^2 \times 10^2$). However the matrix typically has many eigenvalues that are ...
- 21
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answer
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Lanczos memory complexity for dense matrices
Does the Lanczos algorithm remain memory efficient even if the original Hermitian matrix is dense?
- 11
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Looking for a specific version of the Quasi-Minimal Residual (QMR) method
I'm looking for an alternative formulation of Quasi-Minimal Residual (QMR) from Freund and Nachtigal (1994) based on a Lanczos process for complex valued matrices based on $A^H$ instead of $A^T$.
...
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Does the choice of a complex inner product affect Krylov methods?
As far as I understand there are two definitions of the complex inner product:
$$(a,b) = b^H a$$
and
$$(a,b) = a^H b$$
I know some linear algebra libraries such as BLAS and Eigen uses the second one.
...
1
vote
1
answer
95
views
Reason for why apparent acceleration of algebraic multigrid solve by addition of positive definite diagonal matrix
In passing I was told by someone that $K^{\prime}\in\mathbb{R}^{n\times n}$, will be easier to solve by an algebraic multigrid preconditioned conjugate gradient (CG-AMG) solver than $K$, where $K$ is ...
- 189
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Global convergence behavior of several Krylov solvers in scipy.opt
In the context of mechanical simulation, where I solve the stationary action principle directly (i.e. $\nabla S = 0$ for some scalar function $S$), I use the wrapper ...
- 243
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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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