Referring to Krylov Subspaces and the methods of solutions to linear systems of equations which exploit these spaces.
7
votes
2answers
63 views
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 ...
4
votes
1answer
120 views
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 ...
4
votes
3answers
92 views
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 ...
7
votes
3answers
227 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 ...
4
votes
2answers
107 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 ...
8
votes
1answer
138 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 ...
8
votes
1answer
131 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 ...
9
votes
1answer
92 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 ...
6
votes
1answer
126 views
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 ...
12
votes
3answers
367 views
Solving $(G^TA^{-1}G)x = b$ without inverting $A$
I have matrices A and G. A is sparse and is nxn with n very large (can be on the order of several million.) G is an nxm tall matrix with m rather small (1 < m < 1000) and each column can only ...
0
votes
1answer
237 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 ...
8
votes
1answer
197 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 ...
6
votes
2answers
504 views
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. ...
13
votes
3answers
438 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 ...
8
votes
1answer
254 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 ...
4
votes
3answers
453 views
Krylov Subspace Methods for Dense Systems
I am currently researching on the viability of using KS methods for solving large dense systems. What I wish to prove (or disprove) is that methods like CG, BiCG and QMR are as good (if not better) ...
9
votes
2answers
1k views
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 ...
