For questions regarding design and implementation of preconditioners for solving linear systems.
8
votes
2answers
66 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
2answers
72 views
Existence of incomplete cholesky factorization
What is the current state of research on the existence of incomplete cholesky factorizations (in the context of preconditioning) for symmetric positive definite matrices?
I wonder in particular ...
7
votes
1answer
123 views
Extracting diagonal of an approximately diagonal matrix when we don't know its entries
What is a good way to extract the diagonal from a symmetric matrix that is already almost diagonal, but where you don't have the matrix elements (only the ability to apply it to vectors)?
Further ...
5
votes
1answer
138 views
Using algebraic multigrid for preconditioning convection-diffusion operators
I implemented a Navier Stokes based on FEM discretization and PETSc for solving the linear system of equations. To create an efficient solution procedure, I follow the paper "Efficient preconditioning ...
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 ...
8
votes
2answers
112 views
Efficient preconditioner for Augmented Lagrangian
I want to solve a non-linear problem with non-linear equality constrains and I'm using a augmented Lagrangian with a penalty regularization term that, as well known, spoils the condition number of my ...
5
votes
2answers
257 views
Taxonomy of ILU preconditioners
I learned that for BiCGStab solver for sparse linear systems it's pretty much always necessary to use a preconditioner. I realized by now that choosing a good one is problem dependent.
Surfing the ...
6
votes
2answers
168 views
How can I precondition a non-linear problem before linearization?
When I think of solving non-linear equations, I generally think of linearizing first, then applying a preconditioner to the linear matrix. The thought occurred to me that it might be possible to ...
5
votes
2answers
189 views
How does matrix scaling influence linear solvers?
For instance, in MUMPS there is option to scale matrix s.t. all rows/columns have the same norm.
This claims to decrease condition number and improve numerical properties of the matrix: ...
3
votes
3answers
226 views
On Vanilla Preconditioners for solving dense $Ax=b$ iteratively
I am looking for preconditioners which don't assume anything about the matrix or its origins.
I basically want to be able to type in the following in MATLAB and have quick solving time:
...
9
votes
0answers
149 views
Are there any open source inverse-based multilevel ILU implementations?
I am very impressed with the serial performance of multilevel inverse-based ILU preconditioners, particularly for heterogeneous Helmholtz, but I am surprised to not be able to find any open source ...
15
votes
3answers
354 views
What guidelines should I use when searching for good preconditioning methods for a specific problem?
For the solution of large linear systems $Ax=b$ using iterative methods, it is often of interest to introduce preconditioning, e.g. solve instead $M^{-1}(Ax=b)$, where $M$ is here used for ...
9
votes
5answers
655 views
What is the advantage of multigrid over domain decomposition preconditioners, and vice versa?
This is mostly aimed for elliptic PDEs over convex domains, so that I can get a good overview of the two methods.
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 ...
3
votes
1answer
62 views
Applicability of combinatorial and support preconditioner
There are several correspondences between matrices and graphs, e.g., each matrix is the adjacancy matrix of a weighted graph. The terms support preconditioner or combinatorial preconditioner refer to ...
