# Questions tagged [preconditioning]

For questions regarding design and implementation of preconditioners for solving linear systems.

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### Does this partial eigen-expansion have a name?

This question is a follow-up to this one. Let $A\in \mathbb{R}^{n\times n}$ be large, sparse, symmetric and positive definite. Suppose for I already know $m<n$ eigenpairs of $A$, corresponding to ...
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### Best solver/preconditioner for least-squares finite element method

I have seen a lot of literature, lecture videos, etc. on solvers/preconditioners for non-symmetric and/or indefinite systems. However, now I want to solve the mixed poisson/Darcy equation using the ...
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### Do black-box preconditioners for matrix-free methods exist?

Jacobian-Free Newton-Krylov (JFNK) methods, and Krylov methods in general, can be very useful because they don't require explicit storage or construction of a matrix, only the results of matrix-vector ...
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### Solving a large non-hermitian generalised eigenvalue problem from a linear stability analysis using SLEPc

I have a generalised matrix problem: $A x = \lambda B x$ from a spectral method on a linear stability analysis problem. My matrix B is diagonal and positive semi-definite. A is non-hermitian and ...
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### Dual time stepping for fluid dynamics

I'm attempting to implement the Weiss and Smith preconditioner in an existing finite volume code and I am struggling with the idea of dual time stepping. My inner time steps are predictor-corrector, ...
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### What type of matrices is approximate inverse preconditioner $||I - AM||_F$ well suited for?

Take sparse approximate inverse preconditioner $M \approx A^{-1}$ given by solution of $$\underset{M \in S}{\mbox{min}} \; ||I - AM||_F,$$ where $S$ is a set of sparse matrices and $||.||_F$ is the ...
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I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -... 2answers 344 views ### Efficient preconditioner for Augmented Lagrangian I want to solve a non-linear problem with non-linear equality constrains and I'm using a augmented Lagrangian with a penalty regularization term that, as well known, spoils the condition number of my ... 2answers 2k views ### Which preconditioners (and solver) in PETSc for indefinite symmetric systems should I use? My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow): \begin{pmatrix}A & B^T \\ B & C\end{pmatrix} where$C = 0$is the typical case (I have even ... 1answer 469 views ### Extracting diagonal of an approximately diagonal matrix when we don't know its entries What is a good way to extract the diagonal from a symmetric matrix that is already almost diagonal, but where you don't have the matrix elements (only the ability to apply it to vectors)? Further ... 2answers 365 views ### Existence of incomplete cholesky factorization What is the current state of research on the existence of incomplete cholesky factorizations (in the context of preconditioning) for symmetric positive definite matrices? I wonder in particular ... 1answer 631 views ### What is the current state of polynomial preconditioners? I wonder what has happened to polynomial preconditioners. I am interested in them, because they appear to be comparatively elegant from a mathematical perspective, but as far as I have read in surveys ... 1answer 590 views ### Using algebraic multigrid for preconditioning convection-diffusion operators I implemented a Navier Stokes based on FEM discretization and PETSc for solving the linear system of equations. To create an efficient solution procedure, I follow the paper "Efficient ... 2answers 1k views ### Taxonomy of ILU preconditioners I learned that for BiCGStab solver for sparse linear systems it's pretty much always necessary to use a preconditioner. I realized by now that choosing a good one is problem dependent. Surfing the ... 2answers 952 views ### How can I precondition a non-linear problem before linearization? When I think of solving non-linear equations, I generally think of linearizing first, then applying a preconditioner to the linear matrix. The thought occurred to me that it might be possible to ... 3answers 3k views ### What guidelines should I use when searching for good preconditioning methods for a specific problem? For the solution of large linear systems$Ax=b$using iterative methods, it is often of interest to introduce preconditioning, e.g. solve instead$M^{-1}(Ax=b)$, where$M$is here used for left-... 2answers 2k views ### How does matrix scaling influence linear solvers? For instance, in MUMPS there is option to scale matrix s.t. all rows/columns have the same norm. This claims to decrease condition number and improve numerical properties of the matrix: ftp://cuter.rl.... 3answers 621 views ### On Vanilla Preconditioners for solving dense$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: ...
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### 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.