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 ...
1answer
182 views

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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2answers
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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 ...
1answer
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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 ...
1answer
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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, ...
1answer
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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 ...
3answers
439 views

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: ...
5answers
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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.
2answers
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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 ...
1answer
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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 ...