Questions tagged [preconditioning]

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

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165 views

Difference between explicit and implicit preconditioning

What is the difference between an explicit and implicit preconditioner?
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0answers
55 views

Preconditioning matrix with known spectrum

Assume I know all eigenvalues of a matrix $A$ fall into a certain set $\Omega \subset \mathbb{C}$. Is there any way I can exploit this knowledge to design a preconditioner for $A$? Some further ...
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2answers
642 views

Python environments for AMG and Gauss Seidel as solvers instead of preconditioners

I am working on block preconditioning and seemingly it is common to write customised Krylov solvers for them. Within each solver, the individual block linear system with preconditioners are ...
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1answer
615 views

Preconditioning of two step iteration for dense matrices

I would like to solve a dense linear system the form in python $$ L\left(\boldsymbol{x}\right):=\left[\gamma^+\left[\boldsymbol{A}+\frac{1}{2}\boldsymbol{B}^{-1}\right] +\gamma^-\left[\boldsymbol{A}-\...
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403 views

“Cookbook” about iterative linear solvers and preconditioners

I'm using a lot of linear solvers and preconditioners, but mostly, they are magical black boxes to me. Since I'll also have to implement some of them in future, I would like to learn a bit more, ...
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3answers
771 views

role of initial guess for iterative linear solver

Suppose we use a preconditioned iterative solver for a linear system. If the initial state for the solver can be chosen very close to the exact solution - does this reduce requirements for the ...
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1answer
2k views

Solving linear systems with ill-conditioned matrices

As per suggestions of the people from MathOverflow, I'm reposting my question here: I'm currently trying to solve a linear system $Ax = B$, where the matrix $A$ is ill conditioned (i.e. nearly ...
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2answers
378 views

Why does conjugate gradient work with this nonsymmetric preconditioner?

In this previous thread the following multiplicative way to combine symmetric preconditioners $P_1$ and $P_2$ for the symmetric system $Ax=b$ was suggested: \begin{align} P_\text{combo}^{-1} :=& ...
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1answer
564 views

Which preconditioning for large linear elasticity problem?

The problem I want to solve is the displacement formulation of the linear elasticity : $$ \nabla \cdot \sigma = 0 \quad \text{in} \quad \Omega \\ \sigma = \lambda ( \nabla \cdot u ) I + \mu (\nabla \...
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2answers
130 views

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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1answer
191 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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177 views

preconditioned Uzawa method with Petsc

I am trying to improve the resolution of a Stokes problem (P2/P1 on unstructured mesh) defined by the matrix $M$: $M= \begin{pmatrix} A_u & 0 & B_u \\ 0 & A_v & B_v\\ B_u^T & B_v^...
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223 views

Choosing preconditioner for unsymmetric pressure-velocity coupled system

I'm working with pressure-velocity coupled systems. It means that instead of solving 4 different linear systems in segregated approach (1 for pressure and 3 for Ux, Uy, Uz), we can solve only one ...
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74 views

Non-overlaping Domain decomposition - assemble of Laplacian

I am dealing with following 2-dimensional problem in the unit square domain $S_2$ $$- \Delta u (x,y) = f \ \text{in} \ S_2, \hspace{1.5cm} u(x,y) = 0 \ \text{on} \ \partial S_2$$ where $f$ is ...
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1answer
173 views

What iterative method can effectively solve a linear system with this kind of spectrum

I have a linear system with matrix which eigenvalues are uniformly distributed on the unit circle like this: Is it possible to solve this kind of system effectively by iterative method, maybe with ...
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2answers
323 views

Preconditioning symmetric Schur complement

Consider a $2\times 2$ block matrix and a linear system of equations associated to it: \begin{equation} \begin{pmatrix} - A & B \\ B^t & C \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} ...
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1answer
213 views

Best preconditioner for mixed-poisson problem (RT0 elements)

For a very large mixed-poisson problem with lowest order Raviart-Thomas elements (RT0), I plan on using an iterative solver. However, this kind of problem is not positive-definite (saddle point ...
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1answer
238 views

OpenMP threaded nonlinear solver for complex numbers

Problem: I have translated Jacobian-Free Newton-Krylov solver written by C. T. Kelley to Fortran and now want to parallelize it on shared-memory system with OpenMP. In addition, I want to precondition ...
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1answer
818 views

Avoid arithmetic overflow in matrix multiplication

I am solving the following matrix equation for $\mathbf{x}$: $$(J^{\mathbf{T}}J)\mathbf{x}=J^{\mathbf{T}}\mathbf{r}$$ $J$ is $m\times n$ matrix $\mathbf{x}$ is vector of size $n$ $\mathbf{r}$ is ...
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1answer
363 views

Construct a preconditioner for the linear system $Ax = b$ from a different matrix

When I use PETSc to solve my linear systems, I always use the subroutine PetscErrorCode KSPSetOperators(KSP ksp,Mat Amat,Mat Pmat) where ...
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1answer
148 views

Effect of subdomain topologies on overlapping additive Schwarz?

Is there a reference on the effect of subdomain topology on performance of the overlapping additive Schwarz method for (high order) finite elements? For example, taking subdomains to be vertex ...
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1answer
184 views

Guidelines for nested preconditioners

Consider the situation where you want to solve a linear system using a preconditioned Krylov method, but applying the preconditioner itself involves solving an auxiliary system, which is done with ...
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1answer
323 views

Designing a preconditioner for a very Ill-conditionned matrix

I am a physicist with limited numerical methods knowledge and I am trying to speed up the inversion of a very ill-conditioned problem ($rcond>10^{30}$). The same sparse square matrix is used ...
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89 views

Eigenvalue analysis of preconditioned partial differential operator

today, I encountered a confused problem by accident, but I have no ideas to deal with it fully. The question can be described as follows: for example, when we need to use FDM/FEM to discrete the ...
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1answer
281 views

Why should I renormalize physical variables?

I am working with legacy physical codes and I develop new ones based on the output of them. They all use their own internal normalization of variables (for example all distances are divided by the ...
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1answer
174 views

Heuristic help with preconditioning large system ODEs

I'm looking for some general insight on preconditioning. In particular, relevant references/resources/comments would be greatly appreciated. Note, I have been through some of the literature, but am ...
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1answer
2k views

High quality flexible GMRES (FGMRES) implementation

What are the best FGMRES implementations in various languages/frameworks? In particular, are there any good quality Matlab implementations? I am referring to the variation of GMRES where a changing ...
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2answers
825 views

Preconditioning a Krylov method with another Krylov method

In methods like gmres or bicgstab it could be attractive to use another Krylov method as a preconditioner. After all they are easy to implement in a matrix-free way and in a parallel environment. For ...
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1answer
4k views

GPU-accelerated libraries for solving sparse linear systems

There are several libraries to (iteratively) solve large sparse linear equation systems in parallel on a number of CPUs. Our parallel cluster also has attached powerful GPUs, but so far, I did not ...
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2answers
404 views

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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1answer
2k views

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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2answers
473 views

Is there any way to do “double preconditioning”

Question: Suppose that you have two different (factored) preconditioners for a symmetric positive definite matrix $A$: $$A \approx B^TB$$ and $$A \approx C^TC,$$ where the inverses of the factors $B, ...
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138 views

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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477 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

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 -...
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1answer
963 views

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 ...
6
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1answer
581 views

Jacobi preconditioner not reducing condition number?

Let's say you have a general matrix $A$, with diagonal entries $a_{ii} = d>0$. (No assumptions are made about the off-diagonal elements.) Then Jacobi preconditioning doesn't improve condition ...
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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 ...
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2answers
382 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 ...
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1answer
483 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 ...
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1answer
617 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 ...
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1answer
676 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 ...
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360 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 ...
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2answers
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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 ...
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2answers
979 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 ...
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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....
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634 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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733 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 ...
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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-...
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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.
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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 ...