Questions tagged [preconditioning]

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

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Scaling for a nonsymmetric eigenvalue problem

I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in real eigenvalues and ...
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2 votes
2 answers
199 views

Preconditioner for scalar laplacian system

Suppose that I have a large (on the order of 10^6 unknowns) 3D scalar Poisson system which I would like to precondition. The boundary conditions have been treated so that the system is SPD. I.e., $$\...
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Is it necessary to invert precondition matrix for iterative solver?

I was reading these slides about preconditioners. I believe I grasp the idea of how they work but there is something that is still not making sense. If we have the system $Ax=b$ and use a ...
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1 answer
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Computing preconditioner for a non-linear conjugate gradient implementation

Consider the following steps for the $i$-th non-linear conjugate gradient iteration, in the context of 3D electromagnetic inversion, and as discussed in (Newman and Boggs, 2004): (1) set $i = 1$, ...
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2 answers
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Preconditioner for the GMRES method in the Uzawa algorithm

I'm trying to solve \begin{equation}\left\{ \begin{split} \frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\frac1\rho\nabla p&=f\;\;\;\text{in }\Lambda\\ u&=0\;\;\;\text{on }\partial\...
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How to use "fill_factor" in spilu (scipy) [closed]

I am using spilu (incomplete LU decomposition for sparse matrix) in scipy: https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.sparse.linalg.spilu.html However there is an option call "...
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2 votes
1 answer
543 views

Preconditioner for dense matrix "with diagonal predominance"

For a CFD panel-based potential method, I'm trying to reduce the time to solve the linear system. The matrix has the larger values on the diagonal, since the influence of a panel on itself is maximum, ...
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3 votes
1 answer
312 views

SLEPc eigensolvers take long time to converge for large sparse symmetric matrices

I am leveraging the SLEPc library for solving the first $k$ (where $k = 3$ or $4$) eigenvalues and their corresponding vectors for a matrix of size 200,000. The matrix is sparse and symmetric. I ...
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1 answer
232 views

Iterative methods to solve linear system in 3D FEM

I implemented a FEM solver in MATLAB for Poisson's equation in 3D, using hexahedron and sparse matrix for the Laplacian. I was using the backslash but now I have to use a few iterative methods (GMRES ...
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223 views

Preconditioning technique for large sparse non-hermitian matrix

I am attempting to solve a computational acoustics problem that involves solving an underlying sparse matrix. The size of the problem varies with grid size (3D) and fill-in's obviously make direct ...
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Preconditioners and discrete Lagrange multipliers

There is a huge literature on efficient preconditioners for saddle-point problems. In computational physics, the case where the Lagrange multipliers enjoy a weak formulation (say, the Stokes equations)...
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Difference between explicit and implicit preconditioning

What is the difference between an explicit and implicit preconditioner?
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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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2 answers
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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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1 answer
701 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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6 votes
1 answer
416 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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7 votes
3 answers
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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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2 votes
1 answer
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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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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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7 votes
1 answer
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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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5 votes
2 answers
133 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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4 votes
1 answer
208 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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2 votes
0 answers
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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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4 votes
1 answer
284 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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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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10 votes
1 answer
183 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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5 votes
2 answers
353 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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2 votes
1 answer
257 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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1 vote
1 answer
287 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 a shared-memory system with OpenMP. In addition, I want to ...
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1 answer
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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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2 votes
1 answer
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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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5 votes
1 answer
156 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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9 votes
1 answer
197 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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3 votes
1 answer
389 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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1 vote
0 answers
96 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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7 votes
1 answer
303 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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3 votes
1 answer
200 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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6 votes
1 answer
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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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16 votes
2 answers
887 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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4 votes
1 answer
5k 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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11 votes
2 answers
420 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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1 vote
1 answer
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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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15 votes
2 answers
518 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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4 votes
1 answer
146 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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6 votes
3 answers
551 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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8 votes
1 answer
1k 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 ...
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7 votes
1 answer
653 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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13 votes
2 answers
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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5 votes
2 answers
448 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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10 votes
2 answers
567 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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