Questions tagged [preconditioning]

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

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37 views

Solving saddle point problem having non-invertible top-left block with a PETSc nested matrix

My system is a symmetric FE problem with lagrange multipliers: $Z=\begin{pmatrix}A & C^T \\ C & 0\end{pmatrix}$ The matrix $A$ is positive semi-definite, non-invertible. The whole matrix is ...
3
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1answer
52 views

Are there any other better methods than block diagnoal and block upper triangular precondtioner for saddle point problems?

For stokes problems, $$ -\Delta \vec{u} + \nabla p =\vec{f},\qquad \nabla . \vec{u} = 0; $$ with appropriate boundary conditions which guarantee there is a unique solution. Using FDM or FEM, ...
1
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1answer
32 views

Accelerating Conjugate Gradients fitting for small localized kernel (like cubic B-spline)

Question: Is there some pre-conditioner for Conjugate-Gradient (CG) cheap enough, that it is worth using even if my operator is very local (i.e. already has a low number of non-zero elements), as it ...
1
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2answers
453 views

Correct use of scipy's sparse.linalg.spilu

I'm attempting to use scipy's spilu routine as a preconditioner and I'm finding bad performance for my application (solving a global linear system arising from a DG ...
3
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1answer
140 views

Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?

For numerical methods of the Stokes equations, with appropriate boundary: $$-\nabla^{2} \vec{u}+\nabla p=\overrightarrow{0}$$ $$\nabla \cdot \vec{u}=0$$ one may use FDM (finite difference method) ...
2
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2answers
176 views

Why iterative method: AMG preconditioned PCG is slower than Matlab direct method 'A\b'?

Recently, I have met a question that a saying goes that for large linear system: iterative methods are required because of memory problem of direct methods. But when I implement some experiments ...
6
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1answer
288 views

Iterative linear solver for “ugly” saddle point system

I am a graduate student majoring scientific computing. The numeric model I made caused a very ugly-looking saddle-point linear system. It is not symmetric at all and I will attach the sparsity pattern ...
1
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1answer
86 views

Iterative solution of ill-conditioned matrix systems

I want to solve a matrix system of the form $Ax=b$ where $A$ is ill-conditioned. The matrix system comes from a structural simulation problem which was discretized using finite elements. I do not have ...
6
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0answers
61 views

Why not use the preconditioned residual as termination criterion for preconditioned CG?

I have a Poisson equation with wildly varying material parameters (1 .. 1000), wildly varying element sizes (5 nm .. 100 um) and some quite anisotropic (tetrahedral) elements (100 nm x 100 um). I use (...
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0answers
51 views

Kinetic preconditioning

Publication arXiv:0804.2583 describes a method for doing self-consistent iteration without having to diagonalize the Hamiltonian operator at every step. IX. PRECONDITIONING As already ...
3
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1answer
96 views

Optimality of block-Jacobi preconditioner

For a dense $N \times N$ matrix $A$, is the block-Jacobi preconditioner comprising the inverse of the diagonal blocks of $A$ the optimal block-diagonal preconditioner? Could there exist another matrix ...
7
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76 views

How to construct an effective preconditioner for this particular problem

A quick introduction to my problem I am currently developing a method for simulation of water waves in three dimensions based on potential flow theory. The computational bottleneck of the method is ...
2
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1answer
80 views

Re-using LU factorization within iterative (?) setup for a sum of two matrices

So, I would love to make at least some use of my preexisting data, no matter how small, and just out of ideas. Maybe I am just a prisoner of a Kahneman-like theatre-ticket paradox, and don't know ...
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1answer
50 views

PCJACOBI works but the default PCBJACOBI failed in PETSc

I am using PETSc and libmesh to solve a simple linear elastic problem with quite complicated geometry, using linear tetrahedral elements. I am always using the KSP CG as the solver. I noticed that ...
1
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1answer
49 views

Simplest way to precondition Uzawa iteration

I have a diffusion problem with an internal circular dirichlet constraint and a side condition which shall enforce a certain global volume integral. $\nabla(D \nabla u(x)) = 0$ outer boundary ...
2
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1answer
63 views

Right-preconditioning and fixed point linear iterations

Given a linear system $A\textbf{x}=\textbf{b}$, we can express it into the easier-to-solve right-preconditioned form: $$ AM^{-1}\textbf{y}=\textbf{b}, \quad \textbf{y}= M\textbf{x} $$ On the other ...
3
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65 views

Numerical analysis, pivoting and incomplete LU decomposition

When doing LU decomposition, the algorithm will break down if any of the diagonal element $x_{ii}$ is zero. Therefore, we can use pivoting on the matrix such that $x_{ii}$ is no longer zero. That is ...
1
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1answer
138 views

Multigrid preconditioner for conjugate gradient methods

When solving $A*x=b$ using preconditioned conjugate gradient methods one has to solve $z=K^{-1}*r$ for the preconditioning where $K$ is the preconditioner of $A$ and $r$ is the residual vector. ...
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1answer
98 views

what is Sherman-Morrison formula

Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works ...
3
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0answers
71 views

Element Preconditioner

Im just working on a preconditioner for the linear equation system $Ax = b$ arising in FEM for elliptic PDE. $A$ is a s.p.d Matrix with real valued entries. I read something about the element by ...
7
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1answer
365 views

How to directly compute the inverse of an ill-conditioned dense matrix

I know that it is generally a bad idea to compute the inverse matrix directly. However, if it is necessary to compute the inverse of an ill-conditioned invertible dense matrix, then what can I try? ...
8
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156 views

How to find a good preconditioner to the system $(A^T A + \lambda I) x = A^T b$?

The system in the title has a damper factor $\lambda > 0$ and the matrix $A$ is sparse and rectangular, with a structure I can exploit to solve matrix vector products very fast. My current solver, ...
3
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1answer
156 views

Efficient implementation of preconditioners for iterative solvers

I am struggling a bit with the concept of preconditioners for iterative solvers and how to implement them efficiently. The literature mostly provides methods to create a preconditioner matrix $M$ (...
4
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0answers
70 views

Preconditioned residual converges, but true residual doesn't

I'm using Albany w/ Trilinos to solve an elasticity problem with thermal expansion mismatch. I'm using block GMRES with MueLu preconditioning. It works for problem size of several million dofs, but ...
6
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1answer
96 views

Eigenvalue problems with extremely small gaps

I'm interested in numerically diagonalizing a class of structured, symmetric eigenvalue problems with potentially extremely small eigenvalue gaps. The question I have is how to design a numerically ...
6
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3answers
478 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 ...
1
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1answer
159 views

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 $i = 1$, ...
0
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1answer
143 views

How to verify solution to pre-conditioned linear systems solver?

I am solving Ax=b. A has a very large condition number (> O(10^10)) I am using the conjugate gradients method with point jacobi pre-conditioning. I obtained a solution 'x' that "looks" reasonable. ...
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0answers
207 views

Why does PETSc take unexpectedly long to set up its KSP solver with a custom preconditioner? [closed]

I am attempting to solve a large system, $\bf{Ax} = \bf{b}$ with the help of PETSc. Due to the size of the problem, I'm using a matrix-free approach, where $\bf{A}$ is just a shell. I'm also providing ...
8
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2answers
281 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} :=& ...
2
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0answers
50 views

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 ...
2
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2answers
145 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., $$\...
3
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1answer
3k 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 ...
1
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2answers
579 views

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 ...
4
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2answers
212 views

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\...
3
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1answer
233 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 ...
1
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0answers
393 views

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 "...
2
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1answer
318 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, ...
0
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1answer
169 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 ...
4
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0answers
154 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 ...
2
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0answers
66 views

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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1answer
140 views

Difference between explicit and implicit preconditioning

What is the difference between an explicit and implicit preconditioner?
9
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1answer
146 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 ...
3
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0answers
50 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 ...
3
votes
2answers
535 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 ...
1
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1answer
443 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}-\...
0
votes
1answer
491 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 ...
6
votes
1answer
393 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, ...
2
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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 ...
7
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1answer
494 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 \...