Questions tagged [preconditioning]
For questions regarding design and implementation of preconditioners for solving linear systems.
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questions with no upvoted or accepted answers
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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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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 ...
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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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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 ...
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Reference for preconditioning nonlinear conjugate gradient with LU decomposition of jacobian
When solving non-linear systems of equations, @Arnold Neumaier suggested in How can I precondition a non-linear problem before linearization? to use an approximate LU decomposition of an approximate ...
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What is this QR-factorization-based preconditioning called?
I have recently started to delve into someone else's code, and there is a part in there I don't quite understand. The authors of the code use some form of pre-conditioning to speed up the optimization....
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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 ...
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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 ...
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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 ...
user29088
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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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Confusion about preconditioner for incompressible Navier-Stokes equation with implicit-explicit method
Consider the time-dependent Navier-Stokes equation
$$u_t + (u \cdot \nabla) u - \Delta u + \nabla p = f$$
$$\operatorname{div}(u)=0$$
Looking at deal.ii tutorials, I've notice that there are ...
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"black box" preconditioner for shifted linear systems?
Does anyone know of any strategies for creating a preconditioner $P^{-1}_\sigma \approx (A+\sigma I)^{-1}$ given a preconditioner $P^{-1} \approx A^{-1}$, preferably such that the precomputation doesn'...
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Does incomplete LU preconditioning improve the asymptotic scaling of Krylov subspace methods?
It is well known that unpreconditioned Krylov subspace methods applied to the finite-difference-discretised Poisson equation with $n$ grid points per direction require $O(n \, |\log(\varepsilon)|)$ ...
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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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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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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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Hessian-free preconditioner for non linear least squares
I am solving a nonlinear least squares problem using Gauss Newton method. Due to the large dimension of the problem, I use the Hessian-free approach. As a linear solver I use either MINRES or CG. To ...
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Upper bound on condition number in linear preconditioning
I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia:
Consider a matrix splitting $A = M-N$, where $A,M,N$ are all symmetric and positive definite ...
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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 mentioned, ...
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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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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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Normalizing the right-hand side in Jacobi-preconditioned conjugate gradients
I have been reading the following paper: CG versus MINRES: An empirical comparison.
In it a conjugate gradient solver is applied to a system matrix $A$ Jacobi-preconditioned on both sides. ...
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Preconditioning the $[1 \quad-2 \quad 1]$ Finite Difference matrix
Let $A$ be the well known tridiagonal matrix coming from the 1D Finite difference discretization of the Laplacian, with stencil $\frac{[1 \quad-2 \quad 1]}{h^2}$.
The system $Ax = b$ is very large, so ...
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Automatic selection of the SLE solver and preconditioner during simulation
To simulate the physical process necessary to solve the arising systems of linear algebraic equations. The SLE matrix has a highly sparse form. There are a couple dozen non-zero elements in the string,...
