19
votes
Accepted
Why does this preconditioner effectively reduce the condition number of a random SPD matrix?
If the eigenvalues of $A$ are $\lambda_1, \lambda_2, \dots,\lambda_n$, the eigenvalues of $A + \mu I$ are $\lambda_1 + \mu, \lambda_2 + \mu, \dots, \lambda_n + \mu$. It is an easy computation to ...
- 10.6k
15
votes
"Cookbook" about iterative linear solvers and preconditioners
Have a look at Templates for the Solution of Linear Systems:
Building Blocks for Iterative Methods (Barrett et al.). You can find it here.
Here's why I'm recommending this over other references:
the ...
- 2,166
13
votes
Accepted
How to directly compute the inverse of an ill-conditioned dense matrix
Though it is a relatively rare situation when you actually have to calculate an inverse of the matrix, not all techniques were created equally.
I would use the term badly-conditioned instead of ill-...
- 8,592
13
votes
Accepted
When do not use preconditioners for sparse linear system of equations?
In my experience, you always need (or better use) some form of preconditioning. The type and complexity of the precondition would vary depending on the task though.
From Y. Saad, Iterative Methods for ...
- 8,592
12
votes
Accepted
role of initial guess for iterative linear solver
Practical experience shows that trying to get good initial iterates has little value. For example, in the context of solving partial differential equations, if you take the solution from one mesh, ...
- 53.6k
12
votes
Why does this preconditioner effectively reduce the condition number of a random SPD matrix?
The accepted answer is right: you are not making a preconditioner.
To elaborate.
For a matrix $A$, a preconditioner is a matrix $B$ such that $B^{-1}A$ has a smaller condition number than $A$. The ...
- 1,305
8
votes
Accepted
Efficient implementation of preconditioners for iterative solvers
I don't particularly care for the notation $M^{-1}$ precisely because of the confusion you find yourself in. I (and others) simply call the preconditioner $P$.
The point, however, is that for ...
- 53.6k
8
votes
Accepted
Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?
The preconditioner for the FDM method that corresponds to the one you outline for the FEM (i.e., the Sylvester-Wathen approach) will still contain the Schur complement of the FDM matrix. The Schur ...
- 53.6k
6
votes
Does this partial eigen-expansion have a name?
I don't know for sure whether there is an existing name for this method, but @jessechan's suggestion of "truncated eigenvector expansion" sounds perfectly fine to me (and most people would understand ...
- 53.6k
6
votes
Does this partial eigen-expansion have a name?
This is something like a truncated SVD or eigenvector expansion of your solution. If you take
$$x_m = \sum_{j=1}^m \frac{q_j\cdot b}{\lambda_j}q_j$$
with $m=n$, this is the exact solution to $Ax=b$...
- 3,102
6
votes
Accepted
Correct use of scipy's sparse.linalg.spilu
If $\mathbf L$ and $\mathbf U$ give an approximate factorization of $\mathbf A$, you wouldn't want to use $\mathbf P = \mathbf L\cdot \mathbf U$ as a preconditioner (that's approximately $\mathbf A$), ...
- 4,794
6
votes
Accepted
Optimality of block-Jacobi preconditioner
Consider matrix a $2\times 2$ matrix $A$:
$$
A=\left(\begin{array}{cc}
1 & 0\\
2 & 1
\end{array}\right)
$$
Singular values of $A$ are:
$$
\sigma_1 = \sqrt{2}+1,\quad \sigma_2=\sqrt{2}-1
$$
...
- 8,592
6
votes
Accepted
Iterative linear solver for "ugly" saddle point system
You should stick with GMRES, it is the only method that is essentially guaranteed to get a solution here. The real problem appears to be you need a better preconditioner. You could try sticking with ...
- 2,069
5
votes
Accepted
Is it necessary to invert precondition matrix for iterative solver?
OK - so your original equation is $$Ax = b$$
Say you've come up with a good preconditioner for $A$, call this $M$. Also, say you have pre-computed an LU-decomposition for this $M$, i.e.
$$M = L_m ...
- 2,166
5
votes
Is it necessary to invert precondition matrix for iterative solver?
Based on your comments, it looks like your main difficulty is how having an LU factorisation for a given matrix $M$ makes it easy to find a vector $\mathbf{y}$ such that $M\mathbf{y}=\mathbf{b}$ for a ...
- 2,229
5
votes
Accepted
How to verify solution to pre-conditioned linear systems solver?
You've started with a singular linear system of equations $Ax=b$. As a practical matter, it's unlikely that $b$ lies exactly in the range of $A$, so at best you can find a least squares solution that ...
- 18.4k
5
votes
Correct use of scipy's sparse.linalg.spilu
This question has an example of how to create the preconditioner M with a scipy sparse matrix A of shape ...
- 51
5
votes
Accepted
Preconditioner for scalar laplacian system
Preconditioning and iterative solvers are cool, but did you try to solve your problem with some kind of sparse direct solver? If not, try it first.
State–of–the–art preconditioning techniques for ...
- 901
5
votes
Accepted
Right-preconditioning and fixed point linear iterations
Starting with
$AM^{-1} \mathbf{y} = \mathbf{b}$, where $\mathbf{y} \equiv M \mathbf{x}$,
we can manipulate to
$\mathbf{y} - (I-AM^{-1})\mathbf{y} = \mathbf{b}$
Replacing one instance of $\mathbf{...
- 1,193
5
votes
Accepted
Why iterative method: AMG preconditioned PCG is slower than Matlab direct method 'A\b'?
There are several things to consider in this experiment:
Why Matlab sparse direct might be "so fast":
(for your particular test)
In 2D (of course, problem-dependent), your matrix $A$ arising after ...
- 8,592
5
votes
Accepted
Are there any other better methods than block diagnoal and block upper triangular precondtioner for saddle point problems?
Your argument is almost correct, but not quite. You correctly state that the use of one AMG step for $B^{-1}$ and the identity matrix for $S^{-1}$ leads to a constant number of iterations, but that is ...
- 53.6k
5
votes
Where can I find matrices and it's preconditioner for testing?
Generally, preconditioners are considered to be part of the solver, so they are not included in test matrix collections. In fact, preconditioners are rarely constructed as an explicit matrix, making ...
- 615
5
votes
What makes a good preconditioner when only a few approximate iterations are needed?
Edit there seem to be a few quantities that predict the difficulty based on how "flat" the spectrum is. Analysis below corresponds to the purity measure of spectrum flatness, but one could ...
- 2,049
4
votes
Why does conjugate gradient work with this nonsymmetric preconditioner?
In short, orthogonalization of the Krylov vectors occurs with respect to the operator, but not with respect to the preconditioner.
Alright, so say we want to solve $Ax=b$ with preconditioner $B$. ...
- 747
4
votes
Accepted
Which preconditioning for large linear elasticity problem?
You can always try AMG as one preconditioner.
There are several other methods that are used. Look into papers by Johanes Kraus, Neytcheva, Owe Axelsson.
If you use separate displacement ordering ...
- 184
4
votes
role of initial guess for iterative linear solver
It can be even harmful.
In Liesen/Strakos Krylov subspace methods principles and analysis (Chapter 5.8.3) it is reported that a nonzero initial x0 makes a GMRes ...
- 3,408
4
votes
Accepted
What is the difference between Adittive Schwarz as a preprocessor and a solver?
By itself, Schwarz methods are stationary iterations just like Jacobi, Gauss-Seidel, or SOR. They converge to the solution, but often quite slowly.
But, like any other stationary method, one iteration ...
- 53.6k
4
votes
Accepted
Solution of linear system doesn't work, in parallel
ILU is not implemented for non-sequential matrices, see here.
- 233
3
votes
role of initial guess for iterative linear solver
This answer is an addition to the one from Wolfgang Bangerth.
It is certainly not worth to bother with the initial guess for the iterative linear solver if there is any work to be done: coding, ...
- 8,592
3
votes
Accepted
Preconditioner for the GMRES method in the Uzawa algorithm
Please check this paper by Benzi et al. They address this issue and give corresponding references on p. 45.
Shortcut: for the Stokes problem $A = \text{diag}(A_{11},A_{11},\dots,A_{11})$ is just a ...
- 901
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