18 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 ...
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15 votes
Accepted

Why should I renormalize physical variables?

This question is very closely related to (and possibly a duplicate of) Is variable scaling essential when solving some PDE problems numerically?. There are still good practical reasons to ...
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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 ...
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  • 2,116
13 votes
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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-...
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  • 8,451
13 votes
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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 ...
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  • 8,451
12 votes
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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, ...
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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 ...
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9 votes
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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 ...
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8 votes
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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 ...
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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 ...
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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 $Ax=b$...
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  • 3,023
6 votes
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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 $\mathbf A$), ...
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  • 4,326
6 votes
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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 ...
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  • 1,902
5 votes
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Preconditioning symmetric Schur complement

You won't find any black-box scalable solutions because $S$ is typically dense and thus cannot be formed. If your problem comes from a mature research area, there might be experience in the literature ...
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  • 25.3k
5 votes
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High quality flexible GMRES (FGMRES) implementation

Both PETSc and Trilinos have high quality implementations. Both of these libraries also have bindings to other languages.
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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 ...
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  • 2,199
5 votes
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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 ...
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  • 2,116
5 votes
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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 ...
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  • 861
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 ...
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  • 51
5 votes
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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 ...
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5 votes
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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{...
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  • 1,163
5 votes
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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: $$ \sigma_1 = \sqrt{2}+1,\quad \sigma_2=\sqrt{2}-1 $$ ...
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  • 8,451
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 ...
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  • 8,451
5 votes
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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 ...
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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 ...
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4 votes

Avoid arithmetic overflow in matrix multiplication

This isn't a direct answer to your question but rather an alternative approach. It looks like you are solving a least squares (LS) problem using the normal equations. The normal equations are known ...
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  • 5,754
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 $B$. ...
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  • 697
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 ...
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  • 184
4 votes

Best preconditioner for mixed-poisson problem (RT0 elements)

Algebraic Multigrid (AMG) methods have the fundamental limitation that, by and large, their current implementations only work reliable and well for elliptic or similar problems. For saddle point ...
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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 ...
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