Questions tagged [linear-solver]
Referring to methods for solving linear systems of equations.
4
votes
2answers
184 views
How do the properties of a matrix affect the linear system solving
For a general matrix A, there are many properties to describe it: symmetric positive definite or indefinite, condition number, spectrum and so on. I am curious about how these properties affect the ...
11
votes
1answer
644 views
Projecting out the null-space of $A$ from $b$ in $Ax=b$
Given the system $$Ax=b,$$ where $A\in\mathbb{R}^{n\times n}$, I read that, in case Jacobi iteration is used as a solver, the method will not converge if $b$ has a non-zero component in the null-space ...
6
votes
3answers
4k views
How to find QR decomposition of a rectangular matrix in overdetermined linear system solution?
While trying to find cell-centered gradients in finite volume method computation of incompressible fluid flow I get over-determined linear system. This is a well known "cell based least-square" ...
11
votes
4answers
4k views
What are the best Python packages/interfaces to sparse direct solvers?
Please list the Python package (petsc4py, etc...) and the sparse direct solvers it supports. One (community-wiki) answer per package, please.
6
votes
1answer
1k views
How does a Sparse Direct Solver know about dimensionality of a problem being solved?
It is claimed that the time and memory complexities of sparse direct solver are $O(N^2)$ and $O(N^{4/3})$ for 3D problems and $O(N^{1.5})$ and $O(N \log N)$ for 2D, respectively.
But how does a ...
1
vote
3answers
242 views
Unique coordinates (solutions) in a single Gauss-Seidel iteration
I managed to reduce certain computational problem to the Gauss-Seidel solution of the following linear system: $$Ax=Ly,$$ where $A, L\in\mathbb{R}^{n\times n}$ are weighted Laplacian matrices (...
5
votes
2answers
1k views
How does matrix scaling influence linear solvers?
For instance, in MUMPS there is option to scale matrix s.t. all rows/columns have the same norm.
This claims to decrease condition number and improve numerical properties of the matrix: ftp://cuter.rl....
3
votes
2answers
414 views
What numerical methods are recommendable for simulating two phase immiscible fluid flow through a pipe with high capillary pressure?
I'm simulating two phase immiscible drainage (air displacing water) in a rectangular domain of size .6mm x 2.4mm (2 dimensions) using Ansys FLUENT software. I am using an implicit Volume of Fluid ...
3
votes
1answer
68 views
2D Jacobi line maintenance?
Suppose a linear system is given
$$AX=B,$$
where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
8
votes
2answers
126 views
Initial guesses for perturbed linear systems
Suppose you solve a linear system $Au = f$ by an iterative method, e.g. conjugate gradients or Richardson iteration. Then you try to solve a linear system that is slightly perturbed in the matrix and ...
9
votes
2answers
1k views
Safe application of iterative methods on diagonally dominant matrices
Suppose the following linear system is given
$$Lx=c,\tag1$$
where $L$ is the weighted Laplacian known to be positive $semi-$definite with a one dimensional null space spanned by $1_n=(1,\dots,1)\in\...
10
votes
2answers
937 views
Which iterative linear solvers converge for positive semidefinite matrices?
I want to know which of the classic linear solvers (e.g Gauss-Seidel, Jacobi, SOR) are guaranteed to converge for the problem $Ax=b$ where $A$ is positive semi definite and of course $b \in im(A)$
(...
5
votes
2answers
527 views
Recommendation for a good article/book for frontal methods?
Can someone provide an article or book that explains the principle used in frontal solvers? Some examples also may help understand the frontal methods better.Thanks in advance!
10
votes
2answers
1k views
Selection of linear solver for GPGPU computation (OpenCL)
I have already developed a working solution of the Finite Element Method to solve heat transfer problems using GPU and OpenCL using the Conjugate Gradient method. The main disadvantage of this method ...
15
votes
1answer
682 views
Are there any open source inverse-based multilevel ILU implementations?
I am very impressed with the serial performance of multilevel inverse-based ILU preconditioners, particularly for heterogeneous Helmholtz, but I am surprised to not be able to find any open source ...
3
votes
1answer
274 views
How to solve a problem with structure similar to a finite difference discretization of the 2D Poisson equation, but with non-symetric coefficients?
Recently, I've been asking about methods to solve a finite difference discretization of the 2D Poisson equation (see here and here) of the form:
$$U_{i-1,j} + U_{i+1,j} -4U_{i,j} + U_{i,j-1} + U_{i,...
21
votes
2answers
8k views
Libraries for solving sparse linear systems
There are a number of different libraries out there that solve a sparse linear system of equations, however I'm finding it difficult to figure out what the differences are.
As far as I can tell there ...
49
votes
4answers
7k views
What guidelines should I follow when choosing a sparse linear system solver?
Sparse linear systems turn up with increasing frequency in applications. One has a lot of routines to choose from for solving these systems. At the highest level, there is a watershed between direct (...
