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Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

7
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3answers
642 views

Solving shifted linear systems with LU factorization

I am interested in solving a sequence of shifted linear systems $(A+\sigma I)x = b$ for various values of $\sigma$. The matrix $A$ is sparse and not too large, so I have its LU factorization available....
8
votes
2answers
1k views

Taxonomy of ILU preconditioners

I learned that for BiCGStab solver for sparse linear systems it's pretty much always necessary to use a preconditioner. I realized by now that choosing a good one is problem dependent. Surfing the ...
22
votes
3answers
734 views

Solving $(G^TA^{-1}G)x = b$ without inverting $A$

I have matrices $A$ and $G$. $A$ is sparse and is $n\times n$ with $n$ very large (can be on the order of several million.) $G$ is an $n\times m$ tall matrix with $m$ rather small ($1 \lt m \lt 1000$) ...
2
votes
0answers
60 views

Complexity of direct solvers? [duplicate]

Possible Duplicate: How to reorder variables to produce a banded matrix of minimum bandwidth? What is the time and space complexity of direct sparse solvers (e.g., UMFPACK, SUPERLU, PARDISO, etc.)...
2
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0answers
214 views

Is it possible to run a Solver Foundation solver against a model containing linear and non-linear elements?

This is a follow up question to one I made previously about non-linear equations and ranged real numbers in Solver Foundation. I acknowledge that where possible, rewriting a problem that is non-...
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
654 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
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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
243 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
416 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
127 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
956 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
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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
687 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
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2answers
9k 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 (...