Questions tagged [linear-solver]
Referring to methods for solving linear systems of equations.
360
questions
0
votes
1answer
17 views
Sparse direct solver that works inside opened omp parellel region?
Does anyone knows a library that implements sparse direct solver working in already open omp parallel region? The only library that I know it works with this requirement is Pardiso7.2 worth ~8K usd ...
0
votes
0answers
133 views
Solving huge dense square symmetric linear system
I have a linear system of the type
$A x = y$
where A is a dense, square, symmetric, positive definite matrix, $x$ a vector of unknown parameters, and $y$ is a vector of observed quantity.
I know that ...
3
votes
0answers
64 views
Comparing block versus non-block Krylov methods for handling multiple right-hand-sides
Suppose I wish to solve a linear system $AX=B$ iteratively where $A$ is an $m\times m$ matrix and $X,B$ are $m \times s $ matrices (not single vectors). Instead of solving $s$ independent systems I'm ...
1
vote
1answer
90 views
Elementary question on numerical linear algebra
I’ve been facing a problem of solving linear systems Ax=b arising from discretized PDEs (Stokes equations in particular). Nively, it seems that solving Ax=b should not take much more time than simply ...
3
votes
1answer
132 views
Incomplete Cholesky preconditioner for CG efficiency
I am currently solving the harmonic equation using a P1 FEM discretisation. The resulting matrix $A$ is SPD and fairly sparse so I use a preconditioned conjugate gradients (CG) solver to find a ...
2
votes
0answers
92 views
Regularisation of ill-conditioned matrix-vector problem
I have a linear* problem which arises from an integro-differential system, and writes:
$$
(\mathbf{I}+\lambda \mathbf{A})x = b
$$
where $\mathbf{A}$ is a real full matrix, size $n\times n$, but is not ...
2
votes
2answers
197 views
Different sources of error in Finite Element computations
Consider the problem $-\Delta u = f$ in $\Omega$, with $u=0$ on $\partial \Omega$. Suppose that $\Omega$ is a polygon and that we approximate the solutions to the previous problem using Lagrange ...
0
votes
0answers
40 views
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. ...
1
vote
1answer
141 views
Givens rotation algorithm without matrix-matrix multiplication
I would like to implement a givenRotation algorithm without having matrix-matrix multiplication. Matrix-vector is fine or just for looping. I am to decompose a rectangular (m+1)xm Hessenberg matrix.
I ...
4
votes
0answers
120 views
Stable iterative solver for complex symmetric linear systems
I am interested in the iterative solution (preferably Krylov-type solvers) of a problem $\boldsymbol{A}x=b$, with $x,b\in\mathbb{C}^{n\times1}$ and $\boldsymbol{A}\in\mathbb{C}^{n\times n}$. $\...
5
votes
1answer
137 views
Solving absolute value systems
Let $z, b \in \mathbb R^n$, $A \in M_n (\mathbb R)$ and $|z| := (|z_1|, \dots, |z_n|)$. I am searching for an efficient algorithm to solve the absolute value system:
\begin{equation}
z - A |z| = b.
\...
4
votes
0answers
59 views
Solving multiple linear regression in parallel
I am working on a problem where I need to solve approximately 500 Million Linear Regressions (OLS).
What would be the most efficient way to do this (e.g. using GPU or a some framework that can do this ...
1
vote
0answers
50 views
Viable algorithms for efficiently solving a block matrix system with non-uniform sparsity structure
I am trying to use Newton's method to get a stationary solution for a system of equations of the following form:
$$
\begin{Bmatrix}
\frac{\partial x}{\partial t} \\
0
\end{Bmatrix} = \begin{Bmatrix}
f(...
0
votes
1answer
53 views
Efficient solution to linear system involving Kronecker sum in MATLAB
High dimensional finite difference problems often lead to linear systems of the form
$$
A x = b, \qquad A = B_1 \oplus B_2 \oplus \cdots \oplus B_d,
$$
where $\oplus$ denotes the Kronecker sum. $B_i \...
1
vote
1answer
88 views
RCM better than Nested dissection? (For FEM discretizations in 2D and 3D)
I realize this might be a too general question but here goes nothing:
I am trying different re-ordering strategies and checking the fill-in of $A=LU$.
I have 2D ($p=1$, $h=1/40$ on $\Omega = [-1,1]^2$)...
0
votes
0answers
51 views
Equivalence between zero sum games and linear program
It is well known that you can use the algorithm for finding the equilibrium of a Zero-sum game to solve a linear program. In particular, you can take a LP and reduce it to a zero-sum game, and use the ...
4
votes
2answers
268 views
Getting to know about various BLAS implementations
I keep coming across phrases like "highly optimized BLAS kernels" with "architecture-specific optimizations", but have never been able to find what exactly these optimizations are, ...
4
votes
0answers
87 views
hardware agnostic (GPU/CPU) sparse linear algebra C++ solvers framework/technology
I am looking for recommendations on matured C++ solver for Linear Sparse Algebra problems.
The goal is to select between more or less GPU hardware agnostic libraries/frameworks that can be compiled on ...
7
votes
2answers
172 views
solving linear system whose symmetrized matrix is positive definite
Are there iterative methods for the solution of nonsymmetric linear systems $Ax=b$ that can take (theoretical or practical) advantage from knowing that $A+A^T$ is positive definite? These matrices are ...
0
votes
2answers
89 views
Simplest solver for linear equation systems
Normally, this boards sees a lot of traffic about the most efficient and most powerful solvers for huge linear equation systems. But this time, I have the opposite problem:
I need to implement a ...
2
votes
3answers
166 views
Linear solver recommendation(s) for small problems
I am interested in solving many linear systems $Ax = b$, where $A$ is symmetric positive definite and small (i.e. less than 25,000 rows) --- $b$ will be changing.
We can assume that $A$ arises from ...
-1
votes
1answer
80 views
Is there an overview of the runtime speed up of LP/MIP solvers throughout the years?
whenever I read papers on OR that use an LP/MIP approach, they include the time solver used, as well as the version and the year. I would like to know how much faster the same experiment would be ...
3
votes
1answer
191 views
Preconditioning vs. regularization
I used to be more of a numerical linear algebra and computational science person, but recently, I've crossed into stats and machine learning.
For this discussion, let's focus on matrices that are not ...
-1
votes
1answer
98 views
Combinatorial Optimization Problem with Constraints
To explain my problem I'd like to go with an example first.
Suppose I have many bills of grocery purchases in the same shop. On the invoices I can only see the total amount of the invoice and which ...
0
votes
1answer
76 views
Solve for large array of PD matrices
I have N matrices that are positive definite, and I have to solve for a M vectors.
As M is large in my case, doing all solves simultaneously using np.linalg.solve ...
1
vote
1answer
70 views
Algebraic multigrid as solver and as preconditioner
My question is around the efficiency of AMG. In which case AMG can perform better,as solver or as a preconditioner(for example a Krylov space method as CG)? Assume the case of elliptic pdes.
5
votes
1answer
173 views
Solving PDEs in parallel
I have read different approaches on how to solve pdes in parallel which are discretized using finite element method. For example:
Non-overlapping domain decomposition approach as mentioned in https://...
-1
votes
1answer
37 views
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,...
2
votes
0answers
40 views
"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'...
3
votes
0answers
68 views
Kronecker-factored least-squares?
Suppose $a_i,b_i$ are $d\times 1$ matrices, $y_i$ is scalar, and I need to find least-squares solution $w$ of the following system of $n$ equations:
$$y_i=(a_i^T\otimes b_i^T)w$$
Is there a ...
1
vote
1answer
123 views
Complexity of solving an image differential linear system
Define an "image differential linear system" as a linear system $A\mathbf{x}=\mathbf{b}$ wherein $\mathbf{x}$ contains the ($\mathbb{R}$) pixels of an image and each row of $A$ constrains ...
6
votes
1answer
205 views
Symmetric matrix which satisfies conditions of the form $v_i^T X v_i = 0$
I want to solve an underdetermined system of linear equations $A x = b$ with $A \in \mathbb{R}^{n \times r^2}, x \in \mathbb{R}^{r^2}, b \in \mathbb{R}^n$. The matrix $A$ has the following additional ...
1
vote
0answers
102 views
performance comparison between PETSc and SLATE
We want to start a new project to solve a large-scale inverse problem (O(10^6) number of parameters) to invert for subsurface wave speeds. We will use FEM to solve forward and adjoint PDEs. In our ...
3
votes
1answer
130 views
Choice of iterative solver for a sparse asymmetric matrix with symmetric structure
I have a sparse $nxn$ matrix A with pretty interesting structure. It has a block structure with symmetric structure but asymmetric blocks. Expressed mathematically the block $A_{jk} = A_{kj}$ but $A_{...
1
vote
0answers
73 views
Convergence of Conjugate Gradient Algorithm
I am trying to solve a linear elasticity model using finite element discretization in a rectangle domain [0,1]x[0,1]. For the solution of the the linear system $Ku=F$ I am using the CG algorithm. ...
0
votes
0answers
48 views
Open-source iterative solvers robust to noise?
I need to solve $Ax=b$ in about 1 million dimensions. Furthermore, A is only accessible through matrix vector products, and these are noisy/inexact.
Is there any solver with Python interface I can try ...
6
votes
2answers
1k views
When is it easy to invert a sparse matrix?
(Crossposted on cstheory.SE)
When is it easy to invert a sparse matrix? Specifically, I'm wondering about the cases in which matrix inversion has similar cost to sparse matrix multiplication, hence ...
2
votes
1answer
46 views
Solving MX=N where M is structured as a Gaussian 4th-moment tensor
I'm looking to solve numerically the following equation for $(d,d)$ variable $X$, in Einstein summation notation
$$M_{ijkl}X_{kl}=N_{ij}$$
Where $M$ is a $(d,d,d,d)$ 4th-moment tensor of random ...
1
vote
0answers
318 views
Is there any function to calculate condition number of sparse matrix in Eigen libraray?
The function JacobiSVD and BDCSVD can calcuate condtion number of a dense matrix via singular values.
However I need to know condition number of a sparese matrix due to slow computation speed using ...
3
votes
1answer
108 views
Efficient solution to a structured symmetric linear system with condition number estimation
I have a real-valued linear system $Hx = b$ where $H$ is symmetric matrix** (not necessarily positive/negative definite) with a very particular structure:
$$
H = \begin{bmatrix} D && B \\ B^T &...
1
vote
0answers
82 views
2-norm of solution update suddenly becomes zero after a few iterations
I am trying to solve the Poisson equation in 2D for heterostructure devices. I have linearized the equation and discretized it using FDM. I am using BiCGStab to iteratively solve for the solution as ...
1
vote
2answers
328 views
How to use the Thomas-Algorithm to the Heat-diffusion-equation correctly
My post is structured in four parts:
I give you some information about the context my principal questions refer to.
I will tell you what I believe to know about the Thomas Algorithm. If I am wrong ...
0
votes
0answers
65 views
The error in SOR algorithm suddenly falls to zero when it reaches 1e-7 range
I am solving the Poisson equation for heterojunction using Fortran90. I use the SOR algorithm to arrive at the potential profile.
I see the weird behavior where the error (the difference between the $...
0
votes
0answers
36 views
Abnormalities when using SOR to solve the Poisson Equation
I am trying to solve the Poisson equation for Heterostructures using SOR. The equation to solve looks lik
I have discretized the Poisson equation using finite difference and my code is written in ...
0
votes
1answer
119 views
Is it possible to predict solution oscillation before solving the system by looking at coefficient matrix?
Question
When it is about solving a system of equations, is it possible to predict that whether high-frequency noise (e.g. checker-boarding) is likely to appear in the converged solution by looking at ...
4
votes
2answers
167 views
Solve two-player game - minimize the l-infinity norm of a matrix-vector product
I have a matrix $M$ with non-negative real entries, and I would like to minimize the objective function
$$\Phi(v) = \|Mv\|_\infty,$$
where $v$ is constrained to be a probability vector, i.e., $v_1+\...
0
votes
0answers
67 views
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 ...
0
votes
1answer
202 views
Solving large sparse system
I am working on a problem with very large sparse matrices. I'd like to compute $A^{-1} B$, that is a crucial part of converting DAE to ODE (and there is no workaround). Here size of $A$ is 2E+5 x 2E+5 ...
1
vote
1answer
177 views
Efficient way to solve a set of linear equations $Ax=b$ when $A$ is sparse and some elements of $b$ are equal to zero
I have a set of linear equations, $Ax=b$. And about half of the elements in the right-hand side (vector $b$) are equal to zero. My system matrix $A$ is a sparse complex matrix. And $A$ is in the size ...
7
votes
2answers
354 views
Is there an iterative solver for dense matrices with possible zero diagonal entries?
Is there an iterative solver that can handle potentially zero entries on the central diagonal? I am implementing a polynomial fitting algorithm (up to $10^{th}$-order) and my matrix is a "...
