Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

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0answers
32 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 ...
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48 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(...
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1answer
40 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 \...
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1answer
82 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 $\Omega = [-1,1]^2$)...
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47 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 ...
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2answers
205 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, ...
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69 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 ...
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2answers
163 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 ...
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2answers
87 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 ...
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3answers
158 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 ...
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1answer
45 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 ...
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1answer
124 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 ...
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1answer
68 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 ...
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1answer
56 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.
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1answer
133 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://...
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1answer
32 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,...
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0answers
37 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'...
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60 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 ...
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1answer
116 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 ...
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1answer
196 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 ...
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68 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 ...
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45 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 $A_{jk} = A_{kj}$ but $A_{jk} \neq ...
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37 views

How can the choice of coarsening factor affect Multigrid's convergence?

The linear system $Ax=b$ is coming from the discretization of an elliptic PDE. Multigrid method is used in order to solve it. Suppose $c_0$ is the coarsening factor on level 0 and $c_m$ the coarsening ...
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67 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. ...
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43 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 ...
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2answers
449 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 ...
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1answer
42 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 ...
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213 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
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1answer
102 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 &...
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0answers
81 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 ...
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2answers
255 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 ...
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62 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 $...
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32 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 ...
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1answer
107 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 ...
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2answers
155 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+\...
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62 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 ...
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1answer
144 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 ...
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1answer
154 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 ...
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2answers
315 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 "...
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0answers
61 views

Solver for large dense BVP system in python

I have a large system of boundary value problems of the form $$ \frac{d^2 y }{dt^2} = C(t) y + b(t), $$ where the variable $y$ is a vector that has anywhere from 50 to around 500 components, $C$ is a ...
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1answer
131 views

Solving a sparse linear system using transpose of lower triangular matrix without copying

I have a sparse lower-triangular matrix $L$, and a right-hand side $b$, and I'd like to solve the linear system $$L^T x = b$$ but without explicitly creating $L^T$. Ideally, I could write something ...
2
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1answer
99 views

Solving an m x m symmetric linear system involving a matrix multiplication versus an (n+m) x (n+m) system

Suppose that $R$ and $D$ are an $n \times m$ and $m \times m$ matrices. Assume that $m \ll n$ and that $D$ is positive definite. We would like to solve the system $(R^T R + D) x = R^T b$. This ...
3
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1answer
221 views

Is there any reason to scale a matrix before (sparse) Cholesky decomposition?

I have a sparse symmetric positive-definite matrix $M$ and I expect the entries in some rows/columns to have very different orders of magnitude (up to a factor of $10^8$) than the entries in others. ...
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2answers
238 views

Why OpenFOAM uses its own data structures and linear solvers?

I wonder why OpenFOAM code has its own data structures Lists, HashTables, ... etc. when there is the STL in C++? Another ...
3
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1answer
265 views

What happens when I use a conjugate gradient solver with a symmetric positive semi-definite matrix?

I have a symmetric positive semi-definite matrix, i.e., a laplacian and wonder what may happen when I use a CG solver, that is an algorithm for positive definite matrices. What happens when the ...
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2answers
101 views

Solution of symmeric/non-symmetric linear system

I would like to understand what happens in the following: I have a really simple Poisson problem, in 1D, with $u_0 = u_N = 0$. I assembled the stiffness matrix and the right-hand side, and I applied ...
4
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2answers
180 views

More stable method of back substitution?

I've been tinkering a little in Fortran (2008) and wrote the following to solve $Rx=b$ for $R\in\mathbb{R}^{n\times n}$ upper-triangular, $x,b\in\mathbb{R}^n$. My code looks like this: ...
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1answer
109 views

Solving triangular matrix equations on a GPU

Suppose I have these two $N\times N$ lower triangular banded matrices: $A = \begin{bmatrix} a_0 & & \\ a_1 & a_0 & \\ a_2 & a_1 & a_0 \\ a_3 & a_2 & a_1 & a_0 \\ &...
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1answer
75 views

Parallel solution of PDEs

Which is the best approach to solve a PDE in parallel: 1.To split the mesh the mesh in N parts and every processor works on its own part or 2.To take the global linear system Ax=b and solve it in ...
3
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3answers
281 views

Using matrix exponential to solve linear system

Consider the system of linear equations: $$ Ax=b \tag{1} \label{eq1} $$ where $A\in\mathbb F^{n\times n}$, diagonalizable dense matrix, over the field $\mathbb F$ of real or complex numbers, $x\...

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