Questions tagged [linear-solver]

Referring to methods for solving linear systems of equations.

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2answers
685 views

Lost on Matrix Inversion

I try to implement some big matrix inversion. My system configuration is Hardware:- Memory: 62.8GiB, Processor: Intel Xeon(R)CPU E5-2670 v3 @2.30GHZ*48 To implement matrix inversion I am using ...
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1answer
112 views

Use of GPU with respect to CPU

I have research work where I need to compute a matrix inversion. The matrix has a size $31300\times31300$. I am using a universal java matrix package to invert this matrix. But as the dimension of the ...
3
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1answer
223 views

How can a CG solver solve a non positive definite sparse matrix

I am using the CUSP CG solver and I ran it on couple of sparse matrices from the University of Florida sparse matrix collection. The solver was able to solve non positive definite sparse matrices. My ...
2
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1answer
196 views

Can the Power Method be used here?

Given a set of $n$ points on which a triangulation is performed, it is possible to construct coefficients $\lambda_{ij}>0$ such that each point $x_i$ is a convex combination of the points connected ...
7
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1answer
283 views

Solving linear system of the form $ABx=b$

I would like to solve a linear system of the form $ABx=b$ in parallel, where $A$ and $B$ are large, sparse matrices. Currently, I am forming the system matrix $AB$ explicitly, but I would like to ...
5
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2answers
381 views

How "sparse" should a sparse matrix be to see benefits?

I have a matrix, whose size scales as $2^N$ (assume even $N$). In each row of the matrix, only about $2^{N/2}$ of the entries are filled ($N$ can be somewhere between 10 and 40, depending on what's ...
5
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0answers
332 views

conjugate gradient for Newton's method with non positive definite Hessian matrix

I want to minimize a non-linear function $f(x)$ using Newton's method. At each optimization step, I compute a descent direction $d$ to update $x$ using a second-order approximation of $f(x)$: $$ \...
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1answer
239 views

How to use CSDP to express a semidefinite program?

I am trying to use CSDP and am struggling with it. Consider, for example, the following semidefinite program $$\begin{array}{ll} \text{minimize} & 0\\ \text{subject to} & Q - A' Q A - \...
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0answers
125 views

Efficiently solve linear system with matrix quadratic form

Take the system $$A^TCAx=b$$ where $$A\in\mathbb{R}^{n\times m},\;C\in\mathbb{R}^{n\times n},\;x,b\in\mathbb{R}^{m}, \;m\leq n$$ and $$A^TA=I$$ and $$Cy=d$$ can be solved efficiently in general (...
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1answer
94 views

Partitioning SPD matrix with METIS to preserve block SPD-ness

I am using the METIS to partition a matrix and then using domain decomposition to solve the subdomains in parallel using the Restricted Additive Schwarz method. I am currently trying to solve some ...
2
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1answer
668 views

Finite Elements: using preconditioned conjugate gradients with incomplete cholesky decomposition

I have to write a little finite elements code in C. I was asked to implement the conjugate gradients method, which I have done. Now, I am looking to improve further the efficiency of my program by ...
3
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0answers
149 views

Iteratively solving a sparse, ill-conditioned system

I have a sparse (density = 0.2%), ill-conditioned system that I am trying to solve, with no luck. Background I have a sequence of sampled data, where two of every 8 samples have been zeroed due to a ...
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1answer
140 views

Reordering algorithm for minimization of ram usage of a skyline matrix

The stiffness matrix of $Ax=B$ system of linear equations, where $A$ is an $n\times n$ symmetric matrix stored in the form of symmetric skyline matrix, that is associated with a finite element model ...
2
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2answers
356 views

Automatic Differentiation - reverse accumulation of linear system solve

I am studying the reverse mode of automatic differentiation. The reverse mode of automatic differentiation allows the efficient computation of a the derivative of a single dependent variable $y$ with ...
1
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1answer
815 views

Conjugate Gradient for non symmetric matrix

I have a large sparse matrix which is symmetric for the location of non zero values, but the values are different. Could I still use the CG method? I don't have much knowledge of linear algebra, the ...
3
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1answer
111 views

Solve $A^{-1} b$ when one column is replaced

Given square matrix $A_0$, vector $b$, vector $A_0^{-1}b$ and matrices $A_1, A_2, \dots, A_k$, in which each $A_i$ is generated from $A_{i-1}$ by replacing one single column, I would like to find an ...
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0answers
101 views

Acceleration of matrix geometric series

Suppose we want to find $x$ such that: $$x=b+Ax$$ where $A$ is a large sparse square matrix with eigenvalues in the unit circle. There are two representations of the solution: 1) $$x=(I-A)^{-1}b,$$...
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1answer
196 views

Efficiency of parallel direct linear solver

I am currently working on solving a positive definite symmetric systems in parallel. The parallel direct solver I used is MUMPS. However, the performance and efficiency of the parallel direct solver ...
2
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1answer
158 views

Which iterative method and preconditioner from petsc should be used when solving linear algebra in parallel?

I am currently trying to parallelize the incompressible flow solver code. However, when I run the code I realise that the parallel code takes much longer time than sequential code to finish one ...
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1answer
124 views

Questions about iterative projection methods in Saad book

I am reading Chapter 5 of Saad's iterative methods book, and I don't understand section 5.2.1 about the two propositions of optimality results. In the statements of the propositions, what does it mean ...
3
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2answers
408 views

MA57 vs HSL_MA57: symmetric indefinite solvers

What are the differences between MA57 and HSL_MA57 solvers? I'm in an optimization class that will make use of symmetric indefinite factorizations, and I'm trying to learn about the distinction ...
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0answers
60 views

Why do higher order finite elements (Q2) do not perform well for large Peclet number flows, as compared to Q1 finite elments?

I a solving the 2d steady state convection-diffusion problem on the famous flow around a cylinder in rectangular domain benchmark. My numerical results show that with Q1 finite elements, the solver is ...
1
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1answer
56 views

Optimal algorithm choice for mixed diagonal/dense problem

$$ \text{Let}\\ A, B \in \mathbb{C}^{n \times n} \text{ and } \hat{\alpha}, \hat{\beta} \in \mathbb{C}^{n}, \hat{f} \in \mathbb{C}^{2n} \\ \text{Find }\\ \underline{\mathbf{x}} \in \mathbb{C}^{2n} \...
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1answer
85 views

Matlab backslash reordering algorithm

For the linear system $\mathbf A \mathbf x = \mathbf b$ generated from 2D Poisson equation using the standard central finite difference method, $$ \mathbf A = \begin{bmatrix} \mathbf K & -\mathbf ...
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2answers
199 views

How to verify solution to pre-conditioned linear systems solver?

I am solving Ax=b. A has a very large condition number (> O(10^10)) I am using the conjugate gradients method with point jacobi pre-conditioning. I obtained a solution 'x' that "looks" reasonable. ...
2
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1answer
67 views

Implicit solution to Sylvester equation

Suppose a matrix $M\in\mathbb{R}^{n\times n}$ is defined as the solution to a Sylvester equation $$AM+MB=C,$$ for some fixed (known) matrices $A,B,C$. In the regime where $n$ is large, we may with ...
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1answer
74 views

Solving for $C$ in $Q = YCZ$ using least squares in Matlab

I am trying to solve for the matrix $C$ in $Q = YCZ$ in matlab. I have preliminary results but they don't seem realistic. Here, $Q$ is $n \times m-1$, $Y$ is $n \times p$, $C$ is $p \times m$ and $Z$ ...
2
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1answer
219 views

Method to solve linear, first order ODE of generalized matrix matrix form

The equation and its meaning: Consider two sets $(A)_{l=0,...,m_a},$,$(B)_{l=0,...,m_b}$ of hermitian matrices and a set of positive semidefinite matrices $(C)_{l=0,...,m_c}$. Each matrix has the ...
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0answers
62 views

Parallel dense solve with submatrices from mesh refinement with Petsc

For a Bounday Element Method problem I require the solution of a system of linear equations with multiple right-hand sides. Though this is a dense system, I still want to do it via Petsc in parallel. ...
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0answers
43 views

Iterative solver, which balances the equations with the biggest errors

I am trying to understand the mathematical side of an algorithm, which approximates the solution of coulomb's law. Their approach is to balance the two elements with the biggest positive/negative ...
2
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0answers
218 views

Iterative linear solvers compatible with automatic differentiation?

I'm using automatic differentiation on a function that contains a sparse nonsymmetric linear system to be solved. I was using BiCGStab to solve this part of the function, but noticed the derivatives ...
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3answers
354 views

Practical reference on sparse linear solvers for PDEs (Navier-Stokes, Poisson) and on learning PETSc

My background is mainly engineering and applied research and I have been a developer or some CFD software, but mostly at high level without worrying about linear solvers and the like. This has been ...
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2answers
910 views

Linear solvers: How to deal with a singular system? (Poisson equation with Neumann boundary conditions)

The question is in the context of iterative numerical solution of large PDE systems with Finite Differences or Finite Elements: Stating the Poisson equation with Neumann boundary conditions will lead ...
1
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1answer
482 views

Algebraic multigrid in PETSc

Consider a potential/poisson equation on a very large, complicated geometry. Currently, an self-written FEM and linear solvers from NumPy are used. Performance is, of course, not good enough for ...
2
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0answers
61 views

Scaling for a nonsymmetric eigenvalue problem

I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in real eigenvalues and ...
3
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1answer
1k views

Why is the speed of the parts of the LU-decomposition so different?

I know that an easy way to solve the matrix problem $$A\cdot x=b$$ is the LU decomposition $$\begin{split} L,\,U&=\text{lu}(A)\\ y&=\text{solve}(L,\,b)\\ x&=\text{solve}(U,\,y) \end{split}$...
5
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2answers
498 views

Solving Poisson equation with current BC using FEM

I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$ Insulator (Neumann BC) Electrode set at some ...
3
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5answers
10k views

Fast c++ library to solve very big sparse systems

I am working on a project with electrical circuits, where I am trying to compute the voltages at all the nodes of an electrical circuit. I know that the electrical circuit is a perfect grid, so each ...
1
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1answer
220 views

Fast counting of all submatrices of a binary matrix with a full column rank

I have a binary full-rank matrix of size, say, $25 \times 50$. I need to count how many subsets of its columns form matrices with a full column rank, i.e. all the columns in the subset are linearly ...
2
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1answer
1k views

Which is the best subroutine available for solving sparse linear system of equations [closed]

I am trying to solve the system of linear equations: $AX=B$. For this currently I am using Intel MKL Pardiso solver. It works well when the order of $A$ is around $13500\times13500$ and below. Above ...
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0answers
925 views

Tridiagonal Solver in Python

I have a code I'm working on that involves solving a 1D Schrodinger equation using a Crank-Nicolson time step. The code is written in NumPy/SciPy, and I was doing a bit of profiling and discovered ...
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0answers
255 views

How to reshape matrix into row-major order for MKL DSS?

I would like to use MKL to solve a sparse linear system. I chose the DSS (Direct Sparse Solver) interface, which implements the following steps: ...
2
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3answers
2k views

How to get multiple solutions for a optimization problem using any kind of software

I have a optimization problem in which the optimal objective value occurs at multiple point in the feasible space. If I run my problem in LINGO software then it gives me the optimal objective value at ...
0
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1answer
107 views

Writing a non-square linear system in standard form $A\cdot{x}=b$

I have spend the last few days working my way through an interesting paper and I'm building a numerical model so I can apply the method. However, I am getting stuck at an "it can be shown" step. I am ...
4
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1answer
3k views

Iteratively solving 3D Poisson equation in MATLAB

I have written a function that sets up a sparse matrix A and RHS b for the 3D Poisson equation in a relatively efficient way. The set-up is nothing fancy: I have extended the 2D 5-point stencil to an ...
0
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1answer
111 views

Iterative single variable solutions in large linear systems

I have a system where $A$ is a large $n\times n$ marix with fast MVMs. It may have many nonzero entries (albeit in a structured way so as to allow fast MVMs), and is not necessarily diagonally ...
4
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1answer
320 views

Test matrices for large sparse overdetermined system of linear equations

I'm working on some c++ code to solve (conjugate gradient, least squares conjugate gradient, LSQR,..) large sparse overdetermined systems of linear equations. There is a twist to my matrices and the ...
2
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0answers
104 views

Optimization based integration for MPM

I'm considering implementing (just for simplicity) the unconstrained implicit optimization based integration for Material Point Method as described in Chenfanfu Jiang's thesis on MPM (the minimization ...
2
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0answers
254 views

Fast solution of a heptadiagonal linear system

I have a linear system of the form $\mathbf{A}\mathbf{x} = \mathbf{f}$. If the length of the vector $\mathbf{x}$ is $N$, meaning that there are $N$ unknowns, then the matrix $\mathbf{A}$ has seven ...
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0answers
94 views

QR via Householder: less computationally complex variants?

I'm a probabilist and need to do a few computations for a rather big linear least squares problem, so I'm trying to optimize the computation as far as is feasible to me. In computing the QR ...

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