Questions tagged [linear-solver]
Referring to methods for solving linear systems of equations.
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0answers
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Numerical analysis, pivoting and incomplete LU decomposition
When doing LU decomposition, the algorithm will break down if any of the diagonal element $x_{ii}$ is zero. Therefore, we can use pivoting on the matrix such that $x_{ii}$ is no longer zero. That is ...
6
votes
1answer
98 views
numerical solution of an under-determined linear equation in high dimensions
I need to solve a linear regression problem $$Ax=y$$ which is hugely underdetermined. I have around $10^6$ features but only $10^3$ equations. So $A$ is a $1,000\times 1,000,000$ matrix and $y$ a ...
2
votes
2answers
86 views
Asymptotic Complexity of Gaussian Elimination using Complete Pivoting
I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $O(n^3)$.
Is it the same for complete pivoting?
1
vote
1answer
70 views
Conjugate gradient - ill-conditioning and numerical tolerance
I would like to solve system $Ax=b$, where $A$ is SPD, but very ill-conditioned ($\text{cond}(A)>10^{11}$). I am interested in using UNpreconditioned version of the conjugate gradient method.
Is ...
2
votes
1answer
131 views
Which C++ linear algebra library is probably the fastest on solving huge sparse [square matrix] linear system?
I am developing a 2D CFD solver for fluid-particle interaction. To solve Navier-Stokes equations on a grid of size $10000\times 10000$ cells (or >1 million cells), a large linear system $Ax=b$ with $A$...
1
vote
0answers
32 views
GMRES algorithm and Krylov base
I have a question about the precision of the GMRES algorith and its variation a s a function of the size of the Krylov subspace.
I want to solve a Poisson equation using a spectral method.
My problem ...
1
vote
1answer
47 views
Memory and time requirements of the scipy sparse spsolve
I have a system of fairly large set of linear equations (approximately 30K equations). I am using scipy.sparse.spsolve to solve these equations. Initially, I tried ...
0
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1answer
35 views
Can LINCS algorithm be used for colliding molecules?
Supposing that one molecule is static and one is dynamic,
can the dynamic one be solved with LINCS for its shape (angle, bond length) constraints and also keep collisions with static molecule off, ...
2
votes
1answer
80 views
Is steady linear elasticity inherently ill-conditioned?
Compared to the transient PDE for linear elasticity, the steady equations appear to less well-conditioned. Are they inherently ill-conditioned without the transient term?
The condition number for ...
3
votes
1answer
71 views
Condition number of matrix and effects of round off errors
In my numerical linear algebra class, I learned that for some matrices, it could have an element that is a very small number that is approximately 0 (and many orders of magnitude different from all ...
3
votes
1answer
123 views
Derivatives of Approximate Matrix inverses
I am cross posting this question to the mathermatics stack exchange. please find it either at this link, https://math.stackexchange.com/q/2952989/430980, or below:
I have a question concerning the ...
8
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0answers
174 views
Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system
Problem
Solving a non-linear system of equations.
The number of variables is the same as the number of equations.
When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
2
votes
2answers
115 views
Memory/speed tradeoff for many small matrix inverses
Problem
In the case of a finite element code, I have many small (order of 30x30) matrix inverses (or LU factorizations), one per finite element. These matrix inverses never change and must be applied ...
-1
votes
1answer
46 views
How to access solution to linear system in PETSc?
I have just started with PETSc hence it might seem like a very stupid question but I couldn't find any answer in manual.
After Calling KSPSolve, where can I access the soluition for my linear system?
...
2
votes
1answer
110 views
How many operations are needed for LAPACK's zgesv to solve a linear system?
I have a linear system of complex numbers. I am using LAPACK' zgesv (actually I am using intel MKL LAPACKE, but I am assuming the algorithm is the same). No assumption can be made about the system.
I ...
2
votes
2answers
178 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 ...
-1
votes
1answer
79 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 ...
2
votes
1answer
76 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
votes
1answer
122 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 ...
6
votes
1answer
145 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 ...
4
votes
2answers
134 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 ...
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0answers
100 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 $f(x)$:
$$ \...
1
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1answer
184 views
How to use CSDP to express a semidefinite program?
I am trying to use CSDP and am struggling with it. Consider, for example, the following semidefinite program
$$\begin{array}{ll} \text{minimize} & 0\\ \text{subject to} & Q - A' Q A - \...
2
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0answers
99 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 (...
1
vote
1answer
69 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
votes
1answer
116 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 ...
0
votes
0answers
53 views
How do I “push” matlab's lsqr solver to a particular solution?
The background to my problem can be found here: Iteratively solving a sparse, ill-conditioned system
I have a function that now works well. When I give it test data, I recover the expected result. ...
3
votes
0answers
98 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 ...
1
vote
1answer
81 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
votes
2answers
188 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
vote
1answer
84 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
votes
1answer
101 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 ...
2
votes
0answers
66 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)
$$x=(I-A)^{-1}b,$$...
1
vote
1answer
135 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
votes
1answer
97 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 ...
1
vote
1answer
104 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 ...
2
votes
1answer
126 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
55 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
vote
1answer
51 views
Optimal algorithm choice for mixed diagonal/dense problem
$$
\text{Let}\\
A, B \in \mathbb{C}^{n \times n} \text{ and } \hat{\alpha}, \hat{\beta} \in \mathbb{C}^{n}, \hat{f} \in \mathbb{C}^{2n}
\\
\text{Find }\\
\underline{\mathbf{x}} \in \mathbb{C}^{2n} \...
0
votes
1answer
56 views
Matlab backslash reordering algorithm
For the linear system $\mathbf A \mathbf x = \mathbf b$ generated from 2D Poisson equation using the standard central finite difference method,
$$
\mathbf A =
\begin{bmatrix}
\mathbf K & -\mathbf ...
0
votes
1answer
140 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
votes
1answer
52 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 ...
1
vote
1answer
68 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 $Z$ ...
1
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1answer
166 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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vote
0answers
52 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. ...
1
vote
0answers
41 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
votes
0answers
144 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 ...
1
vote
3answers
199 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 ...
1
vote
2answers
409 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
vote
1answer
202 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 ...
