A popular krylov subspace method for solving linear systems of equations, particularly those that exhibit symmetric positive definiteness.
8
votes
2answers
81 views
Genetic algorithm vs conjugate gradient method
I am trying to optimize some force-field parameters in a molecular framework so that the result of simulation comes as close as it can to the experimental structure.
In the past, I have written a ...
11
votes
3answers
230 views
Problems where Conjugate gradient works much better than GMRES
I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD (symmetric-positive-definite) because it requires less ...
3
votes
1answer
100 views
nonlinear conjugate gradient for multivariable functions
For the optimization problem $\underset{\mathbf{x}\in \mathbb{R}^n}{\operatorname{argmin}} f(\mathbf{x})$, we can use the following standard nonlinear conjugate gradient method to find the solution:
...
7
votes
3answers
228 views
Calculating determinant while solving $Ax=b$ using CG
I am solving $Ax=b$ for a huge sparse positive definite matrix $A$ using the conjugate gradient (CG) method. It is possible to compute the determinant of $A$ using the information produced during the ...
4
votes
1answer
97 views
Is there a nonlinear solver similar to CGNR evaluating only $A^TAx$?
First of all, I am quite new to this field and I excuse myself in advance for any stupid content in this question.
In the field of compressed sensing or deblurring I have a nonlinear optimization ...
0
votes
2answers
103 views
Checking for error in conjugate gradient algorithm
What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
5
votes
2answers
264 views
How to remove Rigid Body Motions in Linear Elasticity?
I want to solve $K u = b$ where $K$ is my stiffness matrix. However some constraints may be missing an therefore some rigid body motion may be still present in the system (due to eigenvalue zero). ...
5
votes
0answers
136 views
Estimate extreme eigenvalues with CG
CG may be used to estimate the extremal eigenvalues of a SPD matrix (by computing eigenvalues of tridiagonal matrix associated with the Lanczos algorithm). After a few iterations the largest ...
3
votes
2answers
79 views
How can you explain the following bound on the inner product?
I am reading a paper on stability of CG, and I came across the following statement:
\begin{equation}
\frac{\|A\|\,\|p\|^2}{\langle p,Ap\rangle} \leq \kappa(A)
\end{equation}
where $\kappa(\cdot)$ is ...
2
votes
2answers
255 views
Usage of DifferentiableMultivariateFunction and NonLinearConjugateGradientOptimizer
I try to use the matlab fminunc functionality in java and found the optimisation functions in commons-math. I have absolutely no clue how to use them right because the example tells nothing about the ...
1
vote
1answer
113 views
Gradient descent to stationary, or accumulation point
I recently came across the notion of an accumulation point as a result of a certain gradient descent variation. The following definition was found:
An accumulation point $P$ is such that there are an ...
3
votes
1answer
171 views
Is a checkerboard block decomposition of a matrix useful when solving a linear system with a parallel conjugate gradient method?
According to these lecture notes, a checkerboard block decomposition should exhibit better scalability when applied to parallel matrix-vector multiplication (presumably because of greater cache hit ...
4
votes
2answers
238 views
Sufficient conditions to ensure convergence of the conjugate gradient method
I know that a conjugate gradient method is guaranteed to converge to the solution of a linear system if the matrix is positive definite. I'm working with a family of matrices that have the following ...
13
votes
3answers
441 views
What is the principle behind the convergence of Krylov subspace methods for solving linear systems of equations?
As I understand it, there are two major categories of iterative methods for solving linear systems of equations:
Stationary Methods (Jacobi, Gauss-Seidel, SOR, Multigrid)
Krylov Subspace methods ...
3
votes
2answers
152 views
Looking for a mathematical proof of stability in floating point arithmetic of CG - any reference?
I am looking for a reference - paper, book, discussion, anything that has a mathematical proof for stability of the conjugate gradient method in floating point arithmetic.
Something similar for ...
11
votes
3answers
2k views
BFGS vs. Conjugate Gradient Method
What considerations should I be making when choosing between BFGS and conjugate gradient for optimization? The function I am trying to fit with these variables are exponential functions; however, the ...
4
votes
2answers
180 views
best way to optimize a function with linear/non-linear parameters
I am trying to fit some raw data using a function of the form
$f(r) = \sum_{i=1}^{K} d_kS_k(n_k,\alpha_k,r)$
where
$S_k(n_k,\alpha_k,r) = \frac{\alpha_k ^{n_k+3}}{(n_k+2)!}r^{n_k}\exp(-\alpha_kr)$
...
