A popular krylov subspace method for solving linear systems of equations, particularly those that exhibit symmetric positive definiteness.

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112 views

What are some reasons that Conjugate Gradient iteration does not converge?

I would greatly appreciate it if you could share some reasons the Conjugate Gradient iteration for Ax = b does not converge? My matrix A is symmetric positive definite. Thank you so much! Edit with ...
4
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1answer
155 views

Solving a system of nonlinear PDEs by minimization

I have two coupled nonlinear partial differential equations of the form: $ \begin{align} \dot{u} -f(u,u',u'',v,v',v'')=0 \\ \dot{v} -g(u,u',u'',v,v',v'')=0 \end{align} $ The boundary conditions are ...
4
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1answer
140 views

Linear equation system: Direct solver works, iterative solver does not

I have to solve for x in b = A*x, where a is sparse. This works fine with Matlab's mldivide: x = A \ b. Since I will have to use an iterative algorithm for very large A, I'm currently testing Matlab's ...
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2answers
86 views

SLATEC rouitne dslucs() and MKL correspondence

I am looking for a routine (or set of routines) in the Intel MKL that that can replace dslucs (Incomplete LU BiConjugate Gradient Squared Ax=b Solver) in ...
7
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2answers
351 views

What's the difference between conjugate gradient method and biconjugate gradient method

What's the difference between these two methods? Can a problem be solved by one method will be able to solved by the other? Can both/or one of them be parallelized with OpenMP and/or MPI?
4
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0answers
110 views

Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
3
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1answer
84 views

Would recalculating the residual in the conjugate gradient method help?

The conjugate gradient method suffers from an accumulation of errors as it continues. For this reason it is unwise to use it as a direct solver. My question is, would it help to recalculate the ...
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2answers
1k views

How to solve this system with conjugate gradient algorithm in matlab

CG Algorithm https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!386&v=3 System of equations, the question and the example https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!387&v=3 ...
2
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0answers
57 views

Which non-linear conjugate gradient possess finite termination property

There are many variants of non-linear conjugate gradient method available ( Flatcher-Reeves, Polak-Rebiere, Dai-Yuan). In case of minimization of quadratic function when first search direction is ...
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1answer
191 views

PETSc Krylov Subspace and nullspace

How is the nullspace correction implemented in the PETSc conjugate gradient solver? I searched the source code and documentation, but could not find references on the actual implementation. For ...
3
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2answers
244 views

Gradient descent and conjugate gradient descent

For a project I have to implement these two methods and compare how they perform on different functions. But when I read about it I see that for example the conjugate gradient method is meant to solve ...
8
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2answers
374 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 ...
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3answers
954 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
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1answer
409 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: ...
10
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4answers
385 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
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1answer
135 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
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2answers
172 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 ...
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2answers
996 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
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0answers
217 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
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2answers
93 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
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2answers
675 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 ...
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1answer
190 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
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1answer
317 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
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2answers
446 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 ...
14
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3answers
903 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
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2answers
166 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 ...
15
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3answers
4k 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
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2answers
260 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 $S_k(n_k,\alpha_k,r) = \frac{\alpha_k ^{n_k+3}}{(n_k+2)!}r^{n_k}\exp(-\alpha_kr)$ ...