# Questions tagged [conjugate-gradient]

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

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### Why not use the preconditioned residual as termination criterion for preconditioned CG?

I have a Poisson equation with wildly varying material parameters (1 .. 1000), wildly varying element sizes (5 nm .. 100 um) and some quite anisotropic (tetrahedral) elements (100 nm x 100 um). I use (...
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### Blowup of error in Conjugate Gradient method with periodic Dirichlet Poisson matrix

My problem is that the L2-Norm of the residual for the periodic Poisson matrix $P$ is initially decreasing but starts to blow up after a certain number of iterations. The blowup happens earlier the ...
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### Why does conjugate gradient work with this nonsymmetric preconditioner?

In this previous thread the following multiplicative way to combine symmetric preconditioners $P_1$ and $P_2$ for the symmetric system $Ax=b$ was suggested: \begin{align} P_\text{combo}^{-1} :=& ...
1answer
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### What are the differences between the different gradient-based numerical optimization methods?

I am interested in the specific differences of the following methods: The conjugate gradient method (CGM) is an algorithm for the numerical solution of particular systems of linear equations. The ...
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### 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 ...
1answer
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### Hessian-free and Truncated Newton methods

In this paper on Deep Learning for Machine Learning, the approach is referred to as Hessian-free method. That is because the Hessian is never computed explicitly. Instead, the product of the Hessian ...
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### Is it possible to predict the null space of a structure from contributing elements null spaces?

I am trying to solve an almost incompressible problem with heterogeneous properties by domain decomposition. Solution with CG converges slowly or divergerces completely. My problem becomes ill-...
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### Caveats of Hessian free method

Hessian free iterative optimization techniques like Newton-CG, do not explicitly compute the Hessian but instead approximate the product of the Hessian with a vector through finite difference. The ...
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### Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve Ax=b, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous direction ...
2answers
236 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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?