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

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### How to find a good preconditioner to the system $(A^T A + \lambda I) x = A^T b$?

The system in the title has a damper factor $\lambda > 0$ and the matrix $A$ is sparse and rectangular, with a structure I can exploit to solve matrix vector products very fast. My current solver, ...
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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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### 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 $x_*$ ...
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### Stochastic conjugate directions to improve convergence in narrow valleys

My question concerns a specific statement in this paper: N. N. Schraudolph and T. Graepel, "Conjugate Directions for Stochastic Gradient Descent," in Int. Conf. Artificial Neural Networks, Berlin, ...
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Nocedal and Wright on Conjugate Gradient Methods, p. 123, describe a restart strategy ... whenever two consecutive gradients are far from orthogonal $\qquad {{| \nabla f_k^T \ \nabla f_{k-1} |} \... 0answers 243 views ### Is it possible to predict the null space of a structure from contributing elements null spaces? I am trying to solve an almost incompressible problem with heterogeneous properties by domain decomposition. Solution with CG converges slowly or divergerces completely. My problem becomes ill-... 0answers 42 views ### Subspaces for Iterative methods In the original paper of Conjugate Gradients, the authors mention that if we pick the canonical basis$\{e_1,e_2,\ldots,e_n\}$, to obtain A-orthonormal vectors, we end up with the Gaussian elimination ... 0answers 2k views ### Understanding MATLAB's fmincg optimization function I'm researching numerical optimization. Recently I've come across a variant of a conjugate gradient method named fmincg. The function is written in MATLAB and is ... 0answers 109 views ### Conjugate gradient: the 1-norm of the residual I am trying to solve$Ax=b$using the conjugate gradient method. However, it is important to me to obtain a bound not only on the usual residual$||b-Ax_k||_2$but also on the quantity$||b-Ax_k||_1. ... 0answers 89 views ### How to implement conjugate gradient method to minimize this nonlinear action? Given a 2D stochastic differential equation: \begin{align} \dot{x}_{i}=f_{i}(\textbf{x})+g_{ij}\xi_{j}(t), \end{align} wherei=2$,$g_{ij}g_{jk}=2\epsilon\delta_{ik}$and$f(\textbf{x})=-\nabla\phi(\...
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Richardson's iteration introduce a scalar $\alpha$ to the update formula: $$\textbf{x}^{(k+1)} = \textbf{x}^{(k)} + \alpha \textbf{r}^{(k)}$$ And compute $\alpha$ by minimizing the spectral radius:...
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### 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 ...
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### 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 ...
I am studying optimization methods and I was able to understand and derive the search direction $$p_k = r_{k-1} + \beta p_{k-1}$$ for Conjugate Gradient Method, with  \beta = -\frac{p_{k-1}^...
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 ...