Iterative updates

For the calculation of the LDA ground-state one can proceed either via the direct" methods, i.e. via the glocal minimization of the total free energy with respect to the electronic degrees of freedom, or via the the diagonalization (for large PW basis-sets necessarily iterative diagonalization) of the KS Hamiltonian in combination with an iterative update of chai ge-density and potential. 

Note that the terms depending on the model parameters are diagonal matrices. The full matrices A and depend only on the background conductivity distribution. Therefore, after precomputing full matrices Ae and A for the background model, the iterative updating of F (m) is relatively inexpensive during the inversion process. 

At the constraint-control level, the variables are considered as time-dependent signals. In this work, the constraint controller is designed so as to track the iteratively-updated active constraints by varying the process inputs along the constraint-seeking directions. 

We continue in the ensuing chapters with several tutorials tied together by the theme of how to exploit and/or treat multiple length scales and multiple time scales in simulations. In Chapter 5 Thomas Beck introduces us to real-space and multigrid methods used in computational chemistry. Real-space methods are iterative numerical techniques for solving partial differential equations on grids in coordinate space. They are used because the physical responses from many chemical systems are restricted to localized domains in space. This is a situation that real-space methods can exploit because the iterative updates of the desired functions need information in only a small area near the updated point. 

The models underwent a series of iterative updates. These formed the input for the next phase of model review. 

By iteratively updating w a maximum of F is guaranteed to be found the extracted component y, = w,x with maximum temporal predictability is the least complex signal and thus approaches the simplest source hidden in the mixtures, according to Stone s theorem in the CP learning rule. 

By iteratively updating the moment-tensor and centroid estimates, as well as the corresponding synthetic seismograms and kernels, and inverting the residual signal for additional perturbations in the source parameters, an optimal estimate of the point-source parameters is obtained. 

It has been also demonstrated that a Lagrangian decomposition can significantly reduce the computational burden associated with the solution of monolithic problems. In this chapter, the Optimal Condition Decomposition was applied. This technique facilitates the process required to iteratively update the Lagrangian multipliers. 

The Gibbs sampler, after r iterations, updates to the (r + l)-th iteration... 

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