Chebyshev acceleration

The Gauss-Seidel method can be accelerated by calculating the next approximation and then deliberately overshooting it. Chebyshev acceleration (polynomial extrapolation) may also be used to improve the rate of convergence. 

If it were not for the nonlinear nature of the SCF loop, i.e., if H were a fixed operator, this approach would be equivalent to the well-known Chebyshev accelerated subspace iteration proposed by Bauer [32], and later refined by Rutishauser [33,34]. ... 

A slight variation of this procedure is known as Chebyshev acceleration of SOR and consists of implementing the SOR procedure in two passes, one over the odd matrix elements and one over the even matrix elements with a modification to Eq. (12.76) for the first pass The resulting iterative expressions being ... 

Another modification (frequently combined with SSOR) is the Chebyshev semi-iterative acceleration, which replaces the approximation x + by... 

This assumption is apparently made in all time independent diffusion codes which are accelerated by means of Chebyshev polynomials. (See, for example, [46].) It has not been proved to be true for general heterogeneous reactor models in n dimensions except for the trivial cases of only one lethargy group and homogeneous bare problems. 

As in the case of successive overrelaxation, the efficiency of the application of Chebyshev polynomials in accelerating the outer iterations depends upon the accurate estimation of the particular constant, 5, the dominance ratio for the matrix T. A practical numerical method for estimating <7 is given in [45]. 

After initial testing on small systems, Chelikowsky s group extended their real-space code (now called PARSEC) for a wide range of challenging applications.The applications include quantum dots, semiconductors, nanowires, spin polarization, and molecular dynamics to determine photoelectron spectra, metal clusters, and time-dependent DFT (TDDFT) calculations for excited-state properties. PARSEC calculations have been performed on systems with more than 10,000 atoms. The PARSEC code does not utilize MG methods but rather employs Chebyshev-filtered subspace acceleration and other efficient techniques during the iterative solution process. When possible, symmetries may be exploited to reduce the numbers of atoms treated explicitly. 

We complete this section with a listing of other algorithmic developments in real-space electronic structure. As mentioned above, the PARSEC code has incorporated alternative techniques for accelerating the solution of the eigenvalue problem based on Chebyshev-filtered subspace methods, thus circumventing the need for multiscale methods. Jordan and Mazziotti have developed new spectral difference methods for real-space electronic structure that can yield the same accuracies as the FD representation with... 

Parallel Self-Consistent-Field Calculations via Chebyshev-Filtered Subspace Acceleration. 

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