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Load Flow
Published in Antonio Gómez-Expósito, Antonio J. Conejo, Claudio A. Cañizares, Electric Energy Systems, 2018
Antonio Gómez-Expósito, Fernando L. Alvarado
There are many possible reordering methods, all of which reduce the number of entries in the factored Jacobian. The three main methods proposed by Tinnney and Hart [8] are A priori ordering the nodes according to the valence or degree of the node (valence or degree refers to the number of connections between the node and the neighboring nodes or, equivalently, to the number of nonzero entries in the corresponding Jacobian row or column).Dynamic ordering of the nodes according to their valence. In this case, the valence is defined as the number of connections between the nodes and its neighbors during each step of the elimination process. This method is also known as the minimum degree algorithm.Dynamic ordering of the nodes according to valence, with the valence defined as the number of additional entries (fill-ins) that would occur if the node was eliminated at that step of the elimination process.
Load Flow
Published in Antonio Gómez-Expósito, Antonio J. Conejo, Claudio Cañizares, Electric Energy Systems, 2017
Antonio Gómez-Expósito, Fernando L. Alvarado
There are many possible reordering methods, all of which reduce the number of entries in the factored Jacobian. The three main methods proposed by Tinnney and Hart [8] are A priori ordering the nodes according to the valence or degree of the node (valence or degree refers to the number of connections between the node and the neighboring nodes or, equivalently, to the number of nonzero entries in the corresponding Jacobian row or column).Dynamic ordering of the nodes according to their valence. In this case, the valence is defined as the number of connections between the nodes and its neighbors during each step of the elimination process. This method is also known as the minimum degree algorithm.Dynamic ordering of the nodes according to valence, with the valence defined as the number of additional entries (fill-ins) that would occur if the node was eliminated at that step of the elimination process.
Permutation and Grouping Methods for Sharpening Gaussian Process Approximations
Published in Technometrics, 2018
In the numerical analysis literature on sparse matrix factorizations, it is widely recognized that row-column reordering schemes for large sparse symmetric positive definite matrices are essential for increasing the sparsity of the Cholesky factor (Saad 2003). Finding an optimal such ordering is an NP-complete problem (Yannakakis 1981), and so the algorithms in use are necessarily heuristic, but heuristic algorithms, such as the approximate minimum degree algorithm (Amestoy, Davis, and Duff 1996), have proven to be effective. The goals in Gaussian process approximation are related in that we search for an ordering that produces an approximately sparse inverse Cholesky factor. However, the problem at hand here is more difficult and perhaps less well-defined than sparse matrix factorization; our task is to find the best reordering of observations that produces accurate approximations to the joint density or an approximate likelihood function that delivers efficient parameter estimates, a criterion that depends on the derivative of the approximate and exact likelihood functions with respect to the covariance parameters. Thus, it is extremely unlikely that we will be able to find “optimal” orderings for large datasets. Nevertheless, as in the sparse matrix case, this article shows that heuristically motivated orderings can offer significant improvements in statistical efficiency and model approximation over default orderings.