In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff.[1]
Nested dissection consists of the following steps:
Form an undirected graph in which the vertices represent rows and columns of the system of linear equations, and an edge represents a nonzero entry in the sparse matrix representing the system.
Recursively partition the graph into subgraphs using separators, small subsets of vertices the removal of which allows the graph to be partitioned into subgraphs with at most a constant fraction of the number of vertices.
Perform Cholesky decomposition (a variant of Gaussian elimination for symmetric matrices), ordering the elimination of the variables by the recursive structure of the partition: each of the two subgraphs formed by removing the separator is eliminated first, and then the separator vertices are eliminated.
As a consequence of this algorithm, the fill-in (the set of nonzero matrix entries created in the Cholesky decomposition that are not part of the input matrix structure) is limited to at most the square of the separator size at each level of the recursive partition. In particular, for planar graphs (frequently arising in the solution of sparse linear systems derived from two-dimensional finite element method meshes) the resulting matrix has O(n log n) nonzeros, due to the planar separator theorem guaranteeing separators of size O(√n).[2] For arbitrary graphs there is a nested dissection that guarantees fill-in within a factor of optimal, where d is the maximum degree and m is the number of non-zeros. [3]
^George (1973).
^Lipton, Rose & Tarjan (1979); Gilbert & Tarjan (1986).
In numerical analysis, nesteddissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph...
few shortest paths. This can be approximated using nesteddissections. To compute a nesteddissection, one recursively separates a graph into two parts...
Hamburg. 38: 199–220. doi:10.1007/BF02996932. George, J. Alan (1973), "Nesteddissection of a regular finite element mesh", SIAM Journal on Numerical Analysis...
sparse matrices: Frontal solver — used in finite element methods Nesteddissection — for symmetric matrices, based on graph partitioning Levinson recursion...
problems on these graphs. Separator hierarchies may also be used in nesteddissection, an efficient variant of Gaussian elimination for solving sparse systems...
this concept lies in sparse matrix computations, namely for using nesteddissection to compute the Cholesky factorization of a (symmetric) matrix in parallel...
in the MMGN. Thalamus Deep dissection of brain-stem. Lateral view. Deep dissection of brain-stem. Lateral view. Dissection of brain-stem. Dorsal view...
tissue is removed from the body or plant, and then, often following expert dissection in the fresh state, placed in a fixative which stabilizes the tissues...
He is supposedly hyper-intelligent with an infinity of intricate plans nested within plans, but he sometimes over-thinks things to the point of coming...
hypothesis into question, with many recent phylogenies finding them to be nested within the conifers. Though it is clear they are all related, the exact...
"Prevalence of the parasite Strepsiptera in Polistes as detected by dissection of immatures". Insectes Sociaux. 50 (1): 62–68. doi:10.1007/s000400300010...
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silicone shell pre-filled with viscous silicone gel; structured implants use nested elastomer silicone shells and two saline-filled lumen; and the alternative...
victims of circumstance." Gardner called it "a gripping but gruelling dissection of a relationship that flounders on mismatched desire, conditioned responses...
Therefore, dissections are usually performed in order to determine the sex, and thus the wasp's role in the colony. The regulating behavior, or nest building...
Button, David J.; Barrett, P.M., Paul M; Porro B., Laura (2019). "Digital dissection of the head of the rock dove (Columba livia) using contrast-enhanced computed...
performed as Oenone in the Wooster Group's production of an "exhilarating dissection" of Racine's tragedy Phèdre entitled To You, the Birdie!, at St. Ann's...
announced a conclusion that they had not reached. ... Early on, a meticulous dissection of the body ... confirmed [that] Elvis was chronically ill with diabetes...
positive, depending on the extent of lymph node spread, a radical lymph node dissection will often be performed. If the disease is completely resected, the patient...
Society. Cross-section: C=Back muscles, H=main organ, I=Hunter's organ Dissection, showing the electric organs inside the body. At right, the skin is folded...
carnivorous lifestyle, genetic and fossil evidence places cetaceans as nested within even-toed ungulates, most closely related to hippopotamus within...
and easy to interpret, and are often sold by companies to schools for dissection by students as a lesson in biology and ecology. Owl eggs typically have...
The resulting tangle of blood vessels, often called a nidus (Latin for 'nest'), has no capillaries. It can be extremely fragile and prone to bleeding...
immediate lymph node dissection at presentation had a higher five-year survival rate than patients who underwent a delayed lymph node dissection. This provided...
2020-06-25. Retrieved 2018-04-03. "Examination of "pollen Balls" in the nests of the Alfalfa Leafcutting Bee, Megachile rotundata". United States Department...
Zalisko, Edward J. (2015). Comparative Vertebrate Anatomy: A Laboratory Dissection Guide (Seventh ed.). New York: McGraw-Hill Education. pp. 71–72. ISBN 9780077657055...
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