Nested dissection

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.

Nested dissection consists of the following steps:

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). For arbitrary graphs there is a nested dissection that guarantees fill-in within a ${\displaystyle O(\min\{{\sqrt {d}}\log ^{4}n,m^{1/4}\log ^{3.5}n\})}$ factor of optimal, where d is the maximum degree and m is the number of non-zeros.

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