Solving 1-Laplacians in Nearly Linear Time: Collapsing and Expanding a Topological Ball
نویسندگان
چکیده
Abstract We present an e cient algorithm for solving a linear system arising from the 1-Laplacian of a collapsible simplicial complex with a known collapsing sequence. When combined with a result of Chillingworth, our algorithm is applicable to convex simplicial complexes embedded in R. The running time of our algorithm is nearly-linear in the size of the complex and is logarithmic on its numerical properties. Our algorithm is based on projection operators and combinatorial steps for transferring between them. The former relies on decomposing flows into circulations and potential flows using fast solvers for graph Laplacians, and the latter relates Gaussian elimination to topological properties of simplicial complexes.
منابع مشابه
ar X iv : 0 80 8 . 41 34 v 2 [ cs . D S ] 1 9 Se p 20 08 Spectral Sparsification of Graphs ∗
We introduce a new notion of graph sparsificaiton based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier ...
متن کاملSpectral Sparsification of Graphs
We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier ...
متن کاملSolving multiobjective linear programming problems using ball center of polytopes
Here, we aim to develop a new algorithm for solving a multiobjective linear programming problem. The algorithm is to obtain a solution which approximately meets the decision maker's preferences. It is proved that the proposed algorithm always converges to a weak efficient solution and at times converges to an efficient solution. Numerical examples and a simulation study are used to illu...
متن کاملA linear work, O(n1/6) time, parallel algorithm for solving planar Laplacians
We present a linear work parallel iterative algorithm for solving linear systems involving Laplacians of planar graphs. In particular, if Ax = b, where A is the Laplacian of any planar graph with n nodes, the algorithm produces a vector x̄ such that ||x − x̄||A ≤ ǫ, in O(n log(1/ǫ)) parallel time, doing O(n log(1/ǫ)) work, where c is any positive constant. One of the key ingredients of the solver...
متن کاملExpansion methods for solving integral equations with multiple time lags using Bernstein polynomial of the second kind
In this paper, the Bernstein polynomials are used to approximate the solutions of linear integral equations with multiple time lags (IEMTL) through expansion methods (collocation method, partition method, Galerkin method). The method is discussed in detail and illustrated by solving some numerical examples. Comparison between the exact and approximated results obtained from these methods is car...
متن کامل