## Abstract

We present an asymptotically faster algorithm for solving linear systems in well-structured 3-dimensional truss stiffness matrices. These linear systems arise from linear elasticity problems, and can be viewed as extensions of graph Laplacians into higher dimensions. Faster solvers for the 2-D variants of such systems have been studied using generalizations of tools for solving graph Laplacians [Daitch-Spielman CSC’07, Shklarski-Toledo SIMAX’08]. Given a 3-dimensional truss over n vertices which is formed from a union of k convex structures (tetrahedral meshes) with bounded aspect ratios, whose individual tetrahedrons are also in some sense well-conditioned, our algorithm solves a linear system in the associated stiffness matrix up to accuracy in time O(k^{1}/^{3}n^{5}/^{3} log(1/)). This asymptotically improves the running time O(n^{2}) by Nested Dissection for all k ≪ n. We also give a result that improves on Nested Dissection even when we allow any aspect ratio for each of the k convex structures (but we still require well-conditioned individual tetrahedrons). In this regime, we improve on Nested Dissection for k ≪ n^{1}/44 . The key idea of our algorithm is to combine nested dissection and support theory. Both of these techniques for solving linear systems are well studied, but usually separately. Our algorithm decomposes a 3-dimensional truss into separate and balanced regions with small boundaries. We then bound the spectrum of each such region separately, and utilize such bounds to obtain improved algorithms by preconditioning with partial States of separator-based Gaussian elimination.

Original language | English (US) |
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Title of host publication | STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Monika Henzinger, David Kempe, Ilias Diakonikolas |

Publisher | Association for Computing Machinery |

Pages | 773-786 |

Number of pages | 14 |

ISBN (Electronic) | 9781450355599 |

DOIs | |

State | Published - Jun 20 2018 |

Externally published | Yes |

Event | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: Jun 25 2018 → Jun 29 2018 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 |
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Country/Territory | United States |

City | Los Angeles |

Period | 6/25/18 → 6/29/18 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

- Eigenvalue bounds
- Linear system solver
- Nested dissection
- Preconditioning
- Truss stiffness matrix