Siggraph Asia 2018: Factor Once: Reusing Cholesky Factorizations on Sub-Meshes

A common operation in geometry processing is solving symmetric and positive semi-definite systems on a subset of a mesh with conditions for the vertices at the boundary of the region. This is commonly done by setting up the linear system for the sub-mesh, factorizing the system (potentially applying preordering to improve sparseness of the factors), and then solving by back-substitution. This approach suffers from a comparably high setup cost for each local operation. We propose to reuse factorizations defined on the full mesh to solve linear problems on sub-meshes. We show how an update on sparse matrices can be performed in a particularly efficient way to obtain the factorization of the operator on a sun-mesh significantly outperforming general factor updates and complete refactorization. We analyze the resulting speedup for a variety of situations and demonstrate that our method outperforms factorization of a new matrix by a factor of up to 10 while never being slower in our experiments.

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