Fast Factorization Update for General Elliptic Equations Under Multiple Coefficient Updates

Xiao Liu, Jianlin Xia, Maarten V. de Hoop

Abstract: For discretized elliptic equations, we develop a new factorization update algorithm that is suitable for incorporating coefficient updates with large support and large magnitude in subdomains. When a large number of local updates are involved, in addition to the standard factors in various (interior) subdomains, we precompute some factors in the corresponding exterior subdomains. Exterior boundary maps are constructed hierarchically. The data dependencies among tree-based interior and exterior factors are exploited to enable extensive information reuse. For coefficient updates in a subdomain, only the interior problem in that subdomain needs to be refactorized and there is no need to propagate updates to other tree nodes. The combination of the new interior factors with a chain of existing factors quickly provides the new global factor and thus an effective solution algorithm. The introduction of exterior factors avoids updating higher-level subdomains with large system sizes and makes the idea suitable for handling multiple occurrences of updates. The method can also accommodate the case when the support of updates changes to different subdomains. Numerical tests demonstrate the efficiency and especially the advantage in complexity over a standard factorization update algorithm.