SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1174-A1199, January 2020.
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.

中文翻译：

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多重系数更新下一般椭圆方程的快速分解更新

SIAM科学计算杂志，第42卷，第2期，第A1174-A1199页，2020年1月。
对于离散椭圆方程，我们开发了一种新的因式分解更新算法，该算法适合于在子域中合并具有较大支持和较大幅度的系数更新。当涉及大量本地更新时，除了各个（内部）子域中的标准因子外，我们还预先计算了相应外部子域中的一些因子。外部边界图是分层构建的。利用基于树的内部和外部因素之间的数据依赖性来实现广泛的信息重用。对于子域中的系数更新，仅需要重构该子域中的内部问题，而无需将更新传播到其他树节点。新的内部因素与现有因素链的组合迅速提供了新的全局因素，从而提供了一种有效的求解算法。外部因素的引入避免了使用大系统规​​模来更新更高级别的子域，并使该思想适合于处理多次出现的更新。该方法还可以适应更新支持更改为不同子域的情况。数值测试表明，与标准分解更新算法相比，效率更高，尤其是在复杂性方面具有优势。该方法还可以适应更新支持更改为不同子域的情况。数值测试表明，与标准因数分解更新算法相比，效率更高，尤其是在复杂性方面具有优势。该方法还可以适应更新支持更改为不同子域的情况。数值测试表明，与标准分解更新算法相比，效率更高，尤其是在复杂性方面具有优势。