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We introduce the strong recursive skeletonization factorization (RS-S), a newapproximate matrix factorization based on recursive skeletonization for solvingdiscretizations of linear integral equations associated with elliptic partialdifferential equations in two and three dimensions (and other matrices withsimilar hierarchical rank structure). Unlike previous skeletonization-basedfactorizations, RS-S uses a simple modification of skeletonization, strongskeletonization, which compresses only far-field interactions. This leads to anapproximate factorization in the form of a product of many blockunit-triangular matrices that may be used as a preconditioner ormoderate-accuracy direct solver, with dramatically reduced rank growth. Wefurther combine the strong skeletonization procedure with alternatingnear-field compression to obtain the hybrid recursive skeletonizationfactorization (RS-WS), a modification of RS-S that exhibits reduced storagecost in many settings. Under suitable rank assumptions both RS-S and RS-WSexhibit linear computational complexity, which we demonstrate with a number ofnumerical examples.

A recursive skeletonization factorization based on strong admissibility