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Description
We analyze two parallel numerical strategies for computing selected entries of the matrix inverse of large, sparse, symmetric systems: The selected inverse method and a factorized approximate inverse method. Both techniques are aimed at computations via LU factorizations or incomplete LU (ILU) factorizations. The selected inverse approach exploits the LU/ILU factorization to recover the inverse within the pattern of the (incomplete) LU factorization. In contrast, the factorized approximate inverse method applies a truncated series expansion (Neumann-type), providing an alternative approach at the cost of reduced accuracy. To improve accuracy while keeping sparsity, we introduce low-rank corrections and an adaptive eigenvector deflation strategy. This hybrid approach sets tightening drop tolerances off when possible and helps balance accuracy, fill-in, and computation time. Numerical illustrations for both parallel and sequential computations demonstrate the effectiveness and robustness of our approach.