| Περίληψη: | Motivated by the theoretical understanding of localization phenomena in matrix functions, probing methods have been proposed to approximate selected entries of the matrix inverse. In this thesis we focus on the development of novel
methods to better balance the work of the procedures that compose
algorithms based on probing. We develop multilevel methods that capture the most significant entries of the matrix inverse,
in combination with specialized iterative solvers, to speed-up the solution of the linear system with multiple right-hand sides that is generated by probing.
Sparse approximate inverses are used for preconditioning, to accelerate the convergence of Krylov iterations, while
careful reuse of shared information reduces the number of operations and saves memory.These advancements together with efficient implementations enable the development of a framework for approximating
blocks of entries centered around an a priori sparsity pattern.Numerical examples with matrices arising from discretization of
PDEs and covariance matrices highlight the effectiveness of the
proposed techniques.
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