Multilevel probing methods for approximating selected entries of the matrix inverse on highly parallel heterogeneous architectures

• Κύριος συγγραφέας: Γεωργίου, Βασίλειος
• Άλλοι συγγραφείς: Georgiou, Vasileios
• Γλώσσα: English
• Έκδοση: 2021
• Θέματα: Numerical linear algebra; Selected inversion; High performance computing; Probing method; Υπολογιστική γραμμική άλγεβρα; Υπολογισμός αντιστρόφου; Υπολογισμοί υψηλών επιδόσεων; Μέθοδοι ανίχνευσης

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.