1,721,023 research outputs found
Il funzionale di Totale Variazione nel problema dell'eliminazione del rumore da immagini digitali: dal modello al software matematico
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A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization
No full textWe propose a procedure for building symmetric positive definite band preconditioners for large-scale symmetric, possibly indefinite, linear systems, when the coefficient matrix is not explicitly available, but matrix-vector products involving it can be computed. We focus on linear systems arising in Newton-type iterations within matrix-free versions of projected methods for bound-constrained nonlinear optimization. In this case, the structure and the size of the matrix may significantly change in subsequent iterations, and preconditioner updating algorithms that exploit information from previous steps cannot be easily applied. Our procedure is based on a recursive approach that incrementally improves the quality of the preconditioner, while requiring a modest number of matrix-vector products. A strategy for dynamically choosing the bandwidth of the preconditioners is also presented. Numerical results are provided, showing the performance of our preconditioning technique within a trust-region Newton method for bound-constrained optimization
A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization
No full textWe propose a procedure for building symmetric positive definite band preconditioners for large-scale symmetric, possibly indefinite, linear systems, when the coefficient matrix is not explicitly available, but matrix-vector products involving it can be computed. We focus on linear systems arising in Newton-type iterations within matrix-free versions of projected methods for bound-constrained nonlinear optimization. In this case, the structure and the size of the matrix may significantly change in subsequent iterations, and preconditioner updating algorithms that exploit information from previous steps cannot be easily applied. Our procedure is based on a recursive approach that incrementally improves the quality of the preconditioner, while requiring a modest number of matrix-vector products. A strategy for dynamically choosing the bandwidth of the preconditioners is also presented. Numerical results are provided, showing the performance of our preconditioning technique within a trust-region Newton method for bound-constrained optimization
Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems
Full text linkAlgebraic multilevel preconditioners for algebraic problems arising from the discretization of a class of systems of coupled elliptic partial differential equations (PDEs) are presented. These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods. A convergence theory for the twolevel case is presented
On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods
No full textThe solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. In this paper we discuss the mutual impact of linear algebra and optimization, focusing on interior point methods and on the iterative solution of the KKT system. Three critical issues are addressed: preconditioning, termination control for the inner iterations, and inertia control
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