The course combines a study of the main matrix structures arising in the discretization of integral equations and partial differential equations (PDEs), an innovative type of analysis of the spectral properties of the resulting large matrices, and advanced numerical methods for the solution of the associated linear systems. More specific items are the following: (1) Discrete Fourier Transform and fast Fourier transform with size power of 2, (2) Generalizations of the FFT to any matrix size, (3) Use of FFT for fast circulant, Toeplitz, polynomial computations, (4) Applications to integral equations coming from restorations of blurred signals (images) with noise, (5) Spectral Analysis of classes of matrices coming from approximations of PDEs (Finite Elements, Finite Differences, Isogeometric Analysis, etc) (6) Classical (stationary) iterative solvers, Conjugate Gradient (CG), preconditioned CG, Multigrid, and multi-iterative solvers for specific large linear systems coming from the approximation of PDEs, (7) Matrices coming from approximation of Fractional Differential Equations (FDEs) and the related spectral analysis, and (8) designing of fast solvers in the context of approximated FDEs.

Course Dates

Spring Semester 2017.


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