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Sergey A. Dyachenko
sergeydy AT buffalo DOT edu
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TR 3:30-4:50 pm at 225 Math
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T 5-6 Math Bldg 312 and by appointment
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No mandatory textbook. The course was based primarily on selected material from:
L. N. Trefethen, Spectral Methods in MATLAB.
Y. Saad, Iterative Methods for Sparse Linear Systems.
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This graduate course covered advanced numerical methods with emphasis on two
complementary themes: spectral methods for differential equations and iterative
methods for large sparse linear systems. The spectral-methods portion followed
selected lectures and examples from Trefethen's Spectral Methods in MATLAB;
the sparse-linear-systems portion drew on Saad's
Iterative Methods for Sparse Linear Systems.
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| 1--2 |
Fourier approximation, trigonometric interpolation, spectral differentiation, and the FFT. |
| 3--4 |
Chebyshev approximation, polynomial interpolation, Chebyshev grids, and differentiation matrices. |
| 5--6 |
Spectral and pseudospectral discretization of boundary-value problems and partial differential equations; treatment of periodic and nonperiodic domains. |
| 7--8 |
Time integration for PDEs: explicit and implicit ideas, exponential time differencing, and operator-splitting methods including the split-step method. |
| 9 |
Accuracy, stability, and practical implementation issues for spectral PDE solvers; nonlinear terms and aliasing/dealiasing. |
| 10 |
Large sparse linear systems, Krylov subspaces, residual minimization, and projection methods. |
| 11--12 |
Orthogonal projection methods, including Lanczos and Arnoldi iterations; conjugate-gradient and GMRES-type methods. |
| 13 |
Oblique projection and nonsymmetric Krylov methods, including BiCG, BiCGSTAB, and QMR; preconditioning. |
| 14--15 |
Eigenvalue problems: direct methods, power and inverse iterations, Krylov-based iterative eigensolvers, and project work/discussion. |
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Course grades are based on projects.
Students should decide on a proposed project topic by April 15.
Project ideas and possible topics can be discussed during office hours or by appointment.
Each project includes a 20-minute presentation followed by a written project submission.
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