AMSC/MAPL 460 or AMSC/MAPL 466, or knowledge of basic numerical
analysis (linear equations, nonlinear equations, integration,
interpolation) with permission of instructor. Knowledge of C or
Fortran or Matlab.
Course Description:
Fourier and wavelet transform methods, numerical methods for elliptic
partial differential equations, numerical linear
algebra for sparse matrices, Finite element methods, numerical methods
for time dependent partial differential equations.
Techniques for scientific computation with an introduction to the
theory and software for each topic. Course is part of a two
course sequence (660 and 661), but can be taken independently.
B. Fornberg and T.A. Driscoll,
A fast spectral algorithm
for nonlinear wave equations with linear dispersion
(.pdf), J. Comput. Phys. 155 (1999), 456-467.
basic numerical ideas
and programming practice: sources of error, conditioning and
stability, computer arithmetic, numerical resolution, order of
accuracy, accuracy check;
numerical differentiation, integration, and
interpolation, error expansions and Richardson extrapolation.
Fourier and Wavelet Methods
continuous and discrete Fourier transforms, Nyquist frequency, sampling
theorem, discrete cosine transform, fast Fourier transform (FFT)
algorithm and FFT package;
applications: Fast Poisson solver, discrete convolution and
deconvolution, Wiener filtering,
approximation using truncated series, Gibbs phenomenon, signal compression;
continuous and discrete wavelet transforms, Haar and Daubechies wavelets,
approximation properties, fast wavelet transform;
applications: filtering, signal compression
Elliptic Partial Differential Equations
variational and weak formulations, smooth and nonsmooth solutions;
finite difference methods, convergence; finite element spaces, local
stiffness matrices and assembly of stiffness matrix, adaptive mesh refinement
Sparse matrices
direct methods: symmetric positive definite case: RCM,
min. degree reordering, nested dissection; description of Matlab's algorithms
iterative methods: Jacobi, SOR, conjugate gradient, preconditioning (SSOR,
ILU), GMRES, BiCG, QMR
Time-Dependent Partial Differential Equations
well-posedness, dispersion analysis; time discretization with explicit and
implicit methods, basic facts about stability and convergence; method of
lines for spatial discretization with finite differences or finite elements;
applications: diffusive, dispersive, and hyperbolic equations