Computational Science and Engineering

Table of Contents

1. Applied Linear Algebra:
1.1 Four special matrices
1.2 Differences, derivatives, and boundary conditions
1.3 Elimination leads to K = LDL^T
1.4 Inverses and delta functions
1.5 Eigenvalues and eigenvectors
1.6 Positive definite matrices
1.7 Numerical linear algebra: LU, QR, SVD
1.8 Best basis from the SVD
2. A Framework for Applied Mathematics:
2.1 Equilibrium and the stiffness matrix
2.2 Oscillation by Newton's law
2.3 Least squares for rectangular matrices
2.4 Graph models and Kirchhoff's laws
2.5 Networks and transfer functions
2.6 Nonlinear problems
2.7 Structures in equilibrium
2.8 Covariances and recursive least squares
2.9 Graph cuts and gene clustering
3. Boundary Value Problems:
3.1 Differential equations of equilibrium
3.2 Cubic splines and fourth order equations
3.3 Gradient and divergence
3.4 Laplace's equation
3.5 Finite differences and fast Poisson solvers
3.6 The finite element method
3.7 Elasticity and solid mechanics
4. Fourier Series and Integrals:
4.1 Fourier series for periodic functions
4.2 Chebyshev, Legendre, and Bessel
4.3 The discrete Fourier transform and the FFT
4.4 Convolution and signal processing
4.5 Fourier integrals
4.6 Deconvolution and integral equations
4.7 Wavelets and signal processing
5. Analytic Functions:
5.1 Taylor series and complex integration
5.2 Famous functions and great theorems
5.3 The Laplace transform and z-transform
5.4 Spectral methods of exponential accuracy
6. Initial Value Problems:
6.1 Introduction
6.2 Finite difference methods for ODEs
6.3 Accuracy and stability for u_t = c u_x
6.4 The wave equation and staggered leapfrog
6.5 Diffusion, convection, and finance
6.6 Nonlinear flow and conservation laws
6.7 Fluid mechanics and Navier-Stokes
6.8 Level sets and fast marching
7. Solving Large Systems:
7.1 Elimination with reordering
7.2 Iterative methods
7.3 Multigrid methods
7.4 Conjugate gradients and Krylov subspaces
8. Optimization and Minimum Principles:
8.1 Two fundamental examples
8.2 Regularized least squares
8.3 Calculus of variations
8.4 Errors in projections and eigenvalues
8.5 The Saddle Point Stokes problem
8.6 Linear programming and duality
8.7 Adjoint methods in design.