Computational Science and Engineering I

Course Description

This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.

Lectures

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   Lecture 1 - Positive definite matrices K = A'CA

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   Lecture 2 - One-dimensional applications: A = difference matrix

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   Lecture 3 - Network applications: A = incidence matrix

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   Lecture 4 - Applications to linear estimation: least squares

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   Lecture 5 - Applications to dynamics: eigenvalues of K, solution of Mu'' + Ku = F(t)

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   Lecture 6 - Underlying theory: applied linear algebra

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   Lecture 7 - Discrete vs. continuous: differences and derivatives

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   Lecture 8 - Applications to boundary value problems: Laplace equation

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   Lecture 9 - Solutions of Laplace equation: complex variables

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    Lecture 10 - Delta function and Green's function

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    Lecture 11 - Initial value problems: wave equation and heat equation

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    Lecture 12 - Solutions of initial value problems: eigenfunctions

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    Lecture 13 - Numerical linear algebra: orthogonalization and A = QR

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    Lecture 14 - Numerical linear algebra: SVD and applications

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    Lecture 15 - Numerical methods in estimation: recursive least squares and covariance matrix

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    Lecture 16 - Dynamic estimation: Kalman filter and square root filter

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    Lecture 17 - Finite difference methods: equilibrium problems

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    Lecture 18 - Finite difference methods: stability and convergence

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    Lecture 19 - Optimization and minimum principles: Euler equation

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    Lecture 20 - Finite element method: equilibrium equations

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    Lecture 21 - Spectral method: dynamic equations

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    Lecture 22 - Fourier expansions and convolution

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    Lecture 23 - Fast fourier transform and circulant matrices

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    Lecture 24 - Discrete filters: lowpass and highpass

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    Lecture 25 - Filters in the time and frequency domain

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    Lecture 26 - Filter banks and perfect reconstruction

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    Lecture 27 - Multiresolution, wavelet transform and scaling function

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    Lecture 28 - Splines and orthogonal wavelets: Daubechies construction

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    Lecture 29 - Applications in signal and image processing: compression

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    Lecture 30 - Network flows and combinatorics: max flow = min cut

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    Lecture 31 - Simplex method in linear programming

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    Lecture 32 - Nonlinear optimization: algorithms and theory