| 1 |
Introduction, class organization, use of MATLAB® |
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| 2 |
Linear equation sets as basic computational unit, expressing linear equation sets in matrix form, basic matrix operations |
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| 3 |
Determinants and existence/uniqueness of solutions, Gaussian elimination (without pivoting), matrix inversion |
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| 4 |
Floating point representation, Gaussian elimination with partial pivoting, scaling of computations with matrix size, banded and tri-diagonal systems, LU decomposition |
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| 5 |
Introduction to eigenvalues, eigenvalue properties of Hermetian matrices, bilinear forms and positive-definiteness, Cholesky decomposition, Gershorgin’s theorem |
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| 6 |
Eigenvalues of diagonal and triangular matrices, similarity transforms, calculation of eigenvalues from QR decomposition, iteratively estimating the leading eigenvalue |
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| 7 |
Abstraction from linear algebra to linear operator theory, definition of vector space and basis sets, metrics, inner products, norms, Gram-Schmidt orthogonalization |
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| 8 |
Linear transformations and linear operators, eigenvector expansion, matrices as representations of linear operators |
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| 9 |
Zero eigenvalues, null spaces, and operator inversion, singular value decomposition, orthogonalization by SVD |
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| 10 |
Linear spaces of functions, differentiation and integration as linear operators, inner products of functions, orthogonal functions, Hermetian operators, example : solving simple Schrödinger equation as generalized eigenvalue problem |
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| 11 |
Moving beyond linear systems – solving sets of nonlinear equations. Newton’s method, convergence properties, weak line search to improve robustness |
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| 12 |
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Exam I – linear systems (lectures 1-10) |
| 13 |
Example - solving steady state concentrations in a CSTR with a general reaction network model |
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| 14 |
Stability of steady states from eigenvalues of Jacobian, dynamics of general linear set of first order ODE’s |
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| 15 |
Generalized Newton’s methods for minimizing a function, importance of Hessian matrix, local vs. global minima |
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| 16 |
Hessian-free minimization – steepest descent and conjugate gradient methods, conjugate-gradient methods for linear systems, Jacobi, Gauss-Seidel, SOR iterative methods for solving sparse linear systems, computer assignment : find minimum energy conformation of molecule (using Cerius2) and estimate IR spectra from normal mode analysis |
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| 17 |
Parameter estimation as minimization – intro to method of least squares, introduction to probability theory, conditional and joint probabilities, statistical independence |
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| 18 |
Random variables, binomial and Gaussian distributions, Poisson distribution as the ideal chain length distribution from living polymerization, central limit theorem, “normality” of Gaussian distribution, random walks |
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| 19 |
Markov processes, transition probabilities, example - estimating the gel point of a polymer network, Monte Carlo simulation |
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| 20 |
Hypothesis testing |
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| 21 |
Least squares linear regression, analysis of residuals and the Gauss-Markov conditions, confidence intervals of model parameters |
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| 22 |
Continuation of regression discussion |
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| 23 |
Non-linear least squares parameter estimation (topic continued after discussion of ODE/DAE systems) |
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| 24 |
Expressing dynamical systems as sets of first order ODE’s, numerical solution of ODE IVP’s, explicit methods, Euler, Adams-Bashforth, Runge-Kutta, Verlet and velocity Verlet methods |
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| 25 |
Implicit ODE-IVP methods, implicit Euler and its stability vs. explicit method, linearized implicit Euler, stiffness, predictor/corrector methods, Differential-Algebraic Equation (DAE) systems |
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| 26 |
Solving ODE/DAE-IVP’s with Matlab® |
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| 27 |
Parameter estimation in dynamical systems, example : least squares fitting of rate constants from batch reactor data, parametric sensitivity analysis |
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| 28 |
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Exam II – non-linear algebraic equations and probability theory/parameter estimation (lectures 11-22) |
| 29 |
Finite dimensional representation of functions – real space vs. function space discretization, real space methods : polynomial interpolation, numerical quadrature, orthogonal polynomials |
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| 30 |
Function space discretization – Fourier series and transform |
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| 31 |
Diffusion/convection form of PDE’s in chemical engineering, characteristics and PDE types (elliptic, parabolic, hyperbolic), 1-D convection/diffusion problem with central vs. upwind finite differences |
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| 32 |
Linear stability analysis of discretized 1-D convection/diffusion problem, stiffness problems, explicit vs. implicit methods |
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| 33 |
Example, 1-D tubular reactor model with generic reaction network, finite differences in 2-D, 3-D |
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| 34 |
Method of finite volumes, enforcing continuity when solving Navier/Stokes equations |
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| 35 |
Weighted residual methods, orthogonal collocation |
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| 36 |
Finite element method for 1-D generalized transport equation |
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| 37 |
Finite element method for 2-D generalized transport equation, tessellation of irregular geometries |
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| 38 |
Finite element method examples |
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| 39 |
Finite element method examples (continued) |
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ODE/DAE/PDE methods (lectures 23-39) |
FINAL EXAM (lectures 23-39) |