| SES # | TOPICS | READINGS | READINGS TOPICS |
|---|---|---|---|
| L1 | Using MATLAB® to evaluate and plot expressions | pp. 1-25. | Linear Algebra Linear Systems of Algebraic Equations Review of Scalar, Vector, and Matrix Operations Elimination Methods for Solving Linear Systems Existence and Uniqueness of Solutions |
| L2 | Solving systems of linear equations | pp. 25-32 and 36-56. | Linear Algebra Existence and Uniqueness of Solutions Matrix Inversion Matrix Factorization Matrix Norm and Rank Submatricies and Matrix Partitions Example. Modeling a Separation System Sparse and Banded Matricies |
| L3 | Matrix factorization Modularization | Condition Number
| |
| L4 | When algorithms run into problems: Numerical error, ill-conditioning, and tolerances | pp. 61-77. | Nonlinear Algebraic Systems Existence and Uniqueness of Solutions to a Nonlinear Algebraic Equation Iterative Methods and Use of Taylor Series Newton's Method for a Single Equation The Secant Method Bracketing and Bisection Methods Finding Complex Solutions Systems of Multiple Nonlinear Algebraic Equations Newton's Method for Multiple Nonlinear Equations |
| L5 | Introduction to systems of nonlinear equations | pp. 77-85 and 88-99. | Nonlinear Algebraic Equations Estimating the Jacobian and Quasi-Newton Methods Robust Reduced-step Newton's Method The Trust - Region Newton Method Solving Nonlinear Algebraic Systems in MATLAB® Homotopy Example. Steady-state Modeling of a Condensation Polymerization Reactor Bifurcation Analysis |
| L6 | Modern methods for solving nonlinear equations | pp. 104-113. | Matrix Eigenvalue Analysis Orthogonal Matrices Eigenvalues and Eigenvectors Defined Eigenvalues / Eigenvectors of a 2×2 Real Matrix Multiplicity and Formulas for the Trace and Determinant Eigenvalues and the Existence/uniqueness Properties of Linear Systems Estimating Eigenvalues; Gershgorin's Theorem |
| L7 | Introduction to eigenvalues and eigenvectors | pp. 117-123 and 148-149. | Matrix Eigenvalue Analysis Eigenvector Matrix Decomposition and Basis Sets Computing Roots of a Polynomial |
| L8 | Constructing and using the eigenvector basis | pp. 123-126 and 137-141. | Matrix Eigenvalue Analysis Numerical Calculation of Eigenvalues and Eigenvectors in MATLAB® Eigenvalue Problems in Quantum Mechanics |
| L9 | Function space vs. real space methods for partial differential equations (PDEs) | pp. 141-149. | Matrix Eigenvalue Analysis Singular Value Decomposition Computing the Roots of a Polynomial |
| L10 | Function space | pp. 126-134. | Matrix Eigenvalue Analysis Computing Extremal Eigenvalues The QR Method for Computing all Eigenvalues |
| L11 | Numerical calculation of eigenvalues and eigenvectors Singular value decomposition (SVD) | Initial Value Problems Initial Value Problems of Ordinary Differential Equations (ODE-IVPs) Polynomial Interpolation Newton-cotes Integration Linear ODE Systems and Dynamic Stability | |
| Q1 | Quiz 1 | pp. 154-163 and 169-176. | Initial Value Problems Initial Value Problems of Ordinary Differential Equations (ODE-IVPs) Polynomial Interpolation Newton-cotes Integration Linear ODE Systems and Dynamic Stability |
| L12 | Ordinary differential equation - initial value problems (ODE-IVP) and numerical integration | pp. 176-194. | Initial Value Problems Overview of ODE-IVP Solvers in MATLAB® Accuracy and Stability of Single-step Methods Stiff Stability of BDF Methods |
| L13 | Stiffness. MATLAB® ordinary differential equation (ODE) solvers | pp. 195-203. | Initial Value Problems Differential-Algebraic Equation (DAE) Systems |
| L14 | Implicit ordinary differential equation (ODE) solvers Shooting | pp. 212-231. | Numerical Optimization Local Methods for Unconstrained Optimization Problems The Simplex Method Gradient Methods Newton Line Search Methods Trust-region Newton Method Newton Methods for Large Problems Unconstrained Minimizer fminunc in MATLAB® Example. Fitting a Kinetic Rate Law to Time-dependent Data |
| L15 | Differential algebraic equations (DAEs) Introduction: Optimization | pp. 231-246. | Numerical Optimization Lagrangian Methods for Constrained Optimization Constrained Minimizer fmincon in MATLAB® |
| L16 | Unconstrained optimization | pp. 258-270. | Boundary Value Problems (BVPs) BVPs from Conservation Principles Real-space vs. Function-space BVP Methods The Finite Difference Method Applied to a 2-D BVP Extending the Finite Difference Method Chemical Reaction and Diffusion in a Spherical Catalyst Pellet |
| L17 | Constrained optimization | pp. 270-279. | Boundary Value Problems Finite Differences for a Convection/diffusion Equation |
| L18 | Optimization Sensitivity analysis Introduction: Boundary value problems (BVPs) | pp. 282-299. | Boundary Value Problems Numerical Issues for Discretized PDEs with More Than Two Spatial Dimensions The MATLAB® 1-D Parabolic and Elliptic Solver pdepe Finite Differences in Complex Geometries The Finite Volume Method |
| L19 | Boundary value problems (BVPs) lecture 2 | pp. 299-311. | Boundary Value Problems The Finite Element Method (FEM) FEM in MATLAB® Further Study in the Numerical Solution of BVPs |
| L20 | Boundary value problems (BVPs) lecture 3: Finite differences, method of lines, and finite elements | ||
| L21 | TA tutorial on BVPs, FEMLAB® | ||
| L22 | Introduction: Models vs. Data | pp. 372-389 and 325-338. | Bayesian Statistics and Parameter Estimation General Problem Formulation Example. Fitting Kinetic Parameters of a Chemical Reaction Single-response Linear Regression The Bayesian View of Statistical Inference The Least Squares Method Reconsidered Probability Theory and Stochastic Simulation Important Probability Distributions
Random Vectors and Multivariate Distributions
|
| L23 | Models vs. Data lecture 2: Bayesian view | pp. 389-403. | Bayesian Statistics and Parameter Estimation Selecting a Prior for Single-response Data Confidence Intervals From the Approximate Posterior Density |
| L24 | Uncertainties in model predictions | pp. 403-431. | Bayesian Statistics and Parameter Estimation MCMC Techniques in Bayesian Analysis MCMC Computation of Posterior Predictions Applying Eigenvalue Analysis to Experimental Design Bayesian Multi Response Regression Analysis of Composite Data Sets Bayesian Testing and Model Criticism |
| L25 | Conclude models vs. data | ||
| L26 | TA led review | ||
| Q2 | Quiz 2 (lectures 1 - 21) | Probability Theory and Stochastic Simulation Markov Chains and Processes; Monte Carlo Methods Markov Chains Monte Carlo Simulation in Statistical Mechanics Monte Carlo Integration Simulated Annealing | |
| L27 | Models vs. Data recapitulation Monte carlo and molecular dynamics | ||
| L28 | Guest lecture on Monte Carlo / molecular dynamics: Frederick Bernardin | pp. 363-364. | Probability Theory and Stochastic Simulation Genetic Programming |
| L29 | Global optimization Multiple minima | ||
| L30 | Modeling intrinsically stochastic processes multiscale modeling | pp. 338-353. | Probability Theory and Stochastic Simulation Brownian Dynamics and Stochastic Differential Equations (SDEs) |
| L31 | Fluctuation-dissipation theorem | ||
| L32 | Kinetic Monte Carlo and turbulence modeling | ||
| L33 | Operator splitting Strang splitting | Strang Splitting Schwer, Douglas A., Pisi Lu, William H. Green, Jr., and Viriato Semião. "A Consistent-splitting Approach to Computing Stiff Steady-state Reacting Flows With Adaptive Chemistry." Combust Theory Modelling 7 (2003): 383-399. | |
| L34 | Fourier transforms Fast fourier transform (FFT) | pp. 436-452. | Fourier Analysis Fourier Series and Transforms in One Dimension 1-D Fourier Transforms in MATLAB® Convolution and Correlation Fourier Transforms in Multiple Dimensions |
| L35 | Summary: Problem solving | ||
| L36 | TA led final review |
Table of Contents
Study Materials
Readings
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Unless otherwise noted, the readings are from the required text:
Beers, Kenneth J. Numerical Methods for Chemical Engineering: Applications in MATLAB®. New York, NY: Cambridge University Press, November 2006. ISBN: 9780521859714.
In general, the readings were assigned ahead of the corresponding lecture topic.
The calendar below provides information on the course's lecture (L) and quiz (Q) sessions.