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.
| SES # | TOPICS | READINGS | READINGS TOPICS |
|---|---|---|---|
| L1 | Using MATLAB® to evaluate and plot expressions |
pp. 1-25. |
Linear AlgebraLinear 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 AlgebraExistence 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 SystemsExistence 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 EquationsEstimating 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 AnalysisOrthogonal 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 AnalysisEigenvector 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 AnalysisNumerical 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 AnalysisSingular Value Decomposition Computing the Roots of a Polynomial |
| L10 | Function space | pp. 126-134. |
Matrix Eigenvalue AnalysisComputing Extremal Eigenvalues The QR Method for Computing all Eigenvalues |
| L11 |
Numerical calculation of eigenvalues and eigenvectors Singular value decomposition (SVD) |
Initial Value ProblemsInitial 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 ProblemsInitial 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 ProblemsOverview 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 ProblemsDifferential-Algebraic Equation (DAE) Systems |
| L14 |
Implicit ordinary differential equation (ODE) solvers Shooting | pp. 212-231. |
Numerical OptimizationLocal 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 OptimizationLagrangian 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 ProblemsFinite Differences for a Convection/diffusion Equation |
| L18 |
Optimization Sensitivity analysis Introduction: Boundary value problems (BVPs) | pp. 282-299. |
Boundary Value ProblemsNumerical 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 ProblemsThe 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 EstimationGeneral 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 SimulationImportant Probability Distributions
Random Vectors and Multivariate Distributions
|
| L23 | Models vs. Data lecture 2: Bayesian view | pp. 389-403. |
Bayesian Statistics and Parameter EstimationSelecting 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 EstimationMCMC 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 SimulationMarkov 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 SimulationGenetic Programming |
| L29 |
Global optimization Multiple minima | ||
| L30 |
Modeling intrinsically stochastic processes multiscale modeling | pp. 338-353. |
Probability Theory and Stochastic SimulationBrownian Dynamics and Stochastic Differential Equations (SDEs) |
| L31 | Fluctuation-dissipation theorem | ||
| L32 | Kinetic Monte Carlo and turbulence modeling | ||
| L33 |
Operator splitting Strang splitting |
Strang SplittingSchwer, 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 AnalysisFourier 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 |