L1 |
Using MATLAB® to evaluate and plot expressions |
pp. 1-25.
MATLAB® Tutorial
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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
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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
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L3 |
Matrix factorization
Modularization
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Condition Number
Heath, Michael T. Scientific Computing: An Introductory Survey. 2nd ed. New York, NY: McGraw-Hill Companies, Inc., 2002, pp. 5-6 and 52-65. ISBN: 9780072399103.
Recktenwald, Gerald W. Introduction to Numerical Methods with MATLAB®: Implementations and Applications. Upper Saddle River, NJ: Prentice-Hall, 2000, pp. 402-410. ISBN: 9780201308600.
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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
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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
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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
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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
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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
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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
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L10 |
Function space |
pp. 126-134. |
Matrix Eigenvalue Analysis
Computing Extremal Eigenvalues
The QR Method for Computing all Eigenvalues
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L11 |
Numerical calculation of eigenvalues and eigenvectors
Singular value decomposition (SVD)
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Initial Value Problems
Initial Value Problems of Ordinary Differential Equations (ODE-IVPs)
Polynomial Interpolation
Newton-cotes Integration
Linear ODE Systems and Dynamic Stability
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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
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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
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L13 |
Stiffness. MATLAB® ordinary differential equation (ODE) solvers |
pp. 195-203. |
Initial Value Problems
Differential-Algebraic Equation (DAE) Systems
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L14 |
Implicit ordinary differential equation (ODE) solvers
Shooting
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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
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L15 |
Differential algebraic equations (DAEs)
Introduction: Optimization
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pp. 231-246. |
Numerical Optimization
Lagrangian Methods for Constrained Optimization
Constrained Minimizer fmincon in MATLAB®
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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
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L17 |
Constrained optimization |
pp. 270-279. |
Boundary Value Problems
Finite Differences for a Convection/diffusion Equation
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L18 |
Optimization
Sensitivity analysis
Introduction: Boundary value problems (BVPs)
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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
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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
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L20 |
Boundary value problems (BVPs) lecture 3: Finite differences, method of lines, and finite elements |
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L21 |
TA tutorial on BVPs, FEMLAB® |
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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
- Bernoulli Trials
- The Random Walk Problem
- The Binomial Distribution
- The Gaussian (Normal) Distribution
- The Central Limit Theorem of Statistics
- The Gaussian Distribution With Non-zero Mean
- The Poisson Distribution
Random Vectors and Multivariate Distributions
- The Boltzmann Distribution
- The Maxwell Distribution
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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
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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
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L25 |
Conclude models vs. data |
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L26 |
TA led review |
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Q2 |
Quiz 2 (lectures 1 - 21) |
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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
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L27 |
Models vs. Data recapitulation
Monte carlo and molecular dynamics
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L28 |
Guest lecture on Monte Carlo / molecular dynamics: Frederick Bernardin |
pp. 363-364. |
Probability Theory and Stochastic Simulation
Genetic Programming
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L29 |
Global optimization
Multiple minima
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L30 |
Modeling intrinsically stochastic processes
multiscale modeling
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pp. 338-353. |
Probability Theory and Stochastic Simulation
Brownian Dynamics and Stochastic Differential Equations (SDEs)
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L31 |
Fluctuation-dissipation theorem |
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L32 |
Kinetic Monte Carlo and turbulence modeling |
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L33 |
Operator splitting
Strang splitting
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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.
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L34 |
Fourier transforms
Fast fourier transform (FFT)
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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
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L35 |
Summary: Problem solving |
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L36 |
TA led final review |
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