Mathematical Modeling

Math 4428
Spring 2018

​Objectives:
During this course, students will develop their skill in analyzing optimization, dynamical, and probabilistic models. These skills include:

• Translating a real-world problem into a mathematical model
• Determining a solution or possible solutions to the mathematical model
• Relating mathematical findings back to the real-world application
• Assessing the quality of the model and determining ways to improve the model
• Working with a team to analyze, interpret, and communicate the solution to a problem using optimization, dynamical systems, or probabilistic techniques

Handouts:
Syllabus
Schedule

Homework:

1. One variable optimization - introduction to modeling, sensitivity analysis
2. Multivariable optimization - unconstrained optimization, Lagrange multipliers
3. Computational methods for optimization - Newton's Method, linear programming
4. Introduction to dynamical models
5. Analyzing dynamical models - phase portraits and cobweb plots
6. Analyzing dynamical models - bifurcations, numerical approximations, and chaos
7. Introduction to probability models
8. Stochastic models - Markov chains, Markov processes, Monte Carlo simulation

Notes:

1. Optimization
2. Dynamical systems
3. Probability models

Projects:

• Guidelines
• Presentation rubric
• Project descriptions:
  • Optimization
  • Dynamical systems
  • Probability

Matlab code:

• Optimization​:
  • PC profit (1D - sensitivity analysis, 2D - Lagrange multipliers)
  • Newton's Method 
• Dynamical systems:
  • Whaling
  • Lorenz system (script, solver)
  • Backward Euler and Van der Pol's oscillator
• Probability:
  • Monte Carlo rain
  • Ponderosa pines and linear regression