MATH 486/ Math 522 Mathematical Modeling
Instructor: Hemanshu Kaul
Office: 125C, Engineering 1
Phone: (312) 567-3128
E-mail: kaul [at] iit.edu
Time: 1:50pm, Tuesday-Thursday.
Place: at 025, Engg. 1 Bldg.
Office Hours: 4:35pm-5:35pm Tuesday and Thursday; walk-ins; and by appointment. Emailed questions are also encouraged.
TA Office Hours: Adam Rumpf, arumpf [at] hawk.iit.edu, 10am-11:30am, Tuesday and Thursday, at 129, E1 bldg.
The Course Information Handout has extensive description of the course - topics, textbook, student evaluation policy, as well as other relevant information. Read it carefully!
The Aims and Syllabus of this course.
Advice for students:
Understanding Mathematics - a study guide
Excellent advice for math majors and graduate students, by Terry Tao, 2006 Fields medallist. Required reading.
Some of the primary sources of information/discussion for careers in Mathematical Sciences:
MAA - Careers
SIAM - Careers
INFORMS - Careers
AMS - Careers
- Thursday, 1/15 : Check this webpage regularly for homework assignments, announcements, etc.
- Exam # 1 : Thursday, 2/26. Syllabus: Based on topics covered up to and including 2/12 [total 10 lectures] corresponding to HWs#1-#5.
- Exam # 2 : Thursday, 4/16. Syllabus: Based on topics covered from 2/17 up to 4/2 (including both dates) [total 11 lectures] corresponding to HWs#6-#10.
- Final Exam : Friday, May 8th, 8-10am. Syllabus: All topics covered during the semester.
Instructions: The project is an important part of this course - not just in terms of the grade, but for the sake of comprehensive, practical understanding of how to apply modeling framework to an open
ended real-life problem. This is why the problem statements that I have given to you are just
short and open-ended descriptions of the certain real-life situations. You have complete freedom
in mathematical interpretation of the problem and how you “solve” it. The only requirement is
that you use the mathematical modeling process, and justify your model and its conclusions as
they apply to the problem. Its a test of your creativity in formulation of models and solution
methods, and your ability to find and understand relevant mathematical knowledge.
Read carefully through this list of instructions and advice for your project.
Look through the example project report given at the end of this SIAM report in Appendix B on page 50 of the pdf file for an example on how to format and write your project report. The pages 1-50 of this pdf file are also useful as a detailed overview of how to approach the modeling process for a project.
Deadlines for the semester project:
Last week of February: I will send you the list of project topics by email.
2/27/2015: Email me with project team members (2 per team, 3 if necessary; all members must be registered for the same course number)
3/6/2015: Email me with your choice of project topic among the project descriptions sent to you by email.
4/9/2015: Discuss the initial draft and model of your project with me in person. All members of your team must be present.
5/1/2015: Final submission of Project report and associated materials/programs/etc.
The homework problems listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
If you have the 4th edition of the textbook, please compare the problem statements with those in 5th edition before solving them.
You are allowed the use of calculators /computational software, to aid in the basic computational work of the problems. In case of doubt, ask me for a clarification.
You have to explain what you are doing IN WORDS. Its not enough to write down an equation without explaining how/why/what of it - you have to define what the variables mean. It is not acceptable to simply input data into Mathematica/Maple/Matlab and ask it to figure out the best "model" for you. You can use these software to aid in your computations, not to simply solve the whole thing for you.
- Tuesday, 1/13 : Read and understand Examples from Section 1.1.
- Thursday, 1/15 : Read and understand Examples from Sections 1.2 and 1.3.
- Homework #1: Due Thursday, 1/22: Math 486 students submit 5 problems, and Math 522 students submit all 6 problems.
Section 1.1: #3bc, #10, #13. Section 1.2: #2, #3, #9. Solutions distributed on 1/27.
- Tuesday, 1/20 : Read and understand Examples from Sections 1.3 and 1.4.
- Thursday, 1/22 : Read and understand Section 2.1.
- Homework #2: Due Thursday, 1/29: Math 486 students submit all problems from Section 1.3 and 1 out of the 2 problems from Section 1.4; and Math 522 students submit all problems.
Section 1.3: (#1f and #2e); (#3a); (#6 and #10). Section 1.4: (#3, read and understand example 2 first); (#4). Graded HW distributed 2/3.
- Thursday, 1/29 : Read and understand Examples from Sections 2.2, 2.3 and 2.4.
- Homework #3: Due Thursday, 2/5: Math 486 students submit 5 out of 6 problems, and Math 522 students submit all 6 problems.
Section 2.2: #6. Section 2.3: #4, #9(read Example 2 first, do NOT solve part a or b), Project#2. Section 3.1: #5, #7. Graded HW distributed 2/10.
- Tuesday, 2/3 : Read and understand Example 2 from Section 3.4.
- Homework #4: Due Thursday, 2/12: Math 486 students submit 5 (note: 3.4.7ab and 3.4.8 are compulsory) out of 6 problems , and Math 522 students submit all 6 problems.
Section 3.2: #2b, #3. Section 3.3: #4, #8. Section 3.4: #7ab, #8(compare all 4 models from #7 and #8 here). Solutions distributed on 2/19.
- Thursday, 2/12 : Read and understand Example 2 from Section 7.1. Read and understand the examples in Sections 8.1 and 8.5.
- Homework #5 [pdf]. Due Thursday, 2/19. Solutions discussed and distributed on 2/19.
- Thursday, 2/19 : Read and understand Examples from Sections 7.2, 7.3.
- Tuesday, 2/24 : Read and understand Examples from Sections 6.1 and 6.2.
- Homework #6: Due MONDAY, 3/9, before 4:30pm in TA mailbox in 210, E1 bldg. Graded HWs distributed on 3/24.
Section 7.2: #3, #13a (Set each as a Linear program and then solve it graphically by moving the line corresponding to the objective function and seeing where it is maximized/minimized in the geometric feasible region). Section 7.3:#3, #13a (Set each as a Linear program and then solve it graphically by finding the corner points of the geometric feasible region and checking which corner gives the max or min value for the objective function.). Section 6.1: (Math 486 students choose one of these two problems in this section) #2, Project#1. Section 6.2: Project#1.
- Thursday, 3/5 : Read and understand Examples from Sections 6.3 and 5.1.
- Homework #7: Due Thursday, 3/12 Graded HWs distributed on 3/24.
Section 6.3: #2.
- Thursday, 3/12 : Read and understand Examples from Sections 5.3, and Example 1 from Section 5.5.
- Homework #8: Due Thursday, 3/26: Math 486 students submit 5 out 6 problems, and Math 522 students submit all 6 problems. Solutions discussed and distributed on 3/26.
Section 5.1:[For each of these problems, first carefully write all steps of the algorithm as applied to the problem] #3(run the simulation on Mathematica/Matlab for n=100, 200, 300, etc. to get approximate values of pi), #5, #7(compare your answer to actual value of the volume).
Section 5.2: #1b, #2c.
Section 5.3: #2 (First write the algorithm you will be using and then run it on computer).
- Thursday, 3/26 : Read and understand Examples from Sections 11.1, 11.2, and 11.4 not covered in class.
- Homework #9: Due Thursday, 4/2: Math 486 students submit 4 out of 5 problems, and Math 522 students submit all problems. As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.
Solutions distributed on 4/7.
Section 11.1: #4, #6, #7. Section 11.4: #4, #7.
- Thursday, 4/2 : Read and understand Examples from Sections 12.1, 12.2, and 12.5 not covered in class.
- Homework #10: Due Thursday, 4/9: Math 486 students can skip one problem, and Math 522 students submit all problems. As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.
Solutions to be distributed on 4/14.
Section 12.1: [Read Example 1 first] #6. Section 12.2:[Read and understand the context of equation 12.6 first] #3, #6. Section 12.3: #5abc. Section 12.5: #7 [first write down Euler's method applied to this problem].
- Tuesday, 4/21 : Read and understand Examples from Section 13.2 and 13.3 that were not covered in class.
- Homework #11: Due Tuesday, 4/28: [NOTE the DATE] Math 486 and Math 522 students submit all problems. As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.
Section 13.1: #7. Section 13.2: #6 (Use Calculus as well as method of steepest ascent/descent). Section 13.3: #3.
- Tuesday, 1/13 : The process of math modeling - discussion with examples, Principle of proportionality, difference equations - examples from accounting/ finance/ science, discrete time vs. continuous time. (From Section 1.1 and elsewhere)
- Thursday, 1/15 : Approximating change via difference equations - examples, limiting behavior of DDS(discrete dynamical system) and example of drug dosage, modeling births/deaths/resources through non-linear discrete dynamical systems. (From Sections 1.2, 1.3, and elsewhere)
- Tuesday, 1/20 : Equilibrium values and solutions of DDS, Solutions methods and stability of equilibrium values of homogenous and nonhomogenous linear DDS, Systems of discrete dynamical systems via astronaut docking procedure and its analysis. (From Sections 1.3, 1.4, and elsewhere)
- Thursday, 1/22 : Astronaut docking procedure and its analysis, More nonlinear DDS via a interacting species population model, Properties of proportionality. (From Section 1.4, 2.2, and elsewhere)
- Tuesday, 1/27 : proportionality in non-linear or translated linear systems -examples from physics, Geometric similarity - relationship between geometric notions like volume, surface area, etc. in terms of a fundamental dimension, Ideas underlying model fitting - visual model fitting. (From Sections 2.2, 2.3, 3.1, and elsewhere)
- Thursday, 1/29 : Four sources of error in the modeling process, Transforming data to fit linear systems, Chebyshev Approximation Criterion - min max deviation, Writing Chebyshev AC as a linear optimization problem, Minimizing sum of absolute deviations, Least squares criterion - min sum of squares of deviations, Relationship between CAC and LSC models. (From Sections 3.1, 3.2 and 3.3)
- Tuesday, 2/3 : Least squares criterion - Normal equations and critical points in LSC, Fitting a straight line, Fitting a power curve with fixed exponent, fitting a power curve with unknown exponent, Transforming non-linear data.(From Sections 3.3 and 3.4)
- Thursday, 2/5 : Discussion of HW#3 project using geometric similarity and proportionality. Comparing the various LSC models derived using transformations along with model derived from CAC. (From Sections 3.4 and elsewhere)
- Tuesday, 2/10 : Introduction to Linear Optimization - various forms, standard form (how to convert into standard form), examples, integer variables. (From Section 7.1, and elsewhere)
- Thursday, 2/12 : Mixed Integer Linear Programs and its variants, Binary programs and using 0-1 variables to make decisions, Knapsack Problem, Graphs and networks - basic concepts and examples from applications, Graph Coloring and its relation to scheduling and its relation to coloring maps, (Job) Assignment problem, Vertex cover and expression as linear program, Network flow - max flow problem and expressing it using linear program. (From Sections 8.1, 8.3, 8.5, and elsewhere)
- Tuesday, 2/17 : Max Flow as a linear program, Min-cost Flow as a linear program, Eulerian trails in graphs, History and uses of Graph theory in various fields. (From Sections 8.1, 8.5, and elsewhere)
- Thursday, 2/19 : Discussion of some 0-1 variable problems from HW#5. Geometric solutions of Linear programs, Algebraic solution of Linear Program using simplex algorithm, Geometric intuition behind simplex algorithm, Local search algorithms - underlying concepts and thinking of simplex as a local search algorithm, Uses of Linear programs in solving hard optimization problems/ 0-1 Programs. (From Sections 7.2, 7.4, and elsewhere)
- Tuesday, 2/24 : Markov chains - properties and examples, Models derived from Markov chains, Estimating Stationary distribution of a MC, Component reliability - series system and parallel system, Combinations of series and parallel components. (From Sections 6.1, 6.2, and elsewhere)
- Thursday, 2/26 : Mid-term Exam#1.
- Tuesday, 3/3 : Distribution of Projects and related discussion. Linear regression and what it means, Its calculation using SSE, SST, SSR, properties, Plots of residuals, relation between linear regression and correlation. (From Section 6.3)
- Thursday, 3/5 : Monte Carlo algorithm for calculating area under a curve, volume under a surface, etc., the concept of bounding box and its importance. Random point generation - middle square method. Idea underlying Markov Chain Monte Carlo. (From Sections 5.1 and 5.2 and elsewhere)
- Tuesday, 3/10 : Discussion of Mid-term Exam#1.
- Thursday, 3/12 : Finish discussion of Mid-term Exam #1. Random point generation - middle square method, linear congruence method; (Monte Carlo) Modeling probabilistic behavior using random numbers - fair/ unfair coin, roll of fair/unfair die, etc. (From Sections 5.2 and 5.3)
- Tuesday, 3/24 : Ordinary Differential Equations for instantaneous rate of change for continuous problems and as approximate average rate of change in discrete problems, Population growth models, solving the linear ODE, Population growth under limited resources - logistic growth model and its solution. (From Section 11.1 and elsewhere)
- Thursday, 3/26 : Discussion of HW#8. Solutions curves for an ODE and its relation to initial value, Autonomous DE, Equilibrium values - stable and unstable, phase diagram using the behavior of first and second derivatives, Using phase diagram to sketch solutions curves and solutions fields. (From Sections 11.1, 11.4)
- Tuesday, 3/31 : Solution field for the logistic model, Motivation and mathematics for Euler's method for approximate solution to 1st order DE. (From Sections 11.1, 11.4 and 11.5)
- Thursday, 4/2 : Continuous dynamical system - system of differential equations, Euler's method for approximate solution of a system of Differential equations, Ecological modeling problem with competing species of Blue and Fin whales - modified logistic model with competition factor, equilibrium points including the assumption on competition factor to get a non-extinction equilibrium point, stable vs. unstable equilibrium points. (From Sections 12.1, 12.2, 12.3, 12.5, and elsewhere)
- Tuesday, 4/7 : Continuing the Blue and Fin whales model: discretization of the Dynamical system with time step of 1 year and computing the time needed by populations of the two species to achieve the equilibrium populations, Sensitivity analysis with varying values of competition factor, discretization with increasing values of h(=delta t) to illustrate the phenomenon of chaos arising out of discrete Dynamical system. (From elsewhere)
- Thursday, 4/9 : A competitive-hunter model as a simplification of competing species model with underlying exponential rather than logistic growth, Equillibrium points, graphical analysis and interpretation of the model; A predator-prey model as a variation of the competitive-hunter model, equilibrium points, graphical analysis, and interpretation of the model and the periodic fluctuation with time lag of the relative populations of the predator and the prey (From Sections 12.2 and 12.3 and elsewhere)
- Tuesday, 4/14 : Unconstrained multivariable optimization - using calculus; Sensitivity analysis and the sensitivity coefficient and its use in understanding a model. (From Section 13.2 and elsewhere)
- Thursday, 4/16 : Mid-term Exam#2.
- Tuesday, 4/21 : Using the gradient method of steepest ascent/descent to solve unconstrained multivariable optimization, relation between stepsize and gradient, choice of stepsize. Newton's Method for finding root of single equation, or common root of multiple equations, the principle of global method to find approximate region of solution and local method like Newton's to find the exact solution with required accuracy. Constrained multivariable optimization - using Lagrange multipliers. (From Sections 13.2, 13.3, and elsewhere)
- Thursday, 4/23 : Discussion of Mid-term Exam#2.
- Tuesday, 4/28 : A few comments and a textbook for Game Theory. Complex Networks, examples, fundamental properties - degree distribution, average distance between vertices, clustering coefficient; Erdos-Renyi Random graph model, Stanley Milgram's six degrees of separation, Watts-Strogatz small world network model. (From elsewhere)
- Thursday, 4/30 : Dynamics on small-world networks - SI, SIR models; Degree distributions in real-world networks, Power law distribution and fat tail, Scale-free networks , Robustness via removal of arbitrary nodes or high degree nodes, Barabasi-Albert's preferential attachment model, modification of PA model, weighted relationships/ strength of relationship. (From elsewhere)
For an alternate point-of-view and for additional applications, refer to the following books:
M.M.Meerschaert, Mathematical Modeling, Fourth Edition.
H.P. Williams, Model Building in Mathematical Programming, Fifth Edition.
Hillier and Lieberman, Introduction to Operations Research, 7th edition onwards.
MATLAB - getting started at IIT, by Greg Fasshauer
A crash course
Matlab quick reference sheets
Getting Started in MATLAB (pdf)
The MathWorks Student Center
A Very Elementary MATLAB Tutorial
MATLAB 7: Getting Started Guide<\a>
Official MATLAB Site
A Beginner's guide to Mathematica.
Introduction to Mathematica.
Scilab, two inexpensive (FREE)
alternatives to MATLAB
Links for Additional Information:
Wikipedia on Math Models