Mathematical Modeling What is it? (and how do you spell it?)

A Few Words from others … Applied mathematics is concerned with a better understanding of phenomena by the use of mathematical methods. The process of formulating the mathematical model is often called mathematical modeling. (F.Wan) Mathematical Modeling is the applied mathematician at work. (M.S.Klamkin) Applied Mathematics consists of applying mathematics to real-world problems. (J.L.Synge)

Main point: mathematical modeling is a PROCESS ? The question Mathematical Formulation Analysis, Simulation What’s been done before? (Literature search). Get Data. Parameter Estimation? Model Validation Revise?Revise? Communicate Results

Objectives Satisfied ? The Modeling Process thanks to Christine Beveridge at the U of Q, Australia Model Objectives Hypotheses Mathematical Formulation Translate to Algorithm Computer Implementation Parameter Estimation Trial against model construction data set Correct Algorithm ? Code Correct for Algorithm ? ANALYZE Model Or COMPARE Model Output to model test data set Math Formulation Wrong ? Hypothesis wrong for objectives ? Re-evaluate objectives or Give Up DONE NO YES Formulation Verification Calibration YES Model Analysis and Evaluation

These steps consist of: 1.Recognizing the problem (Nature may require you to “dig” for them) –Interpreting the problem, or refining it. 2.Selecting a mathematical framework (stochastic, deterministic, continuous, discrete). Be aware of what’s been done before, and what’s still undone. 3.Finding the appropriate mathematical tools (computation is generally required). 4.The solution may be approximate. 5.Feedback is required: is the approximation ok? Is the question being answered the one that was asked? 6.Communication will be oral AND written, summarized AND detailed.

Validation and Revision is CRUCIAL! Example: A bank has three tellers. Which is better, first available, or form three lines and go to the shortest one? What’s the question? What type of model should you use? What tools might be necessary? How could we validate the model?

Data can be important … Learn about available databases. Look at previous research. What does “parameter estimation” mean? Data is also important in model validation.

How about Simulation? Computers can be helpful, but deep knowledge isn’t always necessary (you may need to collaborate!) Do you always need data before a simulation can be done? How is the simulation used? Does the importance or type of simulation depend on the type of model? (discrete, continuous, stochastic, deterministic)