Transcript Mathematical Modeling

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
Mathematical Formulation
?
The question
R
e
v
i
s
e
?
What’s been done before? (Literature search).
Get Data. Parameter Estimation?
Analysis, Simulation
Model Validation
Communicate Results
thanks to Christine Beveridge at the U of Q, Australia
Hypotheses
Mathematical Formulation
Verification
Translate to Algorithm
Correct
Algorithm
?
YES
Computer Implementation
Code
Correct for
Algorithm
?
Calibration
NO
NO
YES
Parameter Estimation
Trial against model
construction data set
Model Analysis and Evaluation
Formulation
The Modeling Process
Model Objectives
ANALYZE Model
Or
COMPARE Model Output
to model test data set
YES
DONE
Objectives
Satisfied
?
NO
Math
Formulation
Wrong
?
NO
YES
Hypothesis
YES
wrong for
objectives
?
NO
Re-evaluate objectives
or
Give Up
These steps consist of:
1.
2.
3.
4.
5.
6.
Recognizing the problem (Nature may require you to
“dig” for them)
– Interpreting the problem, or refining it.
Selecting a mathematical framework (stochastic,
deterministic, continuous, discrete). Be aware of what’s
been done before, and what’s still undone.
Finding the appropriate mathematical tools (computation
is generally required).
The solution may be approximate.
Feedback is required: is the approximation ok? Is the
question being answered the one that was asked?
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)