Transcript Mathematical Modeling / Computational Science

Mathematical Modeling /
Computational Science
AiS Challenge
Summer Teacher Institute
2002
Richard Allen
Mathematical Modeling
The process of creating a mathematical
representation of some phenomenon in
order to better understand the phenomenon
and to predict its future behavior.
"The success of a mathematical model
depends on how easy it is to use and how
accurately it predicts."
Examples
Newton’s second law: F = m*a
Radioactive decay: N(t) = N(0)*e-k*t
Compound interest: P(t) = P(0)(1 + r/n)nt
Falling rock: x(t) = - g*t2/2 + v0*t + x0
or
t0 = 0; x0 = H; v0 = V
ti+1 = ti + Δt;
vi+1 = vi - (g * Δt); xi+1 = xi + (vi * Δt);
= 0, 1, 2, …
i
Mathematics Modeling Cycle
Mathematical Modeling
A Real-World Problem:
 Model the spread and
control of the Hantavirus.
 Model manufacturing processes to minimize time-to-market and cost.
 Model training times to optimize performance
in sprints/long distance running.
www.challenge.nm.org/Archive
Understand current activity and predict future behavior.
Example: Falling Rock
Determine the motion of a rock dropped
from height, H, above the ground with
initial velocity, V.
A discrete model: Find the position and
velocity of the rock at the equally spaced
times, t0, t1, t2, …; e.g., t0 = 0 sec., t1 = 1
sec., t2 = 2 sec., etc.
|______|______|____________|______
t0
t1
t2
…
tn
Mathematical Modeling
Simplify  Working Model:
Identify and select factors that
describe important aspects of
the Real World Problem; determine those factors that can be
neglected.
Determine governing principles, physical laws.
 Identify model variables; focus on how they are
related.
 State simplifying assumptions.

Example: Falling Rock
Governing principles: d = v*t and v = a*t.
Simplifying assumptions:
Gravity is the only force acting on the body.
 Flat earth.
 No drag (air resistance).
 Rock’s position and velocity above the ground
will be modeled at discrete times (t0, t1, t2, …)
until rock hits the ground.

Mathematical Modeling
Abstract  Mathematical
Model:
Express the Working
Model in
mathematical terms;
write down mathematical equations whose solution describes the Working Model.
There may not be a "best" model; the one to
be used will depend on the questions to be
studied.
Example: Falling Rock
v0
v1
v2
…
vn
x0
x1
x2
…
xn
|______|______|____________|______
t0
t1
t2
…
tn
t0 = 0; x0 = H; v0 = V; Δt = ti+1 - ti
t1= t0 + Δt
t2= t1 + Δt
x1= x0 + (v0*Δt)
x2= x1 + (v1*Δt)
v1= v0 - (g*Δt)
v2= v1 - (g*Δt)
…
Mathematical Modeling
Program  Computational
Model: Implement Mathematical Model in “computer
code”.
If model is simple enough, it may be solved
analytically; otherwise, a computer program is
required.
Example: Falling Rock
Pseudo Code
Input
V, initial velocity; H, initial height
g, acceleration due to gravity
Δt, time step; imax, maximum number of steps
Output
ti, t-value at time step i
xi, height at time ti
vi, velocity at time ti
Example: Falling Rock
Initialize
ti = t0 = 0; vi = v0 = V; xi = x0 = H
print ti, xi, vi
Time stepping: i = 1, imax
ti = ti + Δt
xi = xi - vi*Δt
vi = vi + g*Δt
print ti, xi, vi
if (xi <= 0), xi = 0; quit
Mathematical Modeling
Simulate  Conclusions:
Execute “computer code”
to obtain Results. Formulate
Conclusions.
Verify your computer program; use check
cases.
 Graphs, charts, and other visualization tools
are useful in summarizing results and drawing
conclusions.

Mathematical Modeling
Interpret Conclusions
and compare with Real
World Problem behavior.

If model results do not “agree” with physical
reality or experimental data, reexamine the
Working Model and repeat modeling steps.

Usually, modeling process proceeds through
several iterations until model is“acceptable”.
Example: Falling Rock
To create a more more realistic model of a
falling rock, some of the simplifying
assumptions could be dropped:
Incorporate air resistance, depends on shape of
rock.
 Improve discrete model: approximate velocities
in the midpoint of time intervals instead of the
beginning.
 Reduce the size of Δt.

Mathematics Modeling Cycle
A Virtual Science Laboratory
The site below is a virtual library to visualize
science. It has projects in mechanics,
electricity and magnetism, life sciences,
waves, astrophysics, and optics. It can be
used to motivate the development of
mathematical models for computational
science projects.
Explore science
Computational Science
Computational science seeks to gain an
understanding of science through the use of
mathematical models on high-performance
computers.
Complements the areas of theory and
experiment in science, but does not replace
theory or experiments.
Is the modern tool of scientific investigation.
Computational Science
Is often use in place of experiments when
experiments are too large, too expensive, too
dangerous, or too time consuming.
Generally involves teamwork.
Is a multidisciplinary activity.
Languages include Fortran, C, C++, and
Java; Matlab; Excel.
Computational Science
Has been successful in various application
areas, including:
Seismology
Global ocean/
climate modeling
Materials research
Drug design
Biology
Environment
Manufacturing
Economics
Medicine
Example: Industry 
First jetliner to be digitally designed, "pre-assembled" on
computer, reducing need for costly, full-scale mockup.
Computational modeling improved the quality of work
and reduced changes, errors, and rework.
Example: Biology 
Road maps for the human brain
Cortical regions activated as
a subject remembers letters
x and r.
Real-time MRI technology
may soon be incorporated
into dedicated hardware
bundled with MRI scanners
allowing the use of MRI in
psychiatry, drug evaluation,
and neuro-surgical planning. www.itrd.gov/pubs/blue00/hecc.html
Example: Climate Modeling
This 3-D shaded relief
representation of a portion
of Pennsylvania uses
color to show maximum
daily temperature.
Displaying multiple data
sets at once, and interactively changing the
display, helps users
quickly explore their data,
to formulate or confirm
hypotheses.
www.itrd.gov/pubs/blue00/
hecc.html
Referenced URLs
AiS Challenge Archive site
www.challenge.nm.org/Archive/
Explore Science site
www.explorescience.com
Boeing example
www.boeing.com/commercial/777family/index.html
Road maps for the human brain example
www.itrd.gov/pubs/blue00/hecc.html
Differential to Difference Equations
Many of the equations involving dynamic processes are
formulated as differential equations. To approximate such
an equation, consider the following example:
dP(t)/dt = k*P(t)
can be approximated by
(Pi+1 – Pi)/ Δt = k*Pi, i = 1, 2, …
P0
P1
P2
Pn
|---------|---------|------------------|------------> t
t0
t1
t2
…
tn
Math/Science URLs
www.shodor.org/master/ - Modeling and Simulation Tools
for Educational Reform - Shodor
www.math.montana.edu/frankw/ccp/modeling/topic.htm Math Modeling in a Real and Complex World
www.city.ac.uk/mathematics/X2ApplMaths/index.html Applied Mathematics Class (population models, linear
programming,dynamics, waves)
www.explorescience.com/activities/activity_list.cfm?categ
oryID=11 -Explore Science Multimedia Activities
www.math.duke.edu/education/ccp/index.html - Connected
CurriculumProject - Interactive Learning Materials for
Mathematics and Its Applications
Math/Science URLs
www.shodor.org/interactivate/ - Shodor Education
Foundation) Middle School mathematics interactive tools
(indexed to several math textbooks)
www.mste.uiuc.edu/ - Database of Mathematics and
Science interactive tools and lesson plans
chemviz.ncsa.uiuc.edu/ - Chemistry Visualization tools for
viewing and analyzing molecular structures
theory.uwinnipeg.ca/java/ - Physics and math interactive
tools
mvhs1.mbhs.edu/mvhsproj/cm.html - High School
curriculum modules using modeling, especially with Stella;
all science subjects.
Math/Science URLs
archive.ncsa.uiuc.edu/edu/ICM/ - Middle School level
models and lesson plans using Model-It and Stella
www.hi-ce.org/ - (Center for Highly Interactive Computing
in Education, U. Michigan) Data collection, graphing, and
modeling software tools; project-based science curriculum
mie.eng.wayne.edu/faculty/chelst/informs/ - Math for
decision making in industry and government
illuminations.nctm.org/imath/912/TroutPond/index.html an interesting discrete math project