Mathematical Modeling / Computational Science

AiS Challenge Summer Teacher Institute

2002Richard 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); i = 0, 1, 2, …

Mathematics Modeling Cycle

Mathematical Modeling

A Real-World Problem: Model the spread and

control of the Hantavirus. Model manufacturing pro-

cesses 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; deter- mine 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 equa- tions 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

x1= x0 + (v0*Δt)

v1= v0 - (g*Δt)

t2= t1 + Δt

x2= x1 + (v1*Δt)

v2= v1 - (g*Δt)

…

Mathematical Modeling

Program Computational Model: Implement Mathe- matical Model in “computer code”.

If model is simple enough, it may be solved analytically; otherwise, a computer program is required.

Example: Falling Rock

Pseudo CodeInput

V, initial velocity; H, initial heightg, acceleration due to gravity

Δt, time step; imax, maximum number of steps

Outputti, t-value at time step ixi, height at time tivi, velocity at time ti

Example: Falling Rock

Initializeti = t0 = 0; vi = v0 = V; xi = x0 = H

print ti, xi, vi

Time stepping: i = 1, imaxti = 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

Environment

Economics

Materials research

Drug design

Biology

Manufacturing

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 inter-actively 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 sitewww.explorescience.com

Boeing examplewww.boeing.com/commercial/777family/index.html

Road maps for the human brain examplewww.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?categoryID=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