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  1. Mathematical Modeling / Computational Science AiS Challenge Summer Teacher Institute 2002 Richard Allen

  2. 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."

  3. 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 + x0or t0 = 0; x0 = H; v0 = V ti+1 = ti + Δt; vi+1 = vi - (g * Δt); xi+1 = xi + (vi * Δt); i = 0, 1, 2, …

  4. Mathematics Modeling Cycle

  5. 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. Understand current activity and predict future behavior.

  6. 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

  7. 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.

  8. 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.

  9. 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.

  10. 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) …

  11. 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.

  12. 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

  13. 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

  14. 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.

  15. 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”.

  16. 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.

  17. Mathematics Modeling Cycle

  18. 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

  19. 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.

  20. 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.

  21. Computational Science Has been successful in various application areas, including: • Seismology • Global ocean/ climate modeling • Environment • Economics • Materials research • Drug design • Biology • Manufacturing • Medicine

  22. 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.

  23. 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.

  24. 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. •

  25. Referenced URLs AiS Challenge Archive site Explore Science site Boeing example Road maps for the human brain example

  26. 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

  27. Math/Science URLs • - Modeling and Simulation Tools for Educational Reform  - Shodor • - Math Modeling in a Real and Complex World • - Applied Mathematics Class (population models, linear programming,dynamics, waves) • -Explore Science Multimedia Activities • - Connected CurriculumProject - Interactive Learning Materials for Mathematics and Its Applications

  28. Math/Science URLs • - Shodor Education Foundation) Middle School mathematics interactive tools (indexed to several math textbooks) • - Database of Mathematics and Science interactive tools and lesson plans • - Chemistry Visualization tools for viewing and analyzing molecular structures • - Physics and math interactive tools • - High School curriculum modules using modeling, especially with Stella; all science subjects.

  29. Math/Science URLs • - Middle School level models and lesson plans using Model-It and Stella • - (Center for Highly Interactive Computing in Education, U. Michigan) Data collection, graphing, and modeling software tools; project-based science curriculum • - Math for decision making in industry and government • - an interesting discrete math project