simulating spatial partial differential equations with cellular automata by brian strader adviser:...
TRANSCRIPT
Simulating Spatial Partial Differential Equations with
Cellular Automata
By Brian Strader
Adviser: Dr. Keith Schubert
Committee: Dr. George Georgiou
Dr. Ernesto Gomez
Introduction & Background
Topics Covered
•Partial Differential Equation, Cellular Automata (CA), & Biology
•Converting Differential Equations to CA
•CA Theoretical Constraints
•Convergence Maps & Guidelines
Introduction & Background
Cellular Automata (CA)
•CA Model uses simple rules about changes with time.
•Rules are localized and involve the values of cell neighbors.
•The set of rules are applied to the cells with the matrix after each time period.
Introduction & Background
Conway’s Game of LifeSurvival Rule: 2-3 Neighbors
Death by Overpopulation: 4+ Neighbors
Introduction & Background
Conway’s Game of LifeDeath by Isolation: 1 or Less Neighbors
Birth: 3 Neighbors
Introduction & Background
Spatial Partial Diff. Equations
•Changes with respect to time.
•Part of the equation depends on changes in space.
Introduction & Background
CA Advantages
•Simple Rules - easy to understand
•Discretized
•Local Problem View
•Highly Parallelizable
Converting Differential Equations to CA
Diff. Equation Form
Conditions:for n(u) = up where p <= 1for o(u) = up where p <= 1
Converting Differential Equations to CA
Diff. Equation Form
Conditions:for n(u) = up where p <= 1for o(u) = up where p <= 1
Converting Differential Equations to CA
Diff. Equation Form
Conditions:for n(u) = up where p <= 1for o(u) = up where p <= 1
Converting Differential Equations to CA
Euler’s Methods
Forward Euler’s Method:
Backward Euler’s Method:
Converting Differential Equations to CA
Euler’s Methods
Forward Euler’s Method:
1 2 3 4 5i=1jj-1 j+1
3.2 5.7 7.3 9.2 -7.5i=2jj-1 j+1
CA Theoretical Constraints
Z-Transform
•Time Domain Frequency Domain
•Discrete Form of Laplace Transform and related to the Fourier Transform
•Transformation makes life easier
•zeros when f(z)=0 poles when g(z)=0
CA Theoretical Constraints
Z-Transform
1. Perform z-transform
2. Solve for Uj
3. Find poles and zeros for Uj=f(z)/g(z)
4. Set poles and zeros values of z < 1 to converge
CA Theoretical Constraints
Forward Euler’s Constraints
Forward Euler’s Linear Form:
Zeros Constraint:
CA Theoretical Constraints
Forward Euler’s Constraints
Forward Euler’s Linear Form:
Poles Constraint:
CA Theoretical Constraints
Backward Euler’s Constraints
Backward Euler’s Linear Form:
Zeros Constraint:
CA Theoretical Constraints
Backward Euler’s Constraints
Backward Euler’s Linear Form:
Poles Constraint:
Convergence Maps & Guidelines
CA Sim 1 2 3 4 5i=1
jj-1 j+1
1.1 1.9 2.8 2.6 5.4i=2jj-1 j+1
0.1 0.35 0.27 0.4 0.57i=njj-1 j+1
0.11 0.34 0.27 0.4 0.56i=n-1jj-1 j+1
...
< 10-10
Convergence Maps & Guidelines
CA Sim 1 2 3 4 5i=1
jj-1 j+1
1.1 1.9 2.8 2.6 5.4i=2jj-1 j+1
541 -5623 -897 456 878i=njj-1 j+1
1.2 872 927 -722 -256i=n-1jj-1 j+1
...
> 1010
Convergence Maps & Guidelines
CA Sim 1 2 3 4 5i=1
jj-1 j+1
1.1 1.9 2.8 2.6 5.4i=2jj-1 j+1
1.1 2.1 3 4 5.1i=4000jj-1 j+1
1 2.1 3.1 3.9 5i=3999jj-1 j+1
...
Convergence Maps & Guidelines
Substituting Uj-1 and Uj+1
•Boundary Zero Values
0.11 0.34 0.27 0.4 0.56
jj-1 j+1
0 0
Convergence Maps & Guidelines
Guidelines
If ((upperZero and lowerPole intersects) and (intesection < initial point)) then
htMax = intersection * safetyBuffer;Else
htMax = initial point * safetyBuffer;End
ht = userInput( < htMax);hx=lowerPole(ht);
Conclusion
Guidelines
If ((upperZero and lowerPole intersects) and (intesection < initial point)) then
htMax = intersection * safetyBuffer;Else
htMax = initial point * safetyBuffer;End
ht = userInput( < htMax);hx=lowerPole(ht);
Conclusion
Future Work
• Proofs of Observations
•Quadratic General Form:
•Efficient Parallelization
•Simulation Error
Conclusion
References Paul Rochester. Euler's Numerical Method for Solving Differential Equations. November 2009. http://people.bath.ac.uk/prr20/ma10126webpage.html
Region of Convergence. Wikipedia. November 2009. http://en.wikipedia.org/wiki/Z-transform
Keith Schubert. Cellular automaton for bioverms, October 2008.
Jane Curnutt, Ernesto Gomez, and Keith Evan Schubert. Patterned growth in extreme environments. 2007.
Cell Image - http://askabiologist.asu.edu/research/buildingblocks/images/cell.jpg
Martin Gardner. The fantastic combinations of john conway’s new solitaire game”life”. Scientific American, (223):120–123, 1970.
T.A. Burton, editor. Modeling and Differential Equations in Biology. Pure andApplied Mathematics. Marcel Dekker Inc., 1980.
J. von Hardenberg, E. Meron, M. Shachak, and Y. Zarmi1. Diversity of vegetationpatterns and desertification. Physical Review Letters, 87(19), November 2001.