simulating spatial partial differential equations with cellular automata by brian strader adviser:...

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


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.


Conway’s Game of LifeSurvival Rule: 2-3 Neighbors

Death by Overpopulation: 4+ Neighbors


Conway’s Game of LifeDeath by Isolation: 1 or Less Neighbors

Birth: 3 Neighbors


Conway’s Game of Life

t = 0



t = 1



t = 2



t = 3


Celluar Automata Simulation


Spatial Partial Diff. Equations

•Changes with respect to time.

•Part of the equation depends on changes in space.


Vegetation Patterns


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


Discretization Techniques


Size of hx

Large hx

Small hx


Euler’s Methods

Forward Euler’s Method:


Size of ht


Euler’s Methods

Backward Euler’s Method:


Euler’s Methods


Backward Euler’s Method:


Euler’s Methods


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

General Linear Form


Convergence and Divergence


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


Z-Transform


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


Forward Euler’s Constraints

Forward Euler’s Linear Form:

Zeros Constraint:


Forward Euler’s Constraints

Forward Euler’s Linear Form:

Poles Constraint:


Backward Euler’s Constraints

Backward Euler’s Linear Form:

Zeros Constraint:


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


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


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

...


Forward Convergence Map


Backward Convergence Map


a Parameters


a Parameters

a1


a Parameters

a2


Forward Constraints Poles Constraint:


Backward Constraints


Simulation Speed


a3 Vertical Constraint


a3 Vertical Constraint

Zeros Constraint:


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


Zeros Boundary Constraint


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);


Guidelines Example

Conclusion

Partial Diff -> CA

Conclusion

Theoretical Constraints

Zeros Constraint:

Poles Constraint:

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.

Conclusion

Acknowledgements

and Questions?

simulating spatial partial differential equations with cellular automata by brian strader adviser:...

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