Simulating Spatial Partial Differential Equations with Cellular Automata

1
Simulating Spatial Partial Differential Equations
with Cellular Automata

• By Brian Strader
• Adviser Dr. Keith Schubert
• Committee Dr. George Georgiou
• Dr. Ernesto Gomez

2

• Topics Covered

• Partial Differential Equation, Cellular Automata
  (CA), Biology
• Converting Differential Equations to CA
• CA Theoretical Constraints
• Convergence Maps Guidelines

3

• 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.

4

• Conways Game of Life

Survival Rule 2-3 Neighbors
Death by Overpopulation 4 Neighbors
5

• Conways Game of Life

Death by Isolation 1 or Less Neighbors
Birth 3 Neighbors
6

• Conways Game of Life

t 0
7
Introduction Background

• Conways Game of Life

t 1
8

• Conways Game of Life

t 2
9

• Conways Game of Life

t 3
10

• Celluar Automata Simulation

11

• Celluar Automata Simulation

12

• Spatial Partial Diff. Equations

• Changes with respect to time.
• Part of the equation depends on changes in space.

13

• Vegetation Patterns

14

• CA Advantages

• Simple Rules - easy to understand
• Discretized
• Local Problem View
• Highly Parallelizable

15
Converting Differential Equations to CA

• Diff. Equation Form

Conditions for n(u) up where p lt 1 for o(u)
up where p lt 1
16
Converting Differential Equations to CA

• Diff. Equation Form

Conditions for n(u) up where p lt 1 for o(u)
up where p lt 1
17
Converting Differential Equations to CA

• Diff. Equation Form

Conditions for n(u) up where p lt 1 for o(u)
up where p lt 1
18
Converting Differential Equations to CA

• Discretization Techniques

19
Converting Differential Equations to CA

• Size of hx

Large hx
Small hx
20
Converting Differential Equations to CA

• Eulers Methods

Forward Eulers Method
21
Converting Differential Equations to CA

• Size of ht

22
Converting Differential Equations to CA

• Eulers Methods

Backward Eulers Method
23
Converting Differential Equations to CA

• Eulers Methods

Forward Eulers Method
Backward Eulers Method
24
Converting Differential Equations to CA

• Eulers Methods

Forward Eulers Method
1
2
3
4
5
i1
j
j-1
j1
3.2
5.7
7.3
9.2
-7.5
i2
j
j-1
j1
25
CA Theoretical Constraints

• General Linear Form

26
CA Theoretical Constraints

• Convergence and Divergence

27
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

28
CA Theoretical Constraints

• Z-Transform

29
CA Theoretical Constraints

• Z-Transform

1. Perform z-transform 2. Solve for Uj 3. Find
poles and zeros for Ujf(z)/g(z) 4. Set poles
and zeros values of z lt 1 to converge
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CA Theoretical Constraints

• Forward Eulers Constraints

Forward Eulers Linear Form
Zeros Constraint
31
CA Theoretical Constraints

• Forward Eulers Constraints

Forward Eulers Linear Form
Poles Constraint
32
CA Theoretical Constraints

• Backward Eulers Constraints

Backward Eulers Linear Form
Zeros Constraint
33
CA Theoretical Constraints

• Backward Eulers Constraints

Backward Eulers Linear Form
Poles Constraint
34
Convergence Maps Guidelines

• CA Sim

1
2
3
4
5
i1
j
j-1
j1
1.1
1.9
2.8
2.6
5.4
i2
...
j
j-1
j1
0.11
0.34
0.27
0.4
0.56
in-1
lt 10-10
j
j-1
j1
0.1
0.35
0.27
0.4
0.57
in
j
j-1
j1
35
Convergence Maps Guidelines

• CA Sim

1
2
3
4
5
i1
j
j-1
j1
1.1
1.9
2.8
2.6
5.4
i2
...
j
j-1
j1
1.2
872
927
-722
-256
in-1
gt 1010
j
j-1
j1
541
-5623
-897
456
878
in
j
j-1
j1
36
Convergence Maps Guidelines

• CA Sim

1
2
3
4
5
i1
j
j-1
j1
1.1
1.9
2.8
2.6
5.4
i2
...
j
j-1
j1
1
2.1
3.1
3.9
5
i3999
j
j-1
j1
1.1
2.1
3
4
5.1
i4000
j
j-1
j1
37
Convergence Maps Guidelines

• Forward Convergence Map

38
Convergence Maps Guidelines

• Backward Convergence Map

39
Convergence Maps Guidelines

• a Parameters

40
Convergence Maps Guidelines

• a Parameters

a1
41
Convergence Maps Guidelines

• a Parameters

a2
42
Convergence Maps Guidelines

• Forward Constraints

Poles Constraint
43
Convergence Maps Guidelines

• Backward Constraints

44
Convergence Maps Guidelines

• Simulation Speed

45
Convergence Maps Guidelines

• a3 Vertical Constraint

46
Convergence Maps Guidelines

• a3 Vertical Constraint

Zeros Constraint
47
Convergence Maps Guidelines

• Substituting Uj-1 and Uj1

• Boundary Zero Values

0.11
0.34
0.27
0.4
0.56
0
0
j
j-1
j1
48
Convergence Maps Guidelines

• Zeros Boundary Constraint

49
Convergence Maps Guidelines

• Zeros Boundary Constraint

50
Convergence Maps Guidelines

• Guidelines

If ((upperZero and lowerPole intersects) and
(intesection lt initial point)) then htMax
intersection safetyBuffer Else htMax
initial point safetyBuffer End ht
userInput( lt htMax) hxlowerPole(ht)
51
Convergence Maps Guidelines

• Guidelines Example

52
Conclusion

• Partial Diff -gt CA

53
Conclusion

• Theoretical Constraints

Zeros Constraint
Poles Constraint
54
Conclusion

• Guidelines

If ((upperZero and lowerPole intersects) and
(intesection lt initial point)) then htMax
intersection safetyBuffer Else htMax
initial point safetyBuffer End ht
userInput( lt htMax) hxlowerPole(ht)
55
Conclusion

• Future Work

• Proofs of Observations
• Quadratic General Form
• Efficient Parallelization
• Simulation Error

56
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 K
eith 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/buildingbloc
ks/images/cell.jpg Martin Gardner. The fantastic
combinations of john conways new solitaire
game life. Scientific American, (223)120123,
1970. T.A. Burton, editor. Modeling and
Differential Equations in Biology. Pure
and Applied Mathematics. Marcel Dekker Inc.,
1980. J. von Hardenberg, E. Meron, M. Shachak,
and Y. Zarmi1. Diversity of vegetation patterns
and desertification. Physical Review Letters,
87(19), November 2001.
57
Conclusion

• Acknowledgements
• and Questions?