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Cellular Neural Networks and Least Squares for partial differential problems parallel solving Intercell project ESCAPaDE 2 N. Fressengeas 1 H. Frezza-Buet 2 1 Laboratoire Mat eriaux Optiques, Photonique et Syst` emes Unit e de

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SLIDE 1

Cellular Neural Networks and Least Squares for partial differential problems parallel solving

Intercell project — ESCAPaDE 2

  • N. Fressengeas1
  • H. Frezza-Buet2

1Laboratoire Mat´

eriaux Optiques, Photonique et Syst` emes Unit´ e de Recherche commune ` a l’Universit´ e Paul Verlaine Metz et ` a Sup´ elec

2Informations Multimodalit´

es et Signal Sup´ elec

June 15th, 2009

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 2

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution

Outline

1

Cellular Neural Networks and Partial Differential Equations Cellular Automata and complex systems CNN’s and PDE quantitative resolution

2

Least Squares and Cellular Neural Networks Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

3

Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 3

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

Cellular automata for the modeling of complex systems

Cellular Automaton Discrete system of cells Finite number of cell states Uniform rule for cell update Cellular Neural Network Discrete system of cells Continuous cell states Several cell update rules

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 4

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

Cellular automata for the modeling of complex systems

Cellular Automaton Discrete system of cells Finite number of cell states Uniform rule for cell update Cellular Neural Network Discrete system of cells Continuous cell states Several cell update rules Complexity from simple local rules Forest fires modeling Lattice Gas Automata figure from MathWorld

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 5

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

Qualitative complexity emerges from simple local rules

Automata have a hard time finding quantitatively exact solutions

Forest Fires Modeled successfully With simple rules Obviously with oversimplified heat transfer laws Lattice Gas Automata Very successful But only for statistics

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 6

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

Qualitative complexity emerges from simple local rules

Automata have a hard time finding quantitatively exact solutions

Forest Fires Modeled successfully With simple rules Obviously with oversimplified heat transfer laws Lattice Gas Automata Very successful But only for statistics

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 7

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

Solving ODEs with VLSI designed Runge-Kutta method

An attempt at finding quantitative results with automata

Runge-Kutta and VLSI Transform PDE1 into ODE2 Solve ODE through Runge-Kutta Implement with VLSI3

1PDE:Partial Differential Equation (or System) 2ODE: Ordinary Differential Equation (or System) 3VLSI:Very-Large-Scale Integration (Electronics)

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 8

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

Solving ODEs with VLSI designed Runge-Kutta method

An attempt at finding quantitative results with automata

Runge-Kutta and VLSI Transform PDE1 into ODE2 Solve ODE through Runge-Kutta Implement with VLSI3 Pro’s and con’s Fast Does no better than Runge-Kutta Need dedicated hardware Now superseded by desktop computers

1PDE:Partial Differential Equation (or System) 2ODE: Ordinary Differential Equation (or System) 3VLSI:Very-Large-Scale Integration (Electronics)

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 9

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

The need for a quantitative approach

The design of a CNN1 from a differential problem for an quantitative solution

Current approaches Sniff the Cellular Neural Network Assess its potentialities

1CNN: Cellular Neural Network

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 10

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Cellular Automata and complex systems CNN’s and PDE quantitative resolution

The need for a quantitative approach

The design of a CNN1 from a differential problem for an quantitative solution

Current approaches Sniff the Cellular Neural Network Assess its potentialities A better approach Design the CNN1 from the differential problem Through a systematic method Verify its results

1CNN: Cellular Neural Network

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 11

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Least Squares Finite Element Method

Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . .

A differential problem Aξ = f on ˜ Υ Bξ = 0 on ˜ Γ

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 12

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Least Squares Finite Element Method

Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . .

A differential problem Aξ = f on ˜ Υ Bξ = 0 on ˜ Γ More general Aiξ = fi on Υi

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 13

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Least Squares Finite Element Method

Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . .

A differential problem Aξ = f on ˜ Υ Bξ = 0 on ˜ Γ More general Aiξ = fi on Υi

  • r

Φ(ξ) = 0 on Υ =

i Υi

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 14

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Least Squares Finite Element Method

Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . .

A differential problem Aξ = f on ˜ Υ Bξ = 0 on ˜ Γ More general Aiξ = fi on Υi

  • r

Φ(ξ) = 0 on Υ =

i Υi

Minimize the squared error sum ξ solution ⇒ ξ minimizes

i

  

  • Υi

(Aiξ − fi)2   

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 15

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Least Squares Finite Element Method

Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . .

A differential problem Aξ = f on ˜ Υ Bξ = 0 on ˜ Γ More general Aiξ = fi on Υi

  • r

Φ(ξ) = 0 on Υ =

i Υi

Minimize the squared error sum ξ solution ⇒ ξ minimizes

i

  

  • Υi

(Aiξ − fi)2    Simpler ξ solution ⇒ ξ minimizes

  • Υ

Φ(ξ)2

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 16

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Minimize the residual integral

Minimization method are numerous Back to standard Finite Elements through variational formulation Null derivative and back to Finite Elements . . . Finite Differences over all Υ Newton-related minimization method Towards Cellular Automata and Neural Networks The latter can be implemented through Cellular Automata

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 17

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Minimize the residual integral

Minimization method are numerous Back to standard Finite Elements through variational formulation Null derivative and back to Finite Elements . . . Finite Differences over all Υ Newton-related minimization method Towards Cellular Automata and Neural Networks The latter can be implemented through Cellular Automata

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 18

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Minimize the residual integral

Minimization method are numerous Back to standard Finite Elements through variational formulation Null derivative and back to Finite Elements . . . Finite Differences over all Υ Newton-related minimization method Towards Cellular Automata and Neural Networks The latter can be implemented through Cellular Automata

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 19

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Finite Difference discretisation

Prelude for a finite number of Cells in the Neural Network

Finite Space Replace Υ by a finite set of points Ω Minimize E

  • ˜

ξ

  • =

Ω ˜

Φ

  • ˜

ξ 2 instead of

  • Υ

Φ(ξ)2 Differential problem replaced by multi-variate non linear minimization problem. With a potentially huge number of dimensions. Lost ? Let’s take an example

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 20

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Finite Difference discretisation

Prelude for a finite number of Cells in the Neural Network

Finite Space Replace Υ by a finite set of points Ω Minimize E

  • ˜

ξ

  • =

Ω ˜

Φ

  • ˜

ξ 2 instead of

  • Υ

Φ(ξ)2 Differential problem replaced by multi-variate non linear minimization problem. With a potentially huge number of dimensions. Lost ? Let’s take an example

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 21

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Finite Difference discretisation

Prelude for a finite number of Cells in the Neural Network

Finite Space Replace Υ by a finite set of points Ω Minimize E

  • ˜

ξ

  • =

Ω ˜

Φ

  • ˜

ξ 2 instead of

  • Υ

Φ(ξ)2 Differential problem replaced by multi-variate non linear minimization problem. With a potentially huge number of dimensions. Lost ? Let’s take an example

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 22

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Finite Difference discretisation

Prelude for a finite number of Cells in the Neural Network

Finite Space Replace Υ by a finite set of points Ω Minimize E

  • ˜

ξ

  • =

Ω ˜

Φ

  • ˜

ξ 2 instead of

  • Υ

Φ(ξ)2 Differential problem replaced by multi-variate non linear minimization problem. With a potentially huge number of dimensions. Lost ? Let’s take an example

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 23

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Finite Difference discretisation

Prelude for a finite number of Cells in the Neural Network

Finite Space Replace Υ by a finite set of points Ω Minimize E

  • ˜

ξ

  • =

Ω ˜

Φ

  • ˜

ξ 2 instead of

  • Υ

Φ(ξ)2 Differential problem replaced by multi-variate non linear minimization problem. With a potentially huge number of dimensions. Lost ? Let’s take an example

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 24

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

From the 1D Poisson Equation to Cellular Neural Networks

Part I: Discretisation △V (x) = ∂2V

∂x2 = ρ (x)

Continuous Equation Φ(ξ) = 0 ⇔ ∂2V

∂x2 − ρ (x) = 0

Discrete Equation

1 d2 (Vi−1 − 2Vi + Vi+1) − ρi = 0

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 25

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

From the 1D Poisson Equation to Cellular Neural Networks

Part I: Discretisation △V (x) = ∂2V

∂x2 = ρ (x)

Continuous Equation Φ(ξ) = 0 ⇔ ∂2V

∂x2 − ρ (x) = 0

Discrete Equation

1 d2 (Vi−1 − 2Vi + Vi+1) − ρi = 0

Continuous Residue Integral

  • Υ

Φ(ξ)2 =

  • Υ

∂2V ∂x2 − ρ (x) 2 dx Discrete Residue Sum E

  • ˜

ξ

  • N−1
  • i=2

Vi−1 − 2Vi + Vi+1 d2 − ρi 2

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 26

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

From the 1D Poisson Equation to Cellular Neural Networks

Part I: Discretisation △V (x) = ∂2V

∂x2 = ρ (x)

Continuous Equation Φ(ξ) = 0 ⇔ ∂2V

∂x2 − ρ (x) = 0

Discrete Equation

1 d2 (Vi−1 − 2Vi + Vi+1) − ρi = 0

Continuous Residue Integral

  • Υ

Φ(ξ)2 =

  • Υ

∂2V ∂x2 − ρ (x) 2 dx Discrete Residue Sum E

  • ˜

ξ

  • N−1
  • i=2

Vi−1 − 2Vi + Vi+1 d2 − ρi 2 Optimization problem Minimize the discrete sum for the parameters V (this example) the discrete ˜ ξ

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 27

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Various minimization methods can be chosen

The general framework of local optimization

all methods derive from the steepest gradient ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

A given method ⇔ a given η Fixed η : steepest gradient η = H−1|˜

ξk (E) : Newton Method

. . . Locality problem All these require access to the whole numeric space ˜ ξ Killer issue for InterCell

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-28
SLIDE 28

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Various minimization methods can be chosen

The general framework of local optimization

all methods derive from the steepest gradient ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

A given method ⇔ a given η Fixed η : steepest gradient η = H−1|˜

ξk (E) : Newton Method

. . . Locality problem All these require access to the whole numeric space ˜ ξ Killer issue for InterCell

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-29
SLIDE 29

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Various minimization methods can be chosen

The general framework of local optimization

all methods derive from the steepest gradient ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

A given method ⇔ a given η Fixed η : steepest gradient η = H−1|˜

ξk (E) : Newton Method

. . . Locality problem All these require access to the whole numeric space ˜ ξ Killer issue for InterCell

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-30
SLIDE 30

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Making computation local only for InterCell

with the help of the stochastic gradient method

The stochastic gradient Replace grad|˜

ξk (E) by i grad|˜ ξk

i (E)

Evaluate the sum in a random order Which cells are involved ? Each cell updates stems from

the gradient of the error with respect to itself

The cells involved are those for which that gradient is not zero It is a close neighborhood depending on the equation itself

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

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SLIDE 31

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Making computation local only for InterCell

with the help of the stochastic gradient method

The stochastic gradient Replace grad|˜

ξk (E) by i grad|˜ ξk

i (E)

Evaluate the sum in a random order Which cells are involved ? Each cell updates stems from

the gradient of the error with respect to itself

The cells involved are those for which that gradient is not zero It is a close neighborhood depending on the equation itself

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-32
SLIDE 32

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

From the 1D Poisson Equation to Cellular Neural Networks

Part II: Local computations △V (x) = ∂2V

∂x2 = ρ (x)

Discrete equation ∀i ∈ N, 1 < i < N, 1 d2 (Vi−1 − 2Vi + Vi+1) − ρi = 0. Discrete Residue Sum E(V ) =

N−1

  • i=2

1 d2 (Vi−1 − 2Vi + Vi+1) − ρi 2 Stochastic Gradient Newton Method

i = 1

  • r

i = N V k+1

i

= V k

i

i = 2 V k+1

i

= 1

5

“ 2V k

i−1 + 4V k i+1 − V k i+2 + d2 ×

“ ρi+1 − 2ρi ”” i = N − 1 V k+1

i

= 1

5

“ 2V k

i+1 + 4V k i−1 − V k i−2 + d2 ×

“ ρi−1 − 2ρi ”” 3 ≤ i ≤ N − 2 V k+1

i

= 1

6

“ −V k

i−2 + 4V k i−1 + 4V k i+1 − V k i+2 + d2 ×

“ ρi−1 − 2ρi + ρi+1””

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-33
SLIDE 33

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

From the 1D Poisson Equation to Cellular Neural Networks

Part II: Local computations △V (x) = ∂2V

∂x2 = ρ (x)

Discrete equation ∀i ∈ N, 1 < i < N, 1 d2 (Vi−1 − 2Vi + Vi+1) − ρi = 0. Discrete Residue Sum E(V ) =

N−1

  • i=2

1 d2 (Vi−1 − 2Vi + Vi+1) − ρi 2 Stochastic Gradient Newton Method

i = 1

  • r

i = N V k+1

i

= V k

i

i = 2 V k+1

i

= 1

5

“ 2V k

i−1 + 4V k i+1 − V k i+2 + d2 ×

“ ρi+1 − 2ρi ”” i = N − 1 V k+1

i

= 1

5

“ 2V k

i+1 + 4V k i−1 − V k i−2 + d2 ×

“ ρi−1 − 2ρi ”” 3 ≤ i ≤ N − 2 V k+1

i

= 1

6

“ −V k

i−2 + 4V k i−1 + 4V k i+1 − V k i+2 + d2 ×

“ ρi−1 − 2ρi + ρi+1””

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-34
SLIDE 34

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

From the 1D Poisson Equation to Cellular Neural Networks

Part II: Local computations △V (x) = ∂2V

∂x2 = ρ (x)

Discrete equation ∀i ∈ N, 1 < i < N, 1 d2 (Vi−1 − 2Vi + Vi+1) − ρi = 0. Discrete Residue Sum E(V ) =

N−1

  • i=2

1 d2 (Vi−1 − 2Vi + Vi+1) − ρi 2 Stochastic Gradient Newton Method

i = 1

  • r

i = N V k+1

i

= V k

i

i = 2 V k+1

i

= 1

5

“ 2V k

i−1 + 4V k i+1 − V k i+2 + d2 ×

“ ρi+1 − 2ρi ”” i = N − 1 V k+1

i

= 1

5

“ 2V k

i+1 + 4V k i−1 − V k i−2 + d2 ×

“ ρi−1 − 2ρi ”” 3 ≤ i ≤ N − 2 V k+1

i

= 1

6

“ −V k

i−2 + 4V k i−1 + 4V k i+1 − V k i+2 + d2 ×

“ ρi−1 − 2ρi + ρi+1””

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-35
SLIDE 35

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specification of the boundary conditions

Neumann boundary conditions are implicitly included

Let us recall the problem specification Aiξ = fi on Υi Φ(ξ) = 0 on Υ =

i Υi

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-36
SLIDE 36

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specification of the boundary conditions

Neumann boundary conditions are implicitly included

Let us recall the problem specification Aiξ = fi on Υi Φ(ξ) = 0 on Υ =

i Υi

Several differential systems

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-37
SLIDE 37

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specification of the boundary conditions

Neumann boundary conditions are implicitly included

Let us recall the problem specification Aiξ = fi on Υi Φ(ξ) = 0 on Υ =

i Υi

Neumann boundary conditions Another differential problem On another domain Included in the residue sum Several differential systems

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-38
SLIDE 38

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specifying Dirichlet boundary conditions

Dirichlet boundary condition as a differential problem included in the residue sum

Update rule for the Dirichlet boundaries Update rule for a Dirichlet boundary Cell : do not change General update rule : ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

grad|˜

ξk (E) must be null

E must be independent of that particular cell: a constant is Dirichlet boundary specific differential problem: 0=0 This is intuitively correct The value specified on a Dirichlet boundary is part of the solution So just specify a true equation: 0=0 or the Boolean True

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-39
SLIDE 39

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specifying Dirichlet boundary conditions

Dirichlet boundary condition as a differential problem included in the residue sum

Update rule for the Dirichlet boundaries Update rule for a Dirichlet boundary Cell : do not change General update rule : ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

grad|˜

ξk (E) must be null

E must be independent of that particular cell: a constant is Dirichlet boundary specific differential problem: 0=0 This is intuitively correct The value specified on a Dirichlet boundary is part of the solution So just specify a true equation: 0=0 or the Boolean True

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-40
SLIDE 40

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specifying Dirichlet boundary conditions

Dirichlet boundary condition as a differential problem included in the residue sum

Update rule for the Dirichlet boundaries Update rule for a Dirichlet boundary Cell : do not change General update rule : ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

grad|˜

ξk (E) must be null

E must be independent of that particular cell: a constant is Dirichlet boundary specific differential problem: 0=0 This is intuitively correct The value specified on a Dirichlet boundary is part of the solution So just specify a true equation: 0=0 or the Boolean True

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-41
SLIDE 41

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specifying Dirichlet boundary conditions

Dirichlet boundary condition as a differential problem included in the residue sum

Update rule for the Dirichlet boundaries Update rule for a Dirichlet boundary Cell : do not change General update rule : ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

grad|˜

ξk (E) must be null

E must be independent of that particular cell: a constant is Dirichlet boundary specific differential problem: 0=0 This is intuitively correct The value specified on a Dirichlet boundary is part of the solution So just specify a true equation: 0=0 or the Boolean True

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-42
SLIDE 42

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specifying Dirichlet boundary conditions

Dirichlet boundary condition as a differential problem included in the residue sum

Update rule for the Dirichlet boundaries Update rule for a Dirichlet boundary Cell : do not change General update rule : ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

grad|˜

ξk (E) must be null

E must be independent of that particular cell: a constant is Dirichlet boundary specific differential problem: 0=0 This is intuitively correct The value specified on a Dirichlet boundary is part of the solution So just specify a true equation: 0=0 or the Boolean True

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-43
SLIDE 43

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specifying Dirichlet boundary conditions

Dirichlet boundary condition as a differential problem included in the residue sum

Update rule for the Dirichlet boundaries Update rule for a Dirichlet boundary Cell : do not change General update rule : ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

grad|˜

ξk (E) must be null

E must be independent of that particular cell: a constant is Dirichlet boundary specific differential problem: 0=0 This is intuitively correct The value specified on a Dirichlet boundary is part of the solution So just specify a true equation: 0=0 or the Boolean True

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-44
SLIDE 44

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions

Specifying Dirichlet boundary conditions

Dirichlet boundary condition as a differential problem included in the residue sum

Update rule for the Dirichlet boundaries Update rule for a Dirichlet boundary Cell : do not change General update rule : ˜ ξk+1 = ˜ ξk − η.grad|˜

ξk (E)

grad|˜

ξk (E) must be null

E must be independent of that particular cell: a constant is Dirichlet boundary specific differential problem: 0=0 This is intuitively correct The value specified on a Dirichlet boundary is part of the solution So just specify a true equation: 0=0 or the Boolean True

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-45
SLIDE 45

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

And now, how do I do it ?

How can I practically use the LSFEM method on InterCell ?

I need a Cellular Automaton I now know how to determine the update rules I can use InterCell software suite (Booz and ParXXL) to program my automaton Problem : I am not such a geek in C++ I want to concentrate on numerics, not C++

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-46
SLIDE 46

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

And now, how do I do it ?

How can I practically use the LSFEM method on InterCell ?

I need a Cellular Automaton I now know how to determine the update rules I can use InterCell software suite (Booz and ParXXL) to program my automaton Problem : I am not such a geek in C++ I want to concentrate on numerics, not C++

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-47
SLIDE 47

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

And now, how do I do it ?

How can I practically use the LSFEM method on InterCell ?

I need a Cellular Automaton I now know how to determine the update rules I can use InterCell software suite (Booz and ParXXL) to program my automaton Problem : I am not such a geek in C++ I want to concentrate on numerics, not C++

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-48
SLIDE 48

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

Formal computing software solution

Remove the geek from the numeric loop

Formal computing Formal computing can help me for this LSFEM stuff But :

I have not completely understood all the maths What are those text files for automaton specification ? I need to re-do it all over again for each differential system

From the differential system to the automaton I need a software suite that computes the automaton from the differential system

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-49
SLIDE 49

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

Formal computing software solution

Remove the geek from the numeric loop

Formal computing Formal computing can help me for this LSFEM stuff But :

I have not completely understood all the maths What are those text files for automaton specification ? I need to re-do it all over again for each differential system

From the differential system to the automaton I need a software suite that computes the automaton from the differential system

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-50
SLIDE 50

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

Formal computing software solution

Remove the geek from the numeric loop

Formal computing Formal computing can help me for this LSFEM stuff But :

I have not completely understood all the maths What are those text files for automaton specification ? I need to re-do it all over again for each differential system

From the differential system to the automaton I need a software suite that computes the automaton from the differential system

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-51
SLIDE 51

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

ESCAPaDE 2 – software architecture

ESCAPaDE: Ergonomic Solver using Cellular Automata for PArtial Differential Equations

From the differential problem to the automaton specification use Open Source Sagea for formal computing use Sage part of EscaBooz to

Specify your partial differential problem using Sage-Python syntax Compute and write the automaton specification

ahttp://www.sagemath.org

From the automaton specification to Booz–ParXXL C++ code Simply run EscaBooz on the automaton specification make build Your Distributed Cellular Automaton based LSFEM method is up and ready to run

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-52
SLIDE 52

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

ESCAPaDE 2 – software architecture

ESCAPaDE: Ergonomic Solver using Cellular Automata for PArtial Differential Equations

From the differential problem to the automaton specification use Open Source Sagea for formal computing use Sage part of EscaBooz to

Specify your partial differential problem using Sage-Python syntax Compute and write the automaton specification

ahttp://www.sagemath.org

From the automaton specification to Booz–ParXXL C++ code Simply run EscaBooz on the automaton specification make build Your Distributed Cellular Automaton based LSFEM method is up and ready to run

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-53
SLIDE 53

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

3D Poisson Equation with fixed charge

Not so easy because of 3D, space variable charge and Dirichlet boundaries

Poisson 3D △V (x, y, z) = ρ (x, y, z) General Update Rule V ← 1 42        V       

12 12 12 12 −2 −2 −2 −2 −2 −2 −1 −1 −1 −1 −1

       + d2.ρ   

1 1 1 1 1 −6

         

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-54
SLIDE 54

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

3D Poisson Equation with fixed charge

Not so easy because of 3D, space variable charge and Dirichlet boundaries

Poisson 3D △V (x, y, z) = ρ (x, y, z) Sample results

5 10 15 5 10 15 2.0 1.5 1.0 0.5 0.0

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-55
SLIDE 55

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

Wave envelope equation

In a waveguide and without paraxial approximation

Wave envelope equation

∂A ∂z − i 2k △A = ik n δnA

Waveguide and input intensity

10 20 30 10 20 30 0.25 0.5 0.75 1 10 20 30

Output intensity

10 20 30 10 20 30 0.5 1 1.5 10 20 30

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-56
SLIDE 56

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

Elliptic equation

For instance: dislocation propagation, wave equation. . .

Elliptic Equation

∂a ∂t = −v ∗ ∂a ∂x

v=0.1

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-57
SLIDE 57

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

Elliptic equation

For instance: dislocation propagation, wave equation. . .

Elliptic Equation

∂a ∂t = −v ∗ ∂a ∂x

v=0.5

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2

slide-58
SLIDE 58

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Automated formal computing automaton specification Software architecture Sample results

All the details of LSFEM on Cellular Neural Networks

arXiv:math-ph/0610037v6 Cellular Neural Networks and Least Squares for partial differential problems parallel solving

Nicolas Fressengeas

Laboratoire Mat´ eriaux Optiques, Photonique et Syst` emes University Paul Verlaine of Metz and Sup´ elec 2, rue Edouard Belin, 57070 Metz Cedex, France email: nicolas.fressengeas@univ-metz.fr

Herv´ e Frezza-Buet

Information, Multimodality and Signal, Sup´ elec 2, rue Edouard Belin, 57070 Metz, France email: Herve.Frezza-Buet@supelec.fr Abstract This paper shows how Cellular Neural Networks (CNN) can be harnessed into solv-

  • N. Fressengeas, H. Frezza-Buet

ESCAPaDE 2