Cellular Neural Networks and Least Squares for partial differential - PowerPoint PPT Presentation

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 Recherche commune ` a l’Universit´ e Paul Verlaine Metz et ` a Sup´ elec 2 Informations Multimodalit´ es et Signal Sup´ elec June 15 th , 2009 N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Least Squares and Cellular Neural Networks Application to automated PDE resolution Outline Cellular Neural Networks and Partial Differential Equations 1 Cellular Automata and complex systems CNN’s and PDE quantitative resolution Least Squares and Cellular Neural Networks 2 Least Squares Finite Element Method Adaptation of LSFEM to automata Boundary conditions Application to automated PDE resolution 3 Automated formal computing automaton specification Software architecture Sample results N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Cellular Automata and complex systems Least Squares and Cellular Neural Networks CNN’s and PDE quantitative resolution Application to automated PDE resolution Cellular automata for the modeling of complex systems Cellular Automaton Cellular Neural Network Discrete system of cells Discrete system of cells Finite number of cell states Continuous cell states Uniform rule for cell update Several cell update rules N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Cellular Automata and complex systems Least Squares and Cellular Neural Networks CNN’s and PDE quantitative resolution Application to automated PDE resolution Cellular automata for the modeling of complex systems Cellular Automaton Cellular Neural Network Discrete system of cells Discrete system of cells Finite number of cell states Continuous cell states Uniform rule for cell update 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

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

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

Cellular Neural Networks and Partial Differential Equations Cellular Automata and complex systems Least Squares and Cellular Neural Networks CNN’s and PDE quantitative resolution Application to automated PDE resolution Solving ODEs with VLSI designed Runge-Kutta method An attempt at finding quantitative results with automata Runge-Kutta and VLSI Transform PDE 1 into ODE 2 Solve ODE through Runge-Kutta Implement with VLSI 3 1 PDE:Partial Differential Equation (or System) 2 ODE: Ordinary Differential Equation (or System) 3 VLSI:Very-Large-Scale Integration (Electronics) N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Cellular Automata and complex systems Least Squares and Cellular Neural Networks CNN’s and PDE quantitative resolution Application to automated PDE resolution Solving ODEs with VLSI designed Runge-Kutta method An attempt at finding quantitative results with automata Runge-Kutta and VLSI Transform PDE 1 into ODE 2 Solve ODE through Runge-Kutta Implement with VLSI 3 Pro’s and con’s Fast Does no better than Runge-Kutta Need dedicated hardware Now superseded by desktop computers 1 PDE:Partial Differential Equation (or System) 2 ODE: Ordinary Differential Equation (or System) 3 VLSI:Very-Large-Scale Integration (Electronics) N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Cellular Automata and complex systems Least Squares and Cellular Neural Networks CNN’s and PDE quantitative resolution Application to automated PDE resolution The need for a quantitative approach The design of a CNN 1 from a differential problem for an quantitative solution Current approaches Sniff the Cellular Neural Network Assess its potentialities 1 CNN: Cellular Neural Network N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Cellular Automata and complex systems Least Squares and Cellular Neural Networks CNN’s and PDE quantitative resolution Application to automated PDE resolution The need for a quantitative approach The design of a CNN 1 from a differential problem for an quantitative solution Current approaches Sniff the Cellular Neural Network Assess its potentialities A better approach Design the CNN 1 from the differential problem Through a systematic method Verify its results 1 CNN: Cellular Neural Network N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Least Squares Finite Element Method Least Squares and Cellular Neural Networks Adaptation of LSFEM to automata Application to automated PDE resolution Boundary conditions Least Squares Finite Element Method Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . . A differential problem A ξ = f on ˜ Υ B ξ = 0 on ˜ Γ N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Least Squares Finite Element Method Least Squares and Cellular Neural Networks Adaptation of LSFEM to automata Application to automated PDE resolution Boundary conditions Least Squares Finite Element Method Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . . A differential problem More general A ξ = f on ˜ Υ A i ξ = f i on Υ i B ξ = 0 on ˜ Γ N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Least Squares Finite Element Method Least Squares and Cellular Neural Networks Adaptation of LSFEM to automata Application to automated PDE resolution Boundary conditions Least Squares Finite Element Method Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . . More general A differential problem A i ξ = f i on Υ i A ξ = f on ˜ Υ or B ξ = 0 on ˜ Γ Φ( ξ ) = 0 on Υ = � i Υ i N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Least Squares Finite Element Method Least Squares and Cellular Neural Networks Adaptation of LSFEM to automata Application to automated PDE resolution Boundary conditions Least Squares Finite Element Method Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . . More general A differential problem A i ξ = f i on Υ i A ξ = f on ˜ Υ or B ξ = 0 on ˜ Γ Φ( ξ ) = 0 on Υ = � i Υ i Minimize the squared error sum   � ( A i ξ − f i ) 2 ξ solution ⇒ ξ minimizes �   i   Υ i N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

Cellular Neural Networks and Partial Differential Equations Least Squares Finite Element Method Least Squares and Cellular Neural Networks Adaptation of LSFEM to automata Application to automated PDE resolution Boundary conditions Least Squares Finite Element Method Bo-Nan Jiang, Springer, 1998 The least-squares finite element method. . . More general A differential problem A i ξ = f i on Υ i A ξ = f on ˜ Υ or B ξ = 0 on ˜ Γ Φ( ξ ) = 0 on Υ = � i Υ i Minimize the squared error sum   � ( A i ξ − f i ) 2 ξ solution ⇒ ξ minimizes �   i   Υ i Simpler � Φ( ξ ) 2 ξ solution ⇒ ξ minimizes Υ N. Fressengeas, H. Frezza-Buet ESCAPaDE 2

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