Low-Cost Global Optimization Application to industrial problems

Transcription

1 Low-Cost Global Optimization Application to Ivorra Benjamin, Isèbe Damien & Mohammadi Bijan Montpellier 2 University Works done in partnership with: Jean-Paul Dufour, Patrick Redont (Montpellier), Laurent Dumas (Paris 6), Olivier Durand (Alcatel), Yves Moreau (CEM2) Juan Santiago, David Hertzog, Heinz Pitsch (Stanford), Larvi Debiane (INRIA), Alexandre Ern (ENPC), Thierry Poinsot (CERFACS), Bouchette Frederic (ISTEEM), Ramos-Del Olmo Angel Manuel (Madrid) May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 1/29

2 Outlines Global optimization : BVP formulation and resolution of optimization problem May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 2/29

3 Outlines Global optimization : BVP formulation and resolution of optimization problem Low-cost sensitivity: Incomplete sensitivity approaches May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 2/29

4 Outlines Global optimization : BVP formulation and resolution of optimization problem Low-cost sensitivity: Incomplete sensitivity approaches Application to : -Shape Optimization of Fast Microfluidic Protein Folding Devices -Temperature and pollution control in a bunsen flame -Optical filters design -Shape optimization of coastal structures. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 2/29

5 Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 3/29

6 Problem Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm min J(x) x Ω ad Where: -x is the optimization parameter. -Ω ad is a compact admissible space. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 4/29

7 Problem Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm min J(x) x Ω ad Where: -x is the optimization parameter. -Ω ad is a compact admissible space. Assumptions: -J C 2 (Ω ad, IR) is coercive - the infimum J m is known. (Facultative we set J m = ) - the problem is admissible: x m Ω ad, s.t. J(x m ) = J m. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 4/29

8 Global Optimization and Dynamical system Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm Most deterministic minimization algorithms can be seen as discretizations of: { M(Θ Z )x ζ (ζ) = d(x(ζ)) x(0) = x 0 May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 5/29

9 Global Optimization and Dynamical system Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm Most deterministic minimization algorithms can be seen as discretizations of: { M(Θ Z )x ζ (ζ) = d(x(ζ)) x(0) = x 0 Where: -x 0 Ω ad the initial condition -ζ is a fictitious time -d a direction in Ω ad -M is a local metric transformation -Θ Z = {ζ, x(ζ), d(x(ζ))/ζ [0, Z]}, Z the elapsed time May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 5/29

10 BVP formulation Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm Previous system give a solution for our optimization problem, if for a given x 0 Ω ad, Z x0 IR such that J(x(Z x0 )) = J m. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 6/29

11 BVP formulation Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm Previous system give a solution for our optimization problem, if for a given x 0 Ω ad, Z x0 IR such that J(x(Z x0 )) = J m. i.e. the following BVP has a solution: M(Θ Z )x ζ (ζ) = d(x(ζ)) x(0) = x 0 J(x(Z x0 )) = J m This BVP is over-determined by x 0. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 6/29

12 BVP formulation Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm Previous system give a solution for our optimization problem, if for a given x 0 Ω ad, Z x0 IR such that J(x(Z x0 )) = J m. i.e. the following BVP has a solution: M(Θ Z )x ζ (ζ) = d(x(ζ)) x(0) = x 0 J(x(Z x0 )) = J m This BVP is over-determined by x 0. Due to hypothesis 3: the problem is admissible: x 0 such that previous BVP has a solution. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 6/29

13 Over-determination deletion Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm We remove the over-determination: We consider x 0 as a new variable such that: With: x 0 argmin v Ωad h(v) h(v) = min t [0,Z v ] (J(x(t, v)) J m) Where: -Z v IR + -x(t, v) the solution of DS found at t starting from v. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 7/29

14 Semi-deterministic algorithm Problem Global Optimization and Dynamical system BVP formulation Over-determination deletion Semi-deterministic algorithm v 1 Ω ad given Find v argmin w Ωad h(w) using Multi-Layers Algorithm: Each layer is a Line search starting from v 1 with random directions. return best v May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 8/29

15 Idea Problem formulation Low-Cost sensitivity approaches May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 9/29

16 Idea Idea Problem formulation Low-Cost sensitivity approaches We recall considered DS: { M(Θ Z )x ζ (ζ) = d(x(ζ)), x(0) = x 0 Associated optimization algorithms only need a descent direction d such that: d. J > ε > 0 May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 10/29

17 Idea Idea Problem formulation Low-Cost sensitivity approaches We recall considered DS: { M(Θ Z )x ζ (ζ) = d(x(ζ)), x(0) = x 0 Associated optimization algorithms only need a descent direction d such that: d. J > ε > 0 We compute d with a low accuracy. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 10/29

18 Problem formulation Example: d = J Idea Problem formulation Low-Cost sensitivity approaches Simulation loop to compute J: J(x) : x q(x) U(q(x)) J(x, q(x), U(q(x))) Where: - x is the shape parameterization - q is the shape geometry - U is the state equation solution May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 11/29

19 Problem formulation Example: d = J Idea Problem formulation Low-Cost sensitivity approaches Simulation loop to compute J: J(x) : x q(x) U(q(x)) J(x, q(x), U(q(x))) Where: - x is the shape parameterization - q is the shape geometry - U is the state equation solution The Jacobian of J is given by: dj dx = J x + J q q x + J U U q q x The last term J U U q q x is the more expensive to compute. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 11/29

20 Problem formulation Example: d = J Idea Problem formulation Low-Cost sensitivity approaches Simulation loop to compute J: J(x) : x q(x) U(q(x)) J(x, q(x), U(q(x))) Where: - x is the shape parameterization - q is the shape geometry - U is the state equation solution The Jacobian of J is given by: dj dx = J x + J q q x + J U U q q x The last term J U U q q x is the more expensive to compute. We want to approximate: J U U q q x May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 11/29

21 Low-Cost sensitivity approaches Idea Problem formulation Low-Cost sensitivity approaches 1- Neglect state variations: Mohammadi, B. and Pironneau, O., Applied Shape Optimization for Fluids, Oxford University Press, 2001 dj dx J x + J q q x May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 12/29

22 Low-Cost sensitivity approaches Idea Problem formulation Low-Cost sensitivity approaches 1- Neglect state variations: Mohammadi, B. and Pironneau, O., Applied Shape Optimization for Fluids, Oxford University Press, 2001 dj dx J x + J q 2- Reduced complexity models: Consider the reduced model: Ũ(x) U(q(x)). state sensitivity is computed using Ũ(x). state is evaluated using U(q(x)). q x May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 12/29

23 Low-Cost sensitivity approaches Idea Problem formulation Low-Cost sensitivity approaches 1- Neglect state variations: Mohammadi, B. and Pironneau, O., Applied Shape Optimization for Fluids, Oxford University Press, 2001 dj dx J x + J q 2- Reduced complexity models: Consider the reduced model: Ũ(x) U(q(x)). state sensitivity is computed using Ũ(x). state is evaluated using U(q(x)). q x 3- Different discretization level: state sensitivity is computed on coarse meshes. state is evaluated on finer discretizations. Coupled with Mesh adaptation. May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 12/29

24 Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 industrial problems May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 13/29

25 Problem 1 Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 14/29

26 Shape parameterization Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 15/29

27 Example of meshes Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Coarse meshes : 30 secs / Fine meshes : 3 mins Computational difference: Difference of 50%!!! May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 16/29

28 Shape optimization results GA: Evaluations: 5400 / Time: 4 days SDA: Evaluations: 3400 (90 % coarse mesh) / Time: 4 hours Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 17/29

29 Shape optimization results GA: Evaluations: 5400 / Time: 4 days SDA: Evaluations: 3400 (90 % coarse mesh) / Time: 4 hours All cases: mixing time 8µs 1.15µs Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Initial mixer Optimized mixer May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 17/29

30 Convergence history Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 µs History Best element Evolution Iteration May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 18/29

31 Experimental implementation Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Exp optimized mixer Num optimized mixer Average gain of 4µs May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 19/29

32 Problem 2 Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Γ AIR H2 N2 O2 AIR May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 20/29

33 Mesh adaptation Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Initial mesh Optimized mesh May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 21/29

34 Numerical results Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours T [K] NO flux [g.cm -2.s -1 ] Problem 3 16-peaks filters results 500 Problem e r [cm] r [cm] Temperature NOx flux May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 22/29

35 Contours Temperature and NOx mass fraction contour Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Optimized Initial Optimized Initial May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 23/29

36 Problem 3 Initial Spectrum 0 L Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Reflected Spectrum Optical Fiber Apodization of the Fiber Transmitted Spectrum May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 24/29

37 16-peaks filters results Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 Refractive index modulation 10 4 Reflectivity Position along the grating (mm) Refractive index modulation 10 4 Reflectivity Position along the grating (mm) Refractive index modulation 10 4 Reflectivity Position along the grating (mm) wavelength (µm) wavelength (µm) wavelength (µm) Classical Sinc AG Optimized SD Optimized May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 25/29

38 Problem 4 Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 26/29

39 Numerical results Initial and Optimized structure Shape: Problem 1 Shape parameterization Example of meshes Shape optimization results Convergence history Experimental implementation Problem 2 Mesh adaptation Contours Problem 3 16-peaks filters results Problem 4 May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 27/29

40 May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 28/29

41 SD is applicable and improve various optimization s May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 29/29

42 SD is applicable and improve various optimization s SD has been efficient on various May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 29/29

43 SD is applicable and improve various optimization s SD has been efficient on various Low cost sensitivity is useful for gradient computation May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 29/29

44 SD is applicable and improve various optimization s SD has been efficient on various Low cost sensitivity is useful for gradient computation Perspectives: new of gradient simplification, other industrial projects... May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 29/29

45 SD is applicable and improve various optimization s SD has been efficient on various Low cost sensitivity is useful for gradient computation Perspectives: new of gradient simplification, other industrial projects... Partners: Jean-Paul Dufour, Patrick Redont (Montpellier), Laurent Dumas (Paris 6) Olivier Durand (Alcatel), Yves Moreau (CEM2) Juan Santiago, David Hertzog, Heinz Pitsch (Stanford) Larvi Debiane (INRIA), Alexandre Ern (ENPC), Thierry Poinsot (Cerfacs), Bouchette Frederic (ISTEEM), Ramos-Del Olmo Angel Manuel (Madrid)!!! Thank You!!! May 17, ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES - p. 29/29