Global Optimization of a Magnetic Lattice using Genetic Algorithms
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1 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, 2008 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
2 Contents 1 Genetic Algorithm Optimization Algorithm Dominance and Nondominated Sorting Crowding Distance 2 Application on Lattice Optimization ɛ, β x ɛ, Low-high β x 3 Reference Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
3 Motivation: Magnetic Lattice Optimization on ALS We want to optimize brightness: 1 Emittance ɛ x. 2 Match β x to ID(insertion device). by tuning Quad strength: k QF, k QD. Constraints: Tr(M) 2. J x > 0, J E > 0. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
4 There are many ways... k QD 0 4 k QF 4 Deterministic, ɛ(k QF, k QD ), ɛ k Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
5 There are many ways... k QD 0 4 k QF k QD 0 4 k QF 4 4 Deterministic, ɛ(k QF, k QD ), ɛ k Brute force, (k QF, k QD ) Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
6 There are many ways... k QD 0 4 k QF k QD 0 4 k QF 4 k QD 0 P C 2 C 1 C 1 P 4 k QF 4 Deterministic, ɛ(k QF, k QD ), ɛ k Brute force, (k QF, k QD ) Stochastic, Evolutionary 4 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
7 Lattice Optimization Genetic Algorithm GA has been used on DC gun photoinjector [Bazarov and Sinclair, 2005]. Population based. Iterative (generation). Red: violate the constraints. Green: meet the constraints. (k QF, k QD ). Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
8 Multi-Objective Optimization Airline tickets for 2012 London Olympics Cost $ AA1 UA1 NW1 Time? Cost? Weighted sum? Best Not the best So far the best BA1 UA2 Whole picture. Time h Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
9 Multi-Objective Optimization Airline tickets for 2012 London Olympics Cost $ AA1 UA1 NW1 Time? Cost? Weighted sum? Best Not the best So far the best BA1 UA2 Whole picture. Time h Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
10 Objective Space Generation f:0, 10, 76; x:100. Red: violate the constraints Green: meet the constraints Blue: Pareto optimal set Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
11 Optimization: General Form The general form of an optimization problem is: Optimization Minimize/Maximize f m (x), m = 1, 2,..., M; subject to g j 0, j = 1, 2,..., J; h k (x) = 0, k = 1, 2,..., K; x (L) i x i x (U) i, i = 1, 2,..., N; (1) MOGA(Multi-Objective Genetic Algorithm): Multiobjectvie, instead of single objective optimization of a weighted sum. Constraint. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
12 Structure of MOGA/GA(Genetic Algorithm) GA mimics the evolution of nature: 1 Crossover: generate children from parents. 2 Mutation: change the children. 3 Nature select: keep only certain number of population. MOGA (Multi-Objective Genetic Algorithm) 1: Initialize population (first generation, random) 2: repeat 3: crossover: 2 parents 2 children. 4: mutation: change children. 5: calculate f m 6: nature select: sorting 7: until stop(reach maximum generation, find solution,... ) Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
13 History of Evolution Our optimization problem [Yang et al., 2008, Robin et al., 2008]: Optimize: 1 Emittance ɛ. 2 min( β x 1.0 ). Constraint: Tr(M x ) 2, Tr(M y ) 2 max(β x ) 30, max(β y ) 30 max(η x ) 0.4 Parameters: 1 QF,QD,QFA in one cell. 1 Evolution of objective functions. ɛ, β x 1 2 (k QF, k QD ), (k QF, k QFA ), (k QD, k QFA ) Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
14 GA: Initialization You have a lot of freedom to create the first generation. No filter: keep everyone, no need to calculate f m or g i. Apply filter: Check f m. Check g i. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
15 GA: Crossover Generate children from parents: x (,t) x (,t+1). Parents are randomly chosen and used only once. (t generation) There are simple ones: Middle point. e.g. 0.5(x (1,t) + x (2,t) ) Blend(BLX), (1 γ)x (1,t) + γx (2,t). γ has random property, and extend certain range beyond [0, 1]. More complicate ones: 1 Upper/Lower limit of variables. [x (L), x (U) ] 2 Continuous probability distribution. P(x). Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
16 GA: Crossover Children are generated around two parents in certain probability. Probability Density (not normalized) η=2.6 η=1.0 η=0.3 Parent 1 Parent x of children x [ 3, 5] We choose polynomial PDF. Boundary is automatic considered. 2 parents to 2 children for every dimension of parameter space. η c to control the shape of PDF. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
17 GA: Mutation Purpose: keep diversity. For each individual: 1 Random, e.g. x (1,t+1) = x (1,t+1) + (r 0.5). 2 Non-Uniform, e.g. x (1,t+1) = x (1,t+1) + τ(x (U) x (L) )(1 r (1 t/t max) b ) 3 Normally Distributed, x (1,t+1) = x (1,t+1) + N(0, σ). More complicate ones will consider: 1 Boundary 2 Probability Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
18 GA: Mutation The new value due to the mutation also follows certain disstribution. Polynomial PDF. Equal probability go left or right. Boundaries are considered. η m can control the shape of PDF. Probability Density (not normalized) η=3.0 η=1.0 η=0.3 Parent x of children Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
19 Calculate f m, merge parents and children Calculate objective function, here lattice properties. Stable/unstable: betatron resonance ( Tr(M), J E, J x ). Constraint: β x, β y, η x. Elite-preserving: merge parents and children, no difference. The population number are fixed. Good parents are kept. Never went worse from generation to generation. In order to pick the better ones, in multiobjective case, we use nondominated sorting to sort the whole population, and keep only the top half. (Airline ticket example). Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
20 Dominance Domination [Deb, 2001]. f (1) f (2) (dominate, precede) 1 The solution f (1) i is no worse than f (2) i in all m-objectives. 2 The solution f (1) is strickly better than f (2) in at least one objective. f 2 b 1 a 1 f, a 1 f a 0 a 2 f, f a 1 c 1 a 1 a 2, a 1 is better than a 2. a 0 a 1 a 1, b 1 and c 1 are not dominated by each other. f 1 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
21 Dominance Domination [Deb, 2001]. f (1) f (2) (dominate, precede) 1 The solution f (1) i is no worse than f (2) i in all m-objectives. 2 The solution f (1) is strickly better than f (2) in at least one objective. f 2 b 1 a 1 f, a 1 f a 0 a 2 f, f a 1 c 1 a 1 a 2, a 1 is better than a 2. a 0 a 1 a 1, b 1 and c 1 are not dominated by each other. f 1 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
22 Application on Lattice Optimization 3 Parmameters, optimize ɛ and β x 1m Red: violate the constraints, or no physical solution. Green: meet the constraints. Blue: Pareto optimal set, the best solutions so far. 1 Generation 19 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
23 Application on Lattice Optimization 3 Parmameters, optimize ɛ and β x 1m Red: violate the constraints, or no physical solution. Green: meet the constraints. Blue: Pareto optimal set, the best solutions so far. 1 Generation 19 2 Generation 46 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
24 Application on Lattice Optimization 3 Parmameters, optimize ɛ and β x 1m Red: violate the constraints, or no physical solution. Green: meet the constraints. Blue: Pareto optimal set, the best solutions so far. 1 Generation 19 2 Generation 46 3 Generation 66 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
25 Application on Lattice Optimization 3 Parmameters, optimize ɛ and β x 1m Red: violate the constraints, or no physical solution. Green: meet the constraints. Blue: Pareto optimal set, the best solutions so far. 1 Generation 19 2 Generation 46 3 Generation 66 4 Generation 130 Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
26 Application on Lattice Optimization 1 up left, small ɛ. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
27 Application on Lattice Optimization 1 up left, small ɛ. 2 down left, β x and ɛ. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
28 Application on Lattice Optimization 1 up left, small ɛ. 2 down left, β x and ɛ. 3 down right, small β x 1. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
29 Application on Lattice Optimization 6 Parmameters, optimize ɛ and β x 1m/10m 1 Pareto optimal set. Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
30 Application on Lattice Optimization 6 Parmameters, optimize ɛ and β x 1m/10m 1 Pareto optimal set. 2 Twiss Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
31 Application on Lattice Optimization 6 Parmameters, optimize ɛ and β x 1m/10m 1 Pareto optimal set. 2 Twiss 3 Twiss Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
32 Application on Lattice Optimization 6 Parmameters, optimize ɛ and β x 1m/10m 1 Pareto optimal set. 2 Twiss 3 Twiss 4 Movies Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
33 Reference Reference I. V. Bazarov and C. K. Sinclair. Multivariate optimization of a high brightness dc gun photoinjector. Phys. Rev. ST Accel. Beams, 8(3): , Mar doi: /PhysRevSTAB K. Deb. Multi-Objective Optimization using Ebolutionary Algorithms. John Wiley & Sons, Ltd, D. S. Robin, W. Wan, F. Sannibale, and V. P. Suller. Global analysis of all linear stable settings of a storage ring lattice. Phys. Rev. ST Accel. Beams, 11(2):024002, Feb doi: /PhysRevSTAB L. Yang, D. Robin, C. Steier, and W. Wan. Global optimization of the magnetic lattice using genetic algorithms. In EPAC08, Global Optimization of a Magnetic Lattice using Genetic Algorithms Lingyun Yang September 3, / 21
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