# LIACS Natural Computing Group Leiden University Evolutionary Algorithms.

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LIACS Natural Computing Group Leiden University Evolutionary Algorithms

LIACS Natural Computing Group Leiden University2 Overview Introduction: Optimization and EAs Genetic Algorithms Evolution Strategies Theory Examples

LIACS Natural Computing Group Leiden University3 Background I Biology = Engineering (Daniel Dennett)

LIACS Natural Computing Group Leiden University4 Introduction Modeling Simulation Optimization ! ! ! ??? ! ! !???! ! ! ???! ! ! Input: Known (measured)Output: Known (measured)Interrelation: UnknownInput: Will be given Model: Already exists How is the result for the input? Objective: Will be given How (with which parameter settings) to achieve this objective?

LIACS Natural Computing Group Leiden University5 Simulation vs. Optimization Simulator … what happens if? Result Trial & Error Simulator... how do I achieve the best result? Optimal Result Maximization / Minimization If so, multiple objectives Optimizer

LIACS Natural Computing Group Leiden University6 Introduction: Optimization Evolutionary Algorithms

LIACS Natural Computing Group Leiden University7 Optimization f : objective function –High-dimensional –Non-linear, multimodal –Discontinuous, noisy, dynamic M  M 1  M 2 ...  M n heterogeneous Restrictions possible over Good local, robust optimum desired Realistic landscapes are like that! Global Minimum Local, robust optimum

LIACS Natural Computing Group Leiden University8 Illustrative Example: Optimize Efficiency –Initial: –Evolution: 32% Improvement in Efficiency ! Optimization Creating Innovation

LIACS Natural Computing Group Leiden University9 Dynamic Optimization Dynamic Function 30-dimensional 3D-Projection

LIACS Natural Computing Group Leiden University10 Classification of Optimization Algorithms Direct optimization algorithm: Evolutionary Algorithms First order optimization algorithm: e.g., gradient method Second order optimization algorithm: e.g., Newton method

LIACS Natural Computing Group Leiden University11 Iterative Optimization Methods General description: Actual PointNew PointDirectional vectorStep size (scalar) At every Iteration: –Choose direction –Determine step size Direction: –Gradient –Random Step size: –1-dim. optimization –Random –Self-adaptive

LIACS Natural Computing Group Leiden University12 Global convergence with probability one: General, but for practical purposes useless Convergence velocity: Local analysis only, specific (convex) functions The Fundamental Challenge

LIACS Natural Computing Group Leiden University13 Global convergence (with probability 1): General statement (holds for all functions) Useless for practical situations: –Time plays a major role in practice –Not all objective functions are relevant in practice Theoretical Statements

LIACS Natural Computing Group Leiden University14 x1x1 x2x2 f(x 1,x 2 ) (x* 1,x* 2 ) f(x* 1,x* 2 ) An Infinite Number of Pathological Cases ! NFL-Theorem: –All optimization algorithms perform equally well iff performance is averaged over all possible optimization problems. Fortunately: We are not Interested in „all possible problems“

LIACS Natural Computing Group Leiden University15 Convergence velocity: Very specific statements –Convex objective functions –Describes convergence in local optima –Very extensive analysis for Evolution Strategies Theoretical Statements

LIACS Natural Computing Group Leiden University Evolutionary Algorithm Principles

LIACS Natural Computing Group Leiden University17 Model-Optimization-Action Evaluation OptimizerBusiness Process Model Simulation FunctionModel from Data ExperimentSubjectiveFunction(s)

LIACS Natural Computing Group Leiden University18 Evolutionary Algorithms Taxonomy Evolutionary Algorithms Evolution Strategies Genetic Algorithms Other Evolutionary Progr. Differential Evol. GP PSO EDA Real-coded Gas … Mixed-integer capabilities Emphasis on mutation Self-adaptation Small population sizes Deterministic selection Developed in Germany Theory focused on convergence velocity Discrete representations Emphasis on crossover Constant parameters Larger population sizes Probabilistic selection Developed in USA Theory focused on schema processing

LIACS Natural Computing Group Leiden University19 Generalized Evolutionary Algorithm t := 0; initialize(P(t)); evaluate(P(t)); while not terminate do P‘(t) := mating_selection(P(t)); P‘‘(t) := variation(P‘(t)); evaluate(P‘‘(t)); P(t+1) := environmental_selection(P‘‘(t)  Q); t := t+1; od

LIACS Natural Computing Group Leiden University Genetic Algorithms

LIACS Natural Computing Group Leiden University21 Genetic Algorithms: Mutation Mutation by bit inversion with probability p m. p m identical for all bits. p m small (e.g., p m = 1/n). 011101010000001 011100010100001

LIACS Natural Computing Group Leiden University22 Genetic Algorithms: Crossover Crossover applied with probability p c. p c identical for all individuals. k-point crossover: k points chosen randomly. Example: 2-point crossover.

LIACS Natural Computing Group Leiden University23 Genetic Algorithms: Selection Fitness proportional: –f fitness, population size Tournament selection: –Randomly select q << individuals. –Copy best of these q into next generation. –Repeat times. –q is the tournament size (often: q = 2 ).

LIACS Natural Computing Group Leiden University Some Theory

LIACS Natural Computing Group Leiden University25 Convergence Velocity Analysis (1+1)-GA, (1, )-GA, (1+ )-GA For counting ones function Convergence velocity:

LIACS Natural Computing Group Leiden University26 Optimum mutation rate ? Absorption times from transition matrix in block form, using Convergence Velocity Analysis II

LIACS Natural Computing Group Leiden University27 p Convergence Velocity Analysis III P too large: –Exponential P too small: –Almost constant Optimal: –O(l ln l)

LIACS Natural Computing Group Leiden University28 (1,l)-GA (k min = -f a ), (1+l)-GA (k min = 0) : Convergence Velocity Analysis IV

LIACS Natural Computing Group Leiden University29 Convergence Velocity Analysis V (1, )-GA, (1  GA (1, )-ES, (1  ES Conclusion: Unifying, search-space independent theory ?

LIACS Natural Computing Group Leiden University30 Convergence Velocity Analysis VI ( , )-GA, (  GA Theory Experiment

LIACS Natural Computing Group Leiden University Evolution Strategies

LIACS Natural Computing Group Leiden University32 Evolution Strategy – Basics Mostly real-valued search space IR n –also mixed-integer, discrete spaces Emphasis on mutation –n -dimensional normal distribution –expectation zero Different recombination operators Deterministic selection –(  )-selection:Deterioration possible –(  )-selection:Only accepts improvements , i.e.: Creation of offspring surplus Self-adaptation of strategy parameters.

LIACS Natural Computing Group Leiden University33 Representation of search points Simple ES with 1/5 success rule: –Exogenous adaptation of step size  –Mutation: N(0,  ) Self-adaptive ES with single step size: –One  controls mutation for all x i –Mutation: N(0,  )

LIACS Natural Computing Group Leiden University34 Representation of search points Self-adaptive ES with individual step sizes: –One individual  i per x i –Mutation: N i (0,  i ) Self-adaptive ES with correlated mutation: –Individual step sizes –One correlation angle per coordinate pair –Mutation according to covariance matrix: N(0, C)

LIACS Natural Computing Group Leiden University35 Evolution Strategy: Algorithms Mutation

LIACS Natural Computing Group Leiden University36 Operators: Mutation – one  Self-adaptive ES with one step size: –One  controls mutation for all x i –Mutation: N(0,  ) Individual before mutationIndividual after mutation1.: Mutation of step sizes 2.: Mutation of objective variables Here the new  ‘ is used!

LIACS Natural Computing Group Leiden University37 Operators: Mutation – one  Thereby  0 is the so-called learning rate –Affects the speed of the  -Adaptation –  0 bigger: faster but more imprecise –  0 smaller: slower but more precise –How to choose  0 ? –According to recommendation of Schwefel*: *H.-P. Schwefel: Evolution and Optimum Seeking, Wiley, NY, 1995.

LIACS Natural Computing Group Leiden University38 Operators: Mutation – one  Position of parents (here: 5) Offspring of parent lies on the hyper sphere (for n > 10); Position is uniformly distributed Contour lines of objective function

LIACS Natural Computing Group Leiden University39 Pros and Cons: One  Advantages: –Simple adaptation mechanism –Self-adaptation usually fast and precise Disadvantages: –Bad adaptation in case of complicated contour lines –Bad adaptation in case of very differently scaled object variables -100 < x i < 100 and e.g. -1 < x j < 1

LIACS Natural Computing Group Leiden University40 Operators: Mutation – individual  i Self-adaptive ES with individual step sizes: –One  i per x i –Mutation: N i (0,  i ) Individual before MutationIndividual after Mutation1.: Mutation of individual step sizes 2.: Mutation of object variables The new individual  i ‘ are used here!

LIACS Natural Computing Group Leiden University41 Operators: Mutation – individual  i ,  ‘ are learning rates, again –  ‘: Global learning rate –N(0,1): Only one realisation –  : local learning rate –N i (0,1): n realisations –Suggested by Schwefel*: *H.-P. Schwefel: Evolution and Optimum Seeking, Wiley, NY, 1995.

LIACS Natural Computing Group Leiden University42 Position of parents (here: 5) Offspring are located on the hyperellipsoid (für n > 10); position equally distributed. Contour lines Operators: Mutation – individual  i

LIACS Natural Computing Group Leiden University43 Pros and Cons: Individual  i Advantages: –Individual scaling of object variables –Increased global convergence reliability Disadvantages: –Slower convergence due to increased learning effort –No rotation of coordinate system possible Required for badly conditioned objective function

LIACS Natural Computing Group Leiden University44 Operators: Correlated Mutations Individual before mutationIndividual after mutation1.: Mutation of Individual step sizes 2.: Mutation of rotation angles New convariance matrix C‘ used here! Self-adaptive ES with correlated mutations: –Individual step sizes –One rotation angle for each pair of coordinates –Mutation according to covariance matrix: N(0, C) 3.: Mutation of object variables

LIACS Natural Computing Group Leiden University45 Operators: Correlated Mutations Interpretation of rotation angles  ij Mapping onto convariances according to x1x1 x2x2 11 22  12

LIACS Natural Computing Group Leiden University46 Operators: Correlated Mutation  ,  ‘,  are again learning rates   ‘ as before   = 0,0873 (corresponds to 5 degree)  Out of boundary correction:

LIACS Natural Computing Group Leiden University47 Correlated Mutations for ES Position of parents (hier: 5) Offspring is located on the Rotatable hyperellipsoid (for n > 10); position equally distributed. Contour lines

LIACS Natural Computing Group Leiden University48 Operators: Correlated Mutations How to create ? –Multiplication of uncorrelated mutation vector with n(n-1)/2 rotational matrices –Generates only feasible (positiv definite) correlation matrices

LIACS Natural Computing Group Leiden University49 Operators: Correlated Mutations Structure of rotation matrix

LIACS Natural Computing Group Leiden University50 Operators: Correlated Mutations Implementation of correlated mutations nq:= n(n-1)/2; for i:=1 to n do  u[i] :=  [i] * N i (0,1); for k:=1 to n-1 do n1 := n-k; n2 := n; for i:=1 to k do d1 :=  u[n1]; d2:=  u[n2];  u[n2] := d1*sin(  [nq])+ d2*cos(  [nq]);  u[n1] := d1*cos(  [nq])- d2*sin(  [nq]); n2 := n2-1; nq := nq-1; od Generation of the uncorrelated mutation vector Rotations

LIACS Natural Computing Group Leiden University51 Pros and Cons: Correlated Mutations Advantages: –Individual scaling of object variables –Rotation of coordinate system possible –Increased global convergence reliability Disadvantages: –Much slower convergence –Effort for mutations scales quadratically –Self-adaptation very inefficient

LIACS Natural Computing Group Leiden University52 Operators: Mutation – Addendum Generating N(0,1)-distributed random numbers? If w > 1

LIACS Natural Computing Group Leiden University53 Evolution Strategy: Algorithms Recombination

LIACS Natural Computing Group Leiden University54 Operators: Recombination Only for  > 1 Directly after Secektion Iteratively generates offspring: for i:=1 to do choose recombinant r1 uniformly at random from parent_population; choose recombinant r2 <> r1 uniformly at random from parent population; offspring := recombine(r1,r2); add offspring to offspring_population; od

LIACS Natural Computing Group Leiden University55 Operators: Recombination How does recombination work? Discrete recombination: –Variable at position i will be copied at random (uniformly distr.) from parent 1 or parent 2, position i. Parent 1Parent 2Offspring

LIACS Natural Computing Group Leiden University56 Operators: Recombination Intermediate recombination: –Variable at position i is arithmetic mean of Parent 1 and Parent 2, position i. Parent 1Parent 2Offspring

LIACS Natural Computing Group Leiden University57 Operators: Recombination Parent 1Parent 2Offspring … Parent  Global discrete recombination: –Considers all parents

LIACS Natural Computing Group Leiden University58 Operators: Recombination Parent 1Parent 2Offspring … Parent  Global intermediary recombination: –Considers all parents

LIACS Natural Computing Group Leiden University59 Evolution Strategy Algorithms Selection

LIACS Natural Computing Group Leiden University60 Operators: (  )-Selection (  )-Selection means: –  parents produce offspring by (Recombination +) Mutation –These  individuals will be considered together –The  best out of  will be selected („survive“) Deterministic selection –This method guarantees monotony Deteriorations will never be accepted = Actual solution candidate= New solution candidateRecombination may be left out Mutation always exists!

LIACS Natural Computing Group Leiden University61 Operators: (  )-Selection (  )-Selection means: –  parents produce  offspring by (Recombination +) Mutation – offspring will be considered alone –The  best out of offspring will be selected Deterministic selection –The method doesn‘t guarantee monotony Deteriorations are possible The best objective function value in generation t+1 may be worse than the best in generation t.

LIACS Natural Computing Group Leiden University62 Operators: Selection Example: (2,3)-Selection Example: (2+3)-Selection Parents don‘t survive!Parents don‘t survive …… but a worse offspring.… now this offspring survives.

LIACS Natural Computing Group Leiden University63 Exception! Possible occurrences of selection –(1+1)-ES: One parent, one offspring, 1/5-Rule –(1, )-ES: One Parent, offspring Example: (1,10)-Strategy One step size / n self-adaptive step sizes Mutative step size control Derandomized strategy –(  )-ES:  > 1 parents,  offspring Example: (2,15)-Strategy Includes recombination Can overcome local optima –(  + )-strategies: elitist strategies Operators: Selection

LIACS Natural Computing Group Leiden University64 Evolution Strategy: Self adaptation of step sizes

LIACS Natural Computing Group Leiden University65 Self-adaptation No deterministic step size control! Rather: Evolution of step sizes –Biology: Repair enzymes, mutator-genes Why should this work at all? –Indirect coupling: step sizes – progress –Good step sizes improve individuals –Bad ones make them worse –This yields an indirect step size selection

LIACS Natural Computing Group Leiden University66 Self-adaptation: Example How can we test this at all? Need to know optimal step size … –Only for very simple, convex objective functions –Here: Sphere model –Dynamic sphere model Optimum locations changes occasionally : Optimum

LIACS Natural Computing Group Leiden University67 Self-adaptation: Example Objective function value … and smallest step size measured in the population average … Largest … According to theory ff optimal step sizes

LIACS Natural Computing Group Leiden University68 Self-adaptation Self-adaptation of one step size –Perfect adaptation –Learning time for back adaptation proportional n –Proofs only for convex functions Individual step sizes –Experiments by Schwefel Correlated mutations –Adaptation much slower

LIACS Natural Computing Group Leiden University69 Evolution Strategy: Derandomization

LIACS Natural Computing Group Leiden University70 Derandomization Goals: –Fast convergence speed –Fast step size adaptation –Precise step size adaptation –Compromise convergence velocity – convergence reliability Idea: Realizations of N(0,  ) are important! –Step sizes and realizations can be much different from each other –Accumulates information over time

LIACS Natural Computing Group Leiden University71 Derandomzed (1, )-ES Current parent: in generation g Mutation ( k=1,…, ): Offspring k Global step size in generation g Individual step sizes in generation g Selection: Choice of best offspring Best of offspring in generation g

LIACS Natural Computing Group Leiden University72 Derandomized (1, )-ES Accumulation of selected mutations: The particular mutation vector, which created the parent! Also: weighted history of good mutation vectors! Initialization: Weight factor:

LIACS Natural Computing Group Leiden University73 Derandomized (1, )-ES Step size adaptation: Norm of vectorVector of absolute values Regulates adaptation speed and precision

LIACS Natural Computing Group Leiden University74 Derandomized (1, )-ES Explanations: –Normalization of average variations in case of missing selection (no bias): –Correction for small n : 1/(5n) –Learning rates:

LIACS Natural Computing Group Leiden University75 Evolution Strategy: Rules of thumb

LIACS Natural Computing Group Leiden University76 Some Theory Highlights Convergence velocity: For (1, )-strategies: For (  )-strategies (discrete and intermediary recombination): Problem dimensionality Speedup by  is just logarithmic – more processors are only to a limited extend useful to increase  Genetic Repair Effect of recombination! Genetic Repair Effect of recombination!

LIACS Natural Computing Group Leiden University77 … For strategies with global intermediary recombination: Good heuristic for (1, ): 9.5219.03150 9.4118.82140 9.3018.60130 9.1818.36120 9.0518.10110 8.9117.82100 8.7517.5090 8.5717.1580 8.3716.7570 8.1416.2860 7.8715.7450 7.5315.0740 7.1014.2030 6.4912.9920 5.4510.9110  n

LIACS Natural Computing Group Leiden University Mixed-Integer Evolution Strategies

LIACS Natural Computing Group Leiden University79 Mixed-Integer Evolution Strategy Generalized optimization problem:

LIACS Natural Computing Group Leiden University80 Mixed-Integer ES: Mutation Learning rates (global) Geometrical distribution Mutation Probabilities

LIACS Natural Computing Group Leiden University Some Application Examples

LIACS Natural Computing Group Leiden University82 Safety Optimization – Pilot Study Aim: Identification of most appropriate Optimization Algorithm for realistic example! Optimizations for 3 test cases and 14 algorithms were performed (28 x 10 = 280 shots) –Body MDO Crash / Statics / Dynamics –MCO B-Pillar –MCO Shape of Engine Mount NuTech’s ES performed significantly better than Monte- Carlo-scheme, GA, and Simulated Annealing Results confirmed by statistical hypothesis testing

LIACS Natural Computing Group Leiden University83 MDO Crash / Statics / Dynamics Minimization of body mass Finite element mesh –Crash ~ 130.000 elements –NVH~ 90.000 elements Independent parameters: Thickness of each unit: 109 Constraints: 18 AlgorithmAvg. reduction (kg)Max. reduction (kg)Min. reduction (kg) Best so far-6.6-8.3-3.3 NuTech ES-9.0-13.4-6.3

LIACS Natural Computing Group Leiden University84 MCO B-Pillar – Side Crash Minimization of mass & displacement Finite element mesh –~ 300.000 elements Independent parameters: Thickness of 10 units Constraints: 0 ES successfully developed Pareto front Mass Intrusion

LIACS Natural Computing Group Leiden University85 MCO Shape of Engine Mount Mass minimal shape with axial load > 90 kN Finite element mesh –~ 5000 elements Independent parameters: 9 geometry variables Dependent parameters: 7 Constraints: 3 ES optimized mount –less weight than mount optimized with best so far method –geometrically better deformation

LIACS Natural Computing Group Leiden University86 Safety Optimization – Example of use Production Run ! Minimization of body mass Finite element mesh –Crash ~ 1.000.000 elements –NVH ~ 300.000 elements Independent parameters: –Thickness of each unit: 136 Constraints: 47, resulting from various loading cases 180 (10 x 18) shots ~ 12 days No statistical evaluation due to problem complexity

LIACS Natural Computing Group Leiden University87 Safety Optimization – Example of use 13,5 kg weight reduction by NuTech’s ES ~ 2 kg more mass reduction than Best so far method Typically higher convergence velocity of ES ~ 45% less time (~ 3 days saving) for comparable quality needed Still potential of improvements after 180 shots. Reduction of development time from 5 to 2 weeks allows for process integration Best so far Method Generations Mass Initial Value Generations Mass Initial Value NuTech’s Evolution Strategy

LIACS Natural Computing Group Leiden University88 MDO – Optimization Run 107,13% 105,05% 100,00% All Individuals Number of Individuals Fitness Infeasible Feasible

LIACS Natural Computing Group Leiden University89 Optimization of metal stamping process Objective: Minimization of defects in the produced parts. Variables: Geometric parameters and forces. ES finds very good results in short time Computationally expensive simulation

LIACS Natural Computing Group Leiden University90 Bridgeman Casting Process Objective: Max. homogeneity of workpiece after Casting Process Variables: 18 continuous speed variables for Casting Schedule Computationally expensive simulation (up to 32h simulation time) Initial (DoE)GCM (Commercial Gradient Based Method) Evolution Strategy Turbine Blade after Casting

LIACS Natural Computing Group Leiden University91 Algorithm100 OFE 200 OFE 1000 OFE SQP14867.759 Conjugate directions 135 Pattern search 15714748 DSCP/Rose nbrock 883829 Complex strategy 152 111 (1,3)-DES9167 (1,5)-DES5348 (1,7)-DES11058 (1,10)-DES1528028 MAES1226512 Minimize the deviation of temperture at the cube´s Surface to 1000°C ! Input: Temperature Profile (12 Variables 7 Temperatures and 5 Time-Steps) Steel Cube Temperature Control

LIACS Natural Computing Group Leiden University92 - Maximisation of growth speed - Minimisation of diameter differences FLUENT: Simulation of fluid flow CASTS: Calculation of Temperature and concentration field 0 m/s3 m/s 1.5 m/s Reaction Gas TCS Growing High Purity Silicon Rod Optimsation of 15 process parameters: Production time 35 hours  30 hours Siemens C Reactor

LIACS Natural Computing Group Leiden University93 Multipoint Airfoil Optimiziation Objective: Find pressure profiles that are a compromise between two given target pressure distributions under two given flow conditions! Variables: 12 to 18 Bezier points for the airfoil High Lift! Low Drag! Start Cruise

LIACS Natural Computing Group Leiden University94 Traffic Light Control Optimization Objective: Minimization of total delay / number of stops Variables: Green times for next switching schedule Dynamic optimization, depending on actual traffic Better results (3-5%) Higher flexibility than with traditional controllers

LIACS Natural Computing Group Leiden University95 Optimization of elevator control Minimization of passenger waiting times. Better results (~10%) than with traditional controller. Dynamic optimization, depending on actual traffic.

LIACS Natural Computing Group Leiden University96 Optimization of network routing Minimization of end-to-end-blockings under service constraints. Optimization of routing tables for existing, hard-wired networks. 10%-1000% improvement.

LIACS Natural Computing Group Leiden University97 Automatic battery configuration Configuration and optimization of industry batteries. Based on user specifications, given to the system. Internet-Configurator (Baan, Hawker, NuTech).

LIACS Natural Computing Group Leiden University98 Optimization of reactor fueling. Minimization of total costs. Creates new fuel assembly reload patterns. Clear improvements (1%-5%) of existing expert solutions. Huge cost saving.

LIACS Natural Computing Group Leiden University99 Optical Coatings: Design Optimization Nonlinear mixed-integer problem, variable dimensionality. Minimize deviation from desired reflection behaviour. Excellent synthesis method; robust and reliable results. MOC-Problems: anti-reflection coating, dialetric filters Robust design: Sharp peaks vs. Robust peaks

LIACS Natural Computing Group Leiden University100 Example: Intravascular Ultrasound Image Analysis Real-time high-resolution tomographic images from the inside of coronary vessels:

LIACS Natural Computing Group Leiden University101 Intravascular Ultrasound Image Analysis Detected Features in an IVUS Image: Vessel Border Sidebranch Lumen Shadow Plague

LIACS Natural Computing Group Leiden University102 Experimental Results on IVUS images Parameters for the lumen feature detector

LIACS Natural Computing Group Leiden University103 Intravascular Ultrasound Image Analysis: Results On each of 5 data sets algorithm ran for 2804 evaluations – 19,5h of total computing time Significant improvement over expert tuning Performance of the best found MI-ES parameter solutions A paired two-tailed t-test was performed on the difference measurements for each image dataset using a 95% confidence interval (p=0.05) The null-hypothesis is that the mean difference results of the best MI-ES individual and the default parameters are equal.

LIACS Natural Computing Group Leiden University104 Literature H.-P. Schwefel: Evolution and Optimum Seeking, Wiley, NY, 1995. I. Rechenberg: Evolutionsstrategie 94, frommann- holzboog, Stuttgart, 1994. Th. Bäck: Evolutionary Algorithms in Theory and Practice, Oxford University Press, NY, 1996. Th. Bäck, D.B. Fogel, Z. Michalewicz (Hrsg.): Handbook of Evolutionary Computation, Vols. 1,2, Institute of Physics Publishing, 2000.

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