A tool to approximate viability kernels, capture basins and resilience values

Kernel Approximation for VIAbility and Resilience

• Written in Java programming language– Regular grid / active learning algorithm– Capture basins and resilience values are computed

in dim d– Heavy or optimal controllers– Two modes: GUI and batch mode

• http://trac.clermont.cemagref.fr/projets/Kaviar/wiki/download#Download

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Installation and running

http://trac.clermont.cemagref.fr/projets/Kaviar/wiki/download#Download

• Requires the java virtual machine (Sun’s JRE environment 5 or later compulsory)

• 2 set up– .jar file to test the models already implemented– .zip file to implement new models

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GUI mode

• Display window

• Console window

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GUI mode (cont)

Dynamical system settings

Viability constraint set

Control bounds

Time stepFunction of the size of the gridStudy of the dynamics!!!

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GUI mode (cont)

Viability controller config

Algorithm type

Optimization settings

General settings

- Simple: gradient descent from the minimal values of the controls- Double: min and max values- Conjugate gradient- Double conjugate gradient- Newton method

Visualize the individual trajectories

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GUI mode (cont)

SVM configurationStopping criterion of the SVMs

computation algorithmSVM algorithm

- C-SMO: see libSVM- Simple SVM- Balk: automative bandwidth tuning- Soft-Balk: balk with soft margin

Bandwidth- big C : hard margin(no misclassification)- small C: soft margin

Only the gaussian kernel is implementedControl the “smoothness” of the SVM function- Small gamma: smooth- Big gamma: less smooth

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GUI mode (cont)

Execution and control

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GUI mode (cont)

Indicators + logs

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Example on the population problem

• Viability kernel approximation– Play with dt, # time steps, # points (and show

trajectories)• To obtain a “good” approximation, the dt value must be

chosen accordingly the number of points and time steps• Inner approximation sometimes…

– Save the results and reload them

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Example on the population problem

• Controller – Kernel approx with dt = 0.05, 6 time steps, 31 points– A point out of the viability kernel approximation

• x0 = 2, y0 = 0.8, 20 time steps, 3 time steps anticipation, 3 distance SVM, dist(K) = 0.025

– Inside the viability kernel• x0 = 2, y0 = 0.5, 150 time steps

– More time steps anticipation: 15– Bigger SVM value: 30– Same parameters, with 1 time step for the viability

kernel approximation 11

Adding a dynamical system• Creation of a new class file (for instance MyClass.java)

– Extend Dynamic_System if viability kernel approximation– Extend Dynamic_System_Target if capture basins approximation– Extend Dynamic_System_Resilience if resilience values

• In this class, create a main method to add your model and launch the software

Public static void main (String[]args){//initKaviar kaviar = new Kaviar();//Optional: to add default modelsKaviar.addModels(Kaviar.DEFAULT_MODEL); //Optional: to add one of the default models//Kaviar.addModels(Population.class);//replace my model by the name of your modelKaviar.addModels(MyModel.class);//Launch the GUIKaviar.startGUI();

}

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MyClass.java (extends Dynamical_System)

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MyClass.java (extends Dynamical_System_Target)

Previous +

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MyClass.java (extends Dynamical_System_Resilience)

Previous +

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Example on the Abrams&Strogatz model• Dynamics and constraints

– 2 languages A and B in competition, no bilingual people• σA: density of speakers of language A (in % - [0;1]).• Parameter a: volatility of language A (a > 1 leads a scenario of

dominance of 1 language)• Parameter s: prestige of language A (s = 0.5: the two languages are

socially equivalent – [0;1])– Government, institution etc. can play on the prestige of

one language, but modifications take time– We consider that one language is endangered when its

proportion of speakers is less that 20%

– with16

Example on the Abrams&Strogatz model

• Resilience values– Endangered language doesn’t mean that the

language is dead. Is there any action policies that allows the system to recover?

– At which cost?

• λ = 1: measure the time the system is deprived from its property of interest

• λ = c1*time + c2(distance(σA from viability))

• … 17

Example on the Abrams&Strogatz model

• Optimal control– Compute the resilience values with the following

parameters:• dt =0.2, dc = 0.5, double optimization, C0 = 1, C1 = 300,

31 points, 6 time steps, inner approx • a = 2: dominance of one language• a = 0.2: stable coexistence

– Control of the system:• x0 =0.95, y0= 0.95 and x0 =0.7, y0= 0.95

• Optimal control outside the viability kernel• Heavy control once the system is back to the kernel

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Batch mode

• .simu files are needed• Create them following a given template• Use the GUI interface

• java -cp Kaviar-1.1.jar Appli/Batch Conso.simu

• 2 files: .svm + .log files, in the Conso… directory 19

Batch mode• java -cp Kaviar-1.1.jar Appli/Batch Conso.simu -v

• 9*2 files: .svm + .log files, in the Conso… directory 20