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

2. 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

3. Installation and running
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

4. GUI mode
Display window
Console window

5. GUI mode (cont) Dynamical system settings Control bounds Time step
Function of the size of the grid
Study of the dynamics!!!
Viability constraint set

6. GUI mode (cont) Viability controller config General settings
Optimization settings
Algorithm type
Visualize the individual trajectories
- Simple: gradient descent from the minimal values of the controls
- Double: min and max values
- Conjugate gradient
- Double conjugate gradient
- Newton method

7. Stopping criterion of the SVMs computation algorithm
GUI mode (cont)
SVM configuration
Stopping criterion of the SVMs computation algorithm
SVM algorithm
Bandwidth
big C : hard margin(no misclassification)
small C: soft margin
Only the gaussian kernel is implemented
Control the “smoothness” of the SVM function
- Small gamma: smooth
- Big gamma: less smooth
- C-SMO: see libSVM
- Simple SVM
- Balk: automative bandwidth tuning
- Soft-Balk: balk with soft margin

8. GUI mode (cont)
Execution and control

9. GUI mode (cont)
Indicators + logs

10. 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

11. 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

12. 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){
//init
Kaviar kaviar = new Kaviar();
//Optional: to add default models
Kaviar.addModels(Kaviar.DEFAULT_MODEL);
//Optional: to add one of the default models
//Kaviar.addModels(Population.class);
//replace my model by the name of your model
Kaviar.addModels(MyModel.class);
//Launch the GUI
Kaviar.startGUI(); }

13. MyClass.java (extends Dynamical_System)

14. MyClass.java (extends Dynamical_System_Target)
Previous +

15. MyClass.java (extends Dynamical_System_Resilience)
Previous +

16. 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%
with

17. 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)) …

18. 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

19. 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

20. Batch mode java -cp Kaviar-1.1.jar Appli/Batch Conso.simu -v
9*2 files: .svm + .log files, in the Conso… directory