1. Effective gradient-free methods for inverse problems Jyri Leskinen FiDiPro DESIGN project

2. Introduction  Current research  Evolutionary algorithms  Inverse problems  Case study: Electrical Impedance Tomography (EIT)  Future

3. Current research  Inverse problems –Shape reconstruction –Electrical Impedance Tomography (EIT)  Methods –Evolutionary algorithms (GA, DE) –Memetic algorithms –Parallel EAs  Implementation of the Game Theory –Nash GAs, MAs, DEs

4. Evolutionary algorithms  Based on the idea of natural selection (Darwin)  Operate a population of solution candidates (“individuals”)  New solutions by variation (crossover, mutation)  Convergence by selection (parent selection, survival selection)

5. Evolutionary algorithms  Several methods –Genetic algorithms (Holland, 1960s; Goldberg, 1989) –Evolutionary strategies (Rechenberg, 1960s) –Differential evolution (Price & Storn, 1995)

6. Evolutionary algorithms  Simple EA –Generate initial population –Until termination criteria met,  Select parents  Produce new individuals by crossing over the parents  Mutate some of the offspring  Select fittest individuals for the next generation

7. Evolutionary algorithms  Pros: –Global search methods –Easy to implement –Allows difficult objective functions  Cons: –Slow convergence rate –Many objective function evaluations needed

8. Local search methods  Operate on neighborhoods using certain moves  Pros: –Fast convergence rate –Less resource-intensive  Cons: –Converges to the nearest optimum –Gradient methods need “nice” objective function

9. Memetic algorithms  Hybridization of EAs and LSs –Global method –Improved convergence rate  Memetic algorithms –A class of hybrid EAs –Based on the idea of memes (Dawkins) –LS applied during the evolutionary process

10. Memetic algorithms  Simple MA –Generate initial population –Until termination criteria met,  Select parents  Produce new individuals by crossing over the parents  Mutate some of the offspring  Improve offspring by local search  Select fittest individuals for the next generation

11. Memetic algorithms  Typically Lamarckian –Acquired properties inherited –Unnatural  MAs not limited to that! –Parameter tuning –Local search operators as memes  Parameters encoded in chromosomes  Meme populations –etc.

12. Inverse problems  Inverse problem: –Data from a physical system –Construct the original model using available data and simulations  Typical IPs: –Image reconstruction –Electromagnetic scattering –Shape reconstruction

13. Inverse problems  Objective function for example a sum of squares min F(x) = ∑ |x(i) – x * (i)| 2 –x: the vector of values from a simulated solution (forward problem) –x * : the vector of target values

14. Inverse problems  Often difficult to solve because of ill- posedness: the acquired data is not sufficient → the solution is not unique!  Extra information needed; regularization

15. Electrical Impedance Tomography  Used in –Medicine (experimental) –Geophysics –Industrial process imaging  Simple, robust, cost-effective  Poor spatial, good temporal resolution

16. Electrical Impedance Tomography  Data from electrodes on the surface of the object  Inject small current using two of the electrodes  Measure voltages using the other electrodes  Reconstruct internal resistivity distribution from voltage patterns

17. Electrical Impedance Tomography Source: Margaret Cheney et al. (1999)

18. Electrical Impedance Tomography Source: Margaret Cheney et al. (1999)

19. Electrical Impedance Tomography Source: The Open Prosthetics Project (http://openprosthetics.org)

20. Electrical Impedance Tomography  PDE: Complete Electrode Model  Forward problem: calculate voltage values U l using FEM

21.  Inverse problem: minimize F(σ h ) by varying the piecewise constant conductivity distribution σ h Electrical Impedance Tomography

22.  Mathematically hard, non-linear ill- posed problem  Typically solved using Newton-Gauss method + regularization (Tikhonov, …)  Resulting image smoothed, image artifacts

23. Electrical Impedance Tomography

24.  Solution: Reconstruct the image using discrete shapes?  Resulting objective function multimodal, non-smooth  Solution: Use global methods

25. Electrical Impedance Tomography  Simple test case: Recover circular homogeneity (6 control parameters)  Two different memetic algorithms proposed: –Lifetime Learning Local Search (LLLSDE) –Variation Operator Local Search (VOLSDE)

26. Electrical Impedance Tomography  Evolutionary framework based on the self- adaptive control parameter differential evolution (SACPDE)  LLLSDE: –Lamarckian MA –Local search operator Nelder-Mead simplex method  VOLSDE: –Weighting factor F improved by one- dimensional local search

27. Electrical Impedance Tomography  Five algorithms tested (GA, DE, SACPDE, LLLSDE, VOLSDE)  Result: –GA performed poorly –DE better, some failures –LLLSDE best, but the difference to other adaptive methods minimal

28. Electrical Impedance Tomography

29. Now & future  Improve diversity using multiple populations (“island model”)  EAs can be used to find Nash equilibria  Improve convergence rate with virtual Nash games?  Can competitive games sometimes produce better solutions than cooperative games in multi-objective optimization?

30. Thank you for your attention! Questions?