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Effective gradient-free methods for inverse problems Jyri Leskinen FiDiPro DESIGN project Introduction Current research Evolutionary algorithms Inverse problems Case study: Electrical Impedance Tomography (EIT) Future 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 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) Evolutionary algorithms Several methods – Genetic algorithms (Holland, 1960s; Goldberg, 1989) – Evolutionary strategies (Rechenberg, 1960s) – Differential evolution (Price & Storn, 1995) 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 Evolutionary algorithms Pros: – Global search methods – Easy to implement – Allows difficult objective functions Cons: – Slow convergence rate – Many objective function evaluations needed 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 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 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 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. 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 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 Inverse problems Often difficult to solve because of illposedness: the acquired data is not sufficient → the solution is not unique! Extra information needed; regularization Electrical Impedance Tomography Used in – Medicine (experimental) – Geophysics – Industrial process imaging Simple, robust, cost-effective Poor spatial, good temporal resolution 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 Electrical Impedance Tomography Source: Margaret Cheney et al. (1999) Electrical Impedance Tomography Source: Margaret Cheney et al. (1999) Electrical Impedance Tomography Source: The Open Prosthetics Project (http://openprosthetics.org) Electrical Impedance Tomography PDE: Complete Electrode Model Forward problem: calculate voltage values Ul using FEM Electrical Impedance Tomography Inverse problem: minimize F(σh) by varying the piecewise constant conductivity distribution σh Electrical Impedance Tomography Mathematically hard, non-linear ill- posed problem Typically solved using Newton-Gauss method + regularization (Tikhonov, …) Resulting image smoothed, image artifacts Electrical Impedance Tomography Electrical Impedance Tomography Solution: Reconstruct the image using discrete shapes? Resulting objective function multimodal, non-smooth Solution: Use global methods 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) Electrical Impedance Tomography Evolutionary framework based on the selfadaptive control parameter differential evolution (SACPDE) LLLSDE: – Lamarckian MA – Local search operator Nelder-Mead simplex method VOLSDE: – Weighting factor F improved by onedimensional local search 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 Electrical Impedance Tomography 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? Thank you for your attention! Questions?