Title: Effective gradient-free methods for inverse problems
1Effective gradient-free methods for inverse
problems
- Jyri Leskinen
- FiDiPro DESIGN project
2Introduction
- Current research
- Evolutionary algorithms
- Inverse problems
- Case study Electrical Impedance Tomography (EIT)
- Future
3Current 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
4Evolutionary 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)
5Evolutionary algorithms
- Several methods
- Genetic algorithms (Holland, 1960s Goldberg,
1989) - Evolutionary strategies (Rechenberg, 1960s)
- Differential evolution (Price Storn, 1995)
6Evolutionary 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
7Evolutionary algorithms
- Pros
- Global search methods
- Easy to implement
- Allows difficult objective functions
- Cons
- Slow convergence rate
- Many objective function evaluations needed
8Local 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
9Memetic 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
10Memetic 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
11Memetic 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.
12Inverse 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
13Inverse 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
14Inverse problems
- Often difficult to solve because of
ill-posedness the acquired data is not
sufficient ? the solution is not unique! - Extra information needed regularization
15Electrical Impedance Tomography
- Used in
- Medicine (experimental)
- Geophysics
- Industrial process imaging
- Simple, robust, cost-effective
- Poor spatial, good temporal resolution
16Electrical 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
17Electrical Impedance Tomography
Source Margaret Cheney et al. (1999)
18Electrical Impedance Tomography
Source Margaret Cheney et al. (1999)
19Electrical Impedance Tomography
Source The Open Prosthetics Project
(http//openprosthetics.org)
20Electrical Impedance Tomography
- PDE Complete Electrode Model
- Forward problem calculate voltage values Ul
using FEM
21Electrical Impedance Tomography
- Inverse problem minimize F(sh) by varying the
piecewise constant conductivity distribution sh
22Electrical Impedance Tomography
- Mathematically hard, non-linear ill-posed problem
- Typically solved using Newton-Gauss method
regularization (Tikhonov, ) - Resulting image smoothed, image artifacts
23Electrical Impedance Tomography
24Electrical Impedance Tomography
- Solution Reconstruct the image using discrete
shapes? - Resulting objective function multimodal,
non-smooth - Solution Use global methods
25Electrical 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)
26Electrical 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
27Electrical 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
28Electrical Impedance Tomography
29Now 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?