effective gradient-free methods for inverse problems jyri leskinen fidipro design project

Effective gradient-free Effective gradient-free methods for inverse methods for inverse

problemsproblems

Jyri LeskinenJyri Leskinen

FiDiPro DESIGN projectFiDiPro DESIGN project

IntroductionIntroduction

Current researchCurrent research Evolutionary algorithmsEvolutionary algorithms Inverse problemsInverse problems Case study: Electrical Impedance Case study: Electrical Impedance

Tomography (EIT)Tomography (EIT) FutureFuture

Current researchCurrent research

Inverse problemsInverse problems– Shape reconstructionShape reconstruction– Electrical Impedance Tomography (EIT)Electrical Impedance Tomography (EIT)

MethodsMethods– Evolutionary algorithms (GA, DE)Evolutionary algorithms (GA, DE)– Memetic algorithmsMemetic algorithms– Parallel EAsParallel EAs

Implementation of the Game TheoryImplementation of the Game Theory– Nash GAs, MAs, DEsNash GAs, MAs, DEs

Evolutionary algorithmsEvolutionary algorithms

Based on the idea of natural Based on the idea of natural selection (Darwin)selection (Darwin)

Operate a population of solution Operate a population of solution candidates (“individuals”)candidates (“individuals”)

New solutions by variation New solutions by variation (crossover, mutation)(crossover, mutation)

Convergence by selection (parent Convergence by selection (parent selection, survival selection)selection, survival selection)


Several methodsSeveral methods– Genetic algorithms (Holland, 1960s; Genetic algorithms (Holland, 1960s;

Goldberg, 1989)Goldberg, 1989)– Evolutionary strategies (Rechenberg, Evolutionary strategies (Rechenberg,

1960s)1960s)– Differential evolution (Price & Storn, Differential evolution (Price & Storn,

1995)1995)


Simple EASimple EA– Generate initial populationGenerate initial population– Until termination criteria met,Until termination criteria met,

Select parentsSelect parentsProduce new individuals by crossing over Produce new individuals by crossing over

the parentsthe parentsMutate some of the offspringMutate some of the offspringSelect fittest individuals for the next Select fittest individuals for the next

generationgeneration


Pros:Pros:– Global search methodsGlobal search methods– Easy to implementEasy to implement– Allows difficult objective functionsAllows difficult objective functions

Cons:Cons:– Slow convergence rateSlow convergence rate– Many objective function evaluations Many objective function evaluations

neededneeded

Local search methodsLocal search methods

Operate on neighborhoods using Operate on neighborhoods using certain movescertain moves

Pros:Pros:– Fast convergence rateFast convergence rate– Less resource-intensiveLess resource-intensive

Cons:Cons:– Converges to the nearest optimumConverges to the nearest optimum– Gradient methods need “nice” objective Gradient methods need “nice” objective

functionfunction

Memetic algorithmsMemetic algorithms

Hybridization of EAs and LSsHybridization of EAs and LSs– Global methodGlobal method– Improved convergence rateImproved convergence rate

Memetic algorithmsMemetic algorithms– A class of hybrid EAsA class of hybrid EAs– Based on the idea of Based on the idea of memesmemes (Dawkins) (Dawkins)– LS applied during the evolutionary LS applied during the evolutionary

processprocess


Simple MASimple MA– Generate initial populationGenerate initial population– Until termination criteria met,Until termination criteria met,

Select parentsSelect parentsProduce new individuals by crossing over Produce new individuals by crossing over

the parentsthe parentsMutate some of the offspringMutate some of the offspring Improve offspring by local searchImprove offspring by local searchSelect fittest individuals for the next Select fittest individuals for the next

generationgeneration


Typically LamarckianTypically Lamarckian– Acquired properties inheritedAcquired properties inherited– UnnaturalUnnatural

MAs MAs notnot limited to that! limited to that!– Parameter tuningParameter tuning– Local search operators as memesLocal search operators as memes

Parameters encoded in chromosomesParameters encoded in chromosomesMeme populationsMeme populations

– etc.etc.

Inverse problemsInverse problems

Inverse problem:Inverse problem:– Data from a physical systemData from a physical system– Construct the original model using Construct the original model using

available data and simulationsavailable data and simulations Typical IPs:Typical IPs:

– Image reconstructionImage reconstruction– Electromagnetic scatteringElectromagnetic scattering– Shape reconstructionShape reconstruction


Objective function for example a sum Objective function for example a sum of squaresof squares

min min FF((xx) = ∑ |) = ∑ |xx((ii) – ) – xx**((ii)|)|22

– xx: the vector of values from a simulated : the vector of values from a simulated solution (forward problem) solution (forward problem)

– xx**: the vector of target values: the vector of target values


Often difficult to solve because of Often difficult to solve because of ill-ill-posednessposedness: the acquired data is not : the acquired data is not sufficient sufficient →→ the solution is not the solution is not unique!unique!

Extra information needed; Extra information needed; regularizationregularization

Electrical Impedance TomographyElectrical Impedance Tomography

Used inUsed in– Medicine (experimental)Medicine (experimental)– GeophysicsGeophysics– Industrial process imagingIndustrial process imaging

Simple, robust, cost-effectiveSimple, robust, cost-effective Poor spatial, good temporal Poor spatial, good temporal

resolutionresolution


Data from electrodes on the surface Data from electrodes on the surface of the objectof the object

Inject small current using two of the Inject small current using two of the electrodeselectrodes

Measure voltages using the other Measure voltages using the other electrodeselectrodes

Reconstruct internal resistivity Reconstruct internal resistivity distribution from voltage patternsdistribution from voltage patterns


Source: Margaret Cheney et al. (1999)


Source: The Open Prosthetics Project (http://openprosthetics.org)


PDE: Complete Electrode ModelPDE: Complete Electrode Model

Forward problem: calculate voltage Forward problem: calculate voltage values values UUll using FEM using FEM

Inverse problem: minimize Inverse problem: minimize FF((σσhh) by ) by varying the piecewise constant varying the piecewise constant conductivity distribution conductivity distribution σσhh



Mathematically hard, non-linear ill-Mathematically hard, non-linear ill-posed problemposed problem

Typically solved using Newton-Gauss Typically solved using Newton-Gauss method + regularization (Tikhonov, method + regularization (Tikhonov, …)…)

Resulting image smoothed, image Resulting image smoothed, image artifactsartifacts


Solution: Reconstruct the image Solution: Reconstruct the image using discrete shapes?using discrete shapes?

Resulting objective function Resulting objective function multimodal, non-smoothmultimodal, non-smooth

Solution: Use global methodsSolution: Use global methods


Simple test case: Recover circular Simple test case: Recover circular homogeneity (6 control parameters)homogeneity (6 control parameters)

Two different memetic algorithms Two different memetic algorithms proposed:proposed:– Lifetime Learning Local Search (LLLSDE)Lifetime Learning Local Search (LLLSDE)– Variation Operator Local Search Variation Operator Local Search

(VOLSDE)(VOLSDE)


Evolutionary framework based on the self-Evolutionary framework based on the self-adaptive control parameter differential adaptive control parameter differential evolution (SACPDE)evolution (SACPDE)

LLLSDE:LLLSDE:– Lamarckian MALamarckian MA– Local search operator Nelder-Mead simplex Local search operator Nelder-Mead simplex

methodmethod VOLSDE:VOLSDE:

– Weighting factor Weighting factor FF improved by one- improved by one-dimensional local searchdimensional local search


Five algorithms tested (GA, DE, Five algorithms tested (GA, DE, SACPDE, LLLSDE, VOLSDE)SACPDE, LLLSDE, VOLSDE)

Result:Result:– GA performed poorlyGA performed poorly– DE better, some failuresDE better, some failures– LLLSDE best, but the difference to other LLLSDE best, but the difference to other

adaptive methods minimaladaptive methods minimal

Now & futureNow & future

Improve diversity using multiple Improve diversity using multiple populations (“island model”)populations (“island model”)

EAs can be used to find Nash equilibriaEAs can be used to find Nash equilibria Improve convergence rate with virtual Improve convergence rate with virtual

Nash games?Nash games? Can competitive games sometimes Can competitive games sometimes

produce better solutions than cooperative produce better solutions than cooperative games in multi-objective optimization?games in multi-objective optimization?

Thank you for your attention!Thank you for your attention!

Questions?Questions?

effective gradient-free methods for inverse problems jyri leskinen fidipro design project

Documents

generation slide

evolutionary process

regularization slide

evolutionary algorithms

survival selection slide

nice objective function

memetic algorithms simple

good temporal resolution