an agent–oriented hierarchic strategy for solving inverse problems

Int. J. Appl. Math. Comput. Sci., , Vol. , No. , –DOI:

AGENT-ORIENTED HIERARCHIC STRATEGY FOR SOLVING INVERSEPROBLEMS

MACIEJ SMOŁKAa∗, ROBERT SCHAEFERa, MACIEJ PASZYNSKIa, DAVID PARDO b,c,d ,JULEN ALVAREZ-ARAMBERRI b,e

aDepartment of Computer Science, AGH University of Science and Technology, Krakow, Polande-mail: smolka,schaefer,[email protected]

bDepartment of Applied Mathematics, Statistics, and Operational ResearchUniversity of the Basque Country (UPV/EHU), Bilbao, Spain

e-mail: dzubiaur,[email protected] Center for Applied Mathematics (BCAM), Bilbao, Spain

d Ikerbasque (Basque Foundation for Sciences), Bilbao, Spain

eUniversity of Pau (UPPA), France

The paper discusses the complex, agent-oriented Hierarchic Memetic Strategy (HMS) dedicated to solving inverse paramet-ric problems. The strategy goes beyond the idea of two-phase global optimization algorithms. The global search performedby a tree of dependent demes is dynamically alternated with local, steepest descent searches. The strategy offers excep-tionally low computational costs, mainly because the direct solver accuracy (performed by the hp-adaptive Finite ElementMethod) is dynamically adjusted for each inverse search step. The computational cost is further decreased by the strategyemployed for solution inter-processing and fitness deterioration. The HMS efficiency is compared with the results of astandard evolutionary technique as well as with the multi-start strategy on benchmarks that exhibit typical inverse problemsdifficulties. Finally, an HMS application to a real-life engineering problem leading to the identification of oil deposits byinverting magnetotelluric measurements is presented. The HMS applicability to the inversion of magnetotelluric data isalso mathematically verified.

Keywords: Inverse problems, hybrid optimization methods, memetic algorithms, multi-agent systems, magnetotelluricdata inversion.

IntroductionInverse problems form an important area of thecontemporary research related to fundamental problemsin science and engineering. Among its numerousapplications one can find non-invasive geophysicalexploration, tumor characterization, and analysis ofunknown materials.

Parametric inverse problems are usually formulatedas optimization ones, where the objective is to minimizethe misfit between the simulated and measured forwardsolutions. When solving such problems, one usually facessome significant obstacles as ill-conditioning, existenceof multiple local minima (multi-modality), and possiblya low regularity of the misfit functional. All of them

∗Corresponding author

significantly reduce the usefulness of convex optimizationmethods (such as gradient-based ones), as well assimple stochastic mechanisms (Monte Carlo and simpleevolutionary schemes) because of:

• the lack of guarantee of finding all solutions;

• enormous computational cost.

The main goal of this work is to obtain a globalinverse solver capable of finding many global and/or localsolutions (misfit minima) with a satisfactory accuracy andan acceptable computational cost. In order to achievethis goal, the paper combines several different ideas: Hi-erarchic Genetic Strategy (HGS) (see e.g. (Schaefer andKołodziej, 2003) and (Wierzba et al., 2003)) to decreasethe cost of the main part of a global search (which consists

2 M. Smołka et al.

of finding the basins of attraction), Cluster-Based FitnessDeterioration (see e.g. (Beasley et al., 1993; Obuchowicz,1997; Wolny and Schaefer, 2011)), Memetic algorithms(see e.g. (Neri et al., 2012)) composing various techniquesinto a single population-based stochastic strategy inorder to gain efficiency and flexibility and Evolution-ary Multi-Agent System (EMAS) (see e.g. (Cetnarowiczet al., 1996)) allowing for a more flexible, asynchronousprocessing and easy embedding of local convex searches.The work presented in this paper is an extension of thehybrid model involving the global genetic search phasefollowed by the local gradient phase, as applied for theinverse resistivity logging measurement simulations withdirect current electrodes (Gajda-Zagorska et al., 2014), aswell as for the identification of the Young modulus for theimprint nanolithography (Barabasz et al., 2014).

The proposed strategy develops dynamically a treeof dependent populations (demes) searching with variouslevels of accuracy that grow from the root to the leaves.Two types of individuals are utilized: passive individualscontaining candidate solutions only, and active individualsconsisting of computational software agents. Both activeand passive individuals are gathered in demes governed bystructural agents assigned to the nodes of the populationtree. Demes of passive individuals are traditionallyevolved by using common selection and genetic operators.Active individuals (agents) compete inside its deme onewith another to perform their actions producing offspringagents with genotypes determined by traditional geneticoperators or by a convex optimization process. Thesearch accuracy is that associated to the solution of directproblems using an hp-adaptive Finite Element Method(hp-FEM), where h (height) refers to the element size, andp to the polynomial order of approximation, which canbe adapted/modified throughout the computational grid(Demkowicz, 2006).

The low computational cost is obtained primarilyby an economical global search performed by the treeof demes, because the more detailed local searchesare activated mainly in the promising regions foundby the parental populations. The total number ofindividuals is significantly lower than in the case of thetraditional, single population evolutionary search. Themain computational cost decrement is achieved via thecommon inverse and forward error scaling, i.e., therough global search is performed with a low misfitaccuracy whereas the accuracy increases in the refinedsearches performed by branches and leaves. Such apolicy minimizes the number of expensive hp-FEMsolver calls, that are necessary for the accurate misfitevaluation. Finally, the agent-oriented architecture basedon the EMAS idea allows for economic and flexibleinvocation of local gradient methods at least once for asingle solution and only if they outperform the stochasticsearch. Moreover, the agent-based architecture facilitates

the parallel evolution of deme populations.We have verified our strategy by a series of

benchmark problems that mimic the optimizationlandscape obtained in real-life inverse problems. Thefirst phase of benchmark tests leads to establish the setof strategy’s parameters that assure its most economicoperation. In the second phase, we compare results of ourproposed strategy with those of other global optimizationmethods executed using a comparable budget (computerresources). In this paper, we compare HMS with twostandard global optimization methods, i.e. the simpleevolutionary algorithm (SEA) and the multi-start. Wehave already performed a benchmark-based comparisonwith a version of HGS in (Smołka and Schaefer, 2014).

We conclude the paper with the application ofour agent-based system to solve an inverse problemconsisting in identifying the resistivity of the undergroundformation by using magnetotelluric measurements. Themagnetotelluric (MT) method is a passive electromagnetic(EM) exploration technique which allows to determinethe resistivity distribution in the subsurface of thearea of interest on scales varying from few meters tohundreds of kilometers. Commercial applications includehydrocarbon, geothermal, underground water monitoring,and more recently, monitoring CO2 sequestration in thesubsurface. The method does not require artificial powersources. Instead, natural power sources are used toinduce the phenomena. These natural sources are nothingbut electric currents within the ionosphere created bydeformations of the magnetosphere. The motivation forthe choice of the MT problem was the detection ofseveral local minima in an MT inversion performed bymeans of a classical gradient method (Alvarez-Aramberriet al., 2013).

The results of all comparative tests show muchbetter quality and number of extremes found by theproposed agent-based system than by other referencemethods. Single population evolutionary algorithms areable to find only a single solution within the assumedbudget. A multi-start method, that consists in theparallel execution of local gradient-type methods from anumber of random starting points, cannot in general findsatisfactory solutions, because within the assumed budgetonly a small number of local processes can be invoked.

The paper concentrates on the algorithmic aspects ofHMS, so we do not discuss implementation-related issueshere. Some notes on a sample implementation can befound in (Smołka and Schaefer, 2014) and we refer theinterested reader there.

1. HMS ArchitectureThe main idea of the HMS is to provide a globaloptimization tool especially suited for solving difficultinverse problems. The considered problems are difficult

Agent-Oriented Hierarchic Strategy for Solving Inverse Problems 3

because of their inherent multi-modality accompaniedby the nontrivial computational cost of direct problemsolution, which is necessary for evaluating the objectivewhich is the misfit between observed and computed valuesof a quantity of interest. Nevertheless, inverse problemshave also some advantageous features. First, theirtheoretical global minimum value is well-known (andequal to 0). Although in practice this theoretical value isnever attained because of noise, modeling errors etc., weknow at least that the objective function is bounded frombelow by 0 and that its takes values close to this bound.This knowledge can be used e.g. in the constructionof stopping conditions for stochastic evolution. Second,in some important cases, the cost of the direct problemsolution can be modulated by an assumed accuracy ofthe solution: it is the case of hp-FEM direct solvers(Demkowicz et al., 2007).

As a global optimization tool, the HMS triesto combine the high-level exploratory ability with theaccuracy and efficiency of a local optimization method.In contrast to two-phase methods in which local searchesare executed right after the completion of the globalphase, the HMS follows the overall idea of memeticalgorithms, i.e., it intermixes local-optimization-orientedmechanisms into a global stochastic search machinery.The global part follows the multi-population evolutionaryapproach introduced by the HGS (Schaefer and Kołodziej,2003). Namely, the global search is performed bya collection of genetic populations. The populationscan evolve in parallel, but they are not mutuallyindependent. The structure of the dependency relationis hierarchical (i.e. tree-like, see Figure 1) with arestricted number of levels. The HGS proved to have

Level 1

Level 2

Level 3

root deme

branch demes

leaf demes

U1

genetic spaces

low accuracy

high accuracy

U2

U3

Fig. 1. HGS-like evolutionary population tree

considerable exploratory capabilities together with a goodsearch accuracy, especially with floating-point phenotypeencoding (Wierzba et al., 2003). The HMS naturally triesto retain these abilities while going beyond the HGS insome aspects. First of all, it adds local optimization tothe set of operations applied to the genetic individuals.But this is performed carefully in order to avoid thepremature population convergence on one hand and thehigh cost of running instances of a local method frominappropriate points on the other hand. Namely, somegenetic individuals (but not necessarily all of them)receive an identity and some intelligence, hence becoming

independent agents in a multi-agent system (MAS). Thedecision of performing the local search becomes theirown responsibility. Moreover, the demes are managedby special controller agents. The idea of turning apassive genetic individual into an intelligent agent hassome further consequences. We have to redefine thegenetic operations in such a way that they can be appliedto agents. While it is straightforward for the mutationand the crossover (although in this case a new agentis activated), the selection cannot be performed in thesimple genetic (or evolutionary) way. Instead, we followthe lines of the EMAS (Cetnarowicz et al., 1996; Byrskiet al., 2013), thus performing an operation analogousto the tournament selection but realized as a two-agentrendezvous.

In the sequel, we shall present the structure of theHMS starting from a description of HMS agent types.

1.1. HMS Agent Types. Main HMS agent typesalong with their interrelations are shown in Figure 2.Master Agent (MA) is a global system coordinator.Deme Agents (DA) manage evolutionary populations atvarious levels of the deme tree. DA specializationsdiffer primarily in the type of population they can own.Evolutionary Agents (EA) hold simple passive collectionsof chromosomes, whereas Local Agents (LA) coordinategroups of Computational Agents (CA). Any CA, apartfrom holding an immutable genotype, can perform one ofthe available actions, such as a local optimization methodexecution. The primary responsibility of the ObjectiveAgent (OA) is the objective computation, which typicallyinvolves calling an external direct problem solver. In thefollowing, we provide detailed descriptions of the HMSagent types.

MasterAgent

ObjectiveAgentcache

DemeAgentaccuracyLevel

EvolutionaryAgentpassivePopulation

LocalAgent ComputationalAgentgenotype readOnlylifeEnergy

*

1*

ExternalSolver

Fig. 2. HMS agent types (UML class diagram)

Master Agent (MA). As a global system coordinator,it is started as a first agent in the HMS MAS.Its responsibilities include performing the system

4 M. Smołka et al.

initialization such as the activation of other essentialagents, i.e., the Objective Agent (OA) and a Deme Agent(DA) of the deme-tree root. After the initialization, theMaster Agent starts the global loop of deme coordinationand checks if the global stopping condition is satisfied.The deme coordination follows the lines of FIPAContract-Net (FIPA, 2002). It begins by sending a call forproposals (CFP) to all active Deme Agents. Then, MAwaits for DA proposals and accepts those that are not inconflict. This is shown in Algorithm 1. DA proposals are

Algorithm 1. Master Agent (MA) algorithm.1: create OA2: create root location DA3: repeat4: send CFP to all active DAs5: receive proposals from DAs6: accept all non conflicting proposals7: request fitness deterioration from OA8: until global stop condition is satisfied.

in conflict when their corresponding activities cannot beexecuted in parallel. Using the terminology of (Byrskiet al., 2013) they are not local. In the current HMSrealization, all actions are local (note that in contrast to(Byrski et al., 2013), we do not consider the migration),so all DA proposals can be accepted by MA.

Deme Agent (DA). It is a deme-tree node coordinator.Each deme has an associated level of computationalaccuracy stored as a property of the corresponding DemeAgent. In fact, Deme Agent is an abstract class with twodifferent specializations: Evolutionary Agent and LocalAgent.

Evolutionary Agent (EA). This is a simple (passive)evolutionary population owner. Periodically, afterreceiving the permission from the Master Agent, itevolves its population for a fixed number of generations(this sequence of genetic epochs is called a metaepoch)and then sprouts a new deme from the current bestindividual unless the sprout condition is not satisfied(see Algorithm 2). Note that similar agents form thestructure of the Globally Balanced HGS (Jojczyk andSchaefer, 2009). Creating the initial population in Line 2has two different meanings. If an EA contains the treeroot deme, then the initial population is sampled usingthe uniform probability distribution. Otherwise, the EA isitself sprouted using its parent’s best individual as a seed.In such a case, the initial population is sampled using thenormal distribution centered in the seed individual withthe standard deviation depending upon the tree level.

Algorithm 2. Evolutionary Agent (EA) algorithm.1: set accuracy level2: create initial deme population3: repeat4: send proposal to MA5: if MA has accepted proposal then6: for all epochs in metaepoch do7: perform selection8: perform crossover and mutation9: for all created individual do

10: request objective computation from OAwith stored accuracy

11: end for12: end for13: if best individual satisfies sprout condition then14: sprout new child DA from best individual15: end if16: end if17: until local stop condition is satisfied

Local Agent (LA). The Local Agent owns a populationof Computational Agents (CA) and acts as a localscheduler of their actions. Namely, it receives actionproposals from Computational Agents, selects one ofthem according to a probability distribution, sends aproposal to the Master Agent and, if the proposal isaccepted, lets the selected Computational Agent performits action (see Algorithm 3). The Local Agent’sresponsibilities include also some action coordination,such as checking if a selected sprout action is allowed.Creating the initial population is performed analogouslyas an Evolutionary Agent, i.e. by sampling using a properprobability distribution (different for root and non-rootdemes). The only difference is the type of createdindividuals: passive chromosomes in case of EvolutionaryAgent and active Computational Agents in case of LocalAgents.

Computational Agent (CA). It is an active individualof the HMS genetic population. It owns an immutablegenotype consisting of an encoded domain point (achromosome) and a level of computational precision. Theprecision level must be consistent with the owning LocalAgent’s level. The mutable part of a ComputationalAgent’s state includes a nonnegative memetic parametercalled life energy. The life energy is exchanged during aComputational Agent action execution such that the totalenergy remains constant within each deme. Only agentswith the positive life energy are considered active (alive)and take part in the system evolution. There exists aset of actions from which an active Computational Agentchooses one at a time to perform. Which actions areactually available depends on parameters such as the life


Algorithm 3. Local Agent (LA) algorithm.1: set accuracy level2: create initial deme population3: repeat4: send CFP to all active CAs5: receive action proposals from CAs and choose one6: send corresponding proposal to MA7: if MA has accepted the proposal then8: if CA action creates new individual then9: create new CA

10: else if chosen action is SPROUT then11: if sprouting can be performed then12: create new child DA13: end if14: end if15: end if16: until local stop condition is satisfied

energy, the objective value, the precision level and others.Finally, the action is performed only if permitted by theowning Local Agent (see Algorithm 4).

Algorithm 4. Computational Agent (CA) algorithm.1: set accuracy level according to LA’s level2: request objective computation from OA with stored

accuracy3: repeat4: remain temporarily inactive5: until objective value is obtained6: while life energy > 0 do7: receive CFP from owning LA8: choose an available action9: send the corresponding proposal to LA

10: if received permission from LA then11: perform chosen action12: update life energy13: end if14: end whileCA permanently inactive

There is an energy quantum related to eachaction, which is spent (during GET it can sometimesbe gained) by a Computational Agent during theaction execution. Currently, the following actions areconsidered (cf. (Byrski et al., 2013)): GET, MUTATE,CROSSOVER, LOCOPT and SPROUT. In all testsprepared for this paper, we set all action energies to 1 andthe initial CA energy to 5.

The GET action is a two-agent stochastic duel duringwhich the proper quantum energy moves from the loserto the winner. A Computational Agent with a lower(i.e. closer to the global minimum) objective value hasmore chances to win. MUTATE and CROSSOVER arestraightforward counterparts of corresponding genetic (or

evolutionary) operations, as e.g., the normal mutationand the arithmetic crossover. The SPROUT action isinspired by the child branch sprouting operation, which isfundamental in the HGS (Schaefer and Kołodziej, 2003).In the HMS, it produces a new deme together with itsDeme Agent and an initial population of ComputationalAgents. Obviously, SPROUT makes no sense at the leaflevel, where it can be optionally replaced with LOCOPT.The LOCOPT is a local optimization method executionstarted from the agent’s decoded chromosome. In thecurrent realization, LOCOPT is allowed only at the leaves.

The action selection is determined by a givenprobability distribution. The probability of the LOCOPTcan be computed using a formula like the following:

pLOCOPT = p0 + (1− p0)s

1 + objective, (1)

where 0 ≤ p0 < 1 is the guaranteed probability ands > 0 is close to but not equal to 1 to allow selectingof other actions. Thus, the better the objective value,the more strongly LOCOPT is preferred. Other actionsavailable according to the current life energy receive equalremaining probability. The same formula can also be usedfor computing SPROUT probability at non-leaf levels.

Objective Agent (OA). In the real HMS application(i.e. in solving inverse problems), the objective value iscomputed externally by a specialized direct solver. Theresponsibility of an Objective Agent (typically, one inthe whole system) is to provide a proper solver gateway,i.e., to execute the solver process (or several parallelprocesses) properly and to transfer the input data to thesolver and the output back to the HMS. Of special interestfor us is the case of computing the objective by means ofa direct hp-FEM solver when the direct problem solutionis Lipschitz continuous with respect to the parameters.This property is not straightforward and has to be provedin any particular case. In Section 3.4 it is shown forthe magnetotelluric problem (see Remark 1), which isour real-world test case. Then, we can adapt the solveraccuracy to the assumed accuracy of HMS tree demes:see Algorithm 5. Finally, we know the dependencybetween the solver accuracy and the computational costof the direct problem solution (see (Barabasz et al., 2014;Gajda-Zagorska et al., 2014) for details), which in turnis the main unit component of the overall HMS cost.Therefore, we can optimize the overall cost by modulatingthe deme accuracy.

Additional Objective Agent activities may include:caching solver results, solver instance pooling (in thecase of parallel execution) and scheduling objectivecomputations according to a sophisticated optimizingpolicy (e.g., a diffusion-based one (Grochowski et al.,2006)).

6 M. Smołka et al.

Algorithm 5. Objective computation with hp-FEM directsolver (see Subsection 3.4)

1: input: tree level j, parameter value2: compute relative FEM error erel3: while erel < Ratio(j) do4: execute 1 step of hp adaptation5: solve problem on new fine and coarse meshes6: compute new value of erel7: end while8: return objective computed by means of final mesh

A quite special kind of additional OA activityis the deterioration of the objective function. Thelatter is a proper objective modification that leads tothe leveling of central regions of attraction basins ofalready found local and global minima (Beasley et al.,1993; Obuchowicz, 1997). The idea is to discouragethe evolutionary individuals from exploring alreadywell-recognized areas. In this paper, we adopt theCluster-Based Fitness Deterioration technique describedin (Wolny and Schaefer, 2011). Namely, after a requestfrom MA, all gathered objective data (in this case, thepoints at which the objective has been computed) atselected accuracy levels are clustered (currently, usingDBSCAN algorithm (Ester et al., 1996)). Afterwards, foreach recognized cluster, we construct the ellipsoidal clus-ter extension

CE = x ∈ RN : (x− x)TΣ−1(x− x) ≤ 1

determined by its center x and a symmetric matrixΣ, which in our case is the cluster unbiased samplecovariance matrix (cf. (Wolny and Schaefer, 2011)). Then,each cluster extension contributes to the deterioration bythe following formula

f(x) := f(x) + fmaxΨ(x), (2)

where Ψ is a well-known bump function

Ψ(x) =

exp

(1− 1

1−(x−x)T Σ−1(x−x)

)for x ∈ CE

0 otherwise(3)

and fmax is the maximal objective value over the cluster.Ψ is an infinitely smooth function and its support isexactly the considered ellipsoid. Note that in contrastto the standard (cf. (Wolny and Schaefer, 2011)), wedo not make use of simple Gaussian functions becausethey are positive everywhere, which destroys the desiredzero-value feature of inverse problem global minima.

1.2. Population Structure. As stated before, the HMSgenetic population is decomposed into dependent demesforming a dynamically-changing tree of the fixed maximal

depth m. Genetic individuals, i.e. Computing Agents,located at the tree levels close to the root, perform thechaotic and inaccurate search. When approaching theleaves, the search becomes more and more focused andthe accuracy is increased (see Figure 1). The variabilityof the search accuracy results from the diversity of thegenotype encoding precision used at different tree levels.The latter depends on the encoding type. In the case of thebinary encoding (as in the Simple Genetic Algorithm), itcan be achieved by the binary genotype length variation,whereas in the case of the real number encoding (as inthe Simple Evolutionary Algorithm) it can be realized bythe appropriate phenotype scaling. The latter case is usedin the prototype implementation of the HMS, so here wepresent some details. The description follows the onespresented in existing papers (Wierzba et al., 2003; Jojczykand Schaefer, 2009).

In the real number encoding, both phenotypes andgenotypes are vectors from RN . We assume that thesolution domain is a box

D = [a1, b1]× · · · × [aN , bN ]

and we take a sequence of scaling factors ηi ∈ R suchthat η1 > η2 > . . . ηm−1 > ηm = 1. Then, the geneticuniversum at the tree level j is

Uj =

[0,b1 − a1

ηj

]× · · · ×

[0,bN − aN

ηj

](4)

and the encoding mapping at the level j is defined as

D 3 x 7−→xk − akηj

Nk=1

∈ Uj . (5)

Moreover, we define the scaling mapping

scalei,j : Ui 3 x 7→ηiηjx ∈ Uj .

In such genetic universa, the search at lower levels ismore chaotic because the mutation acts stronger, andless precise because of limitations in the real numberrepresentation. It is possible to use various geneticoperators in such an encoding. Among the most importantones, we select the normal mutation

yi = xi +N (0, σmutj ) for i = 1, . . . , N,

where N (0, σmutj ) is a normally-distributed randomvariable with standard deviation σmutj set separately foreach level j, and the arithmetic crossover

yi = x1i + U([0, 1])(x2

i − x1i ) for i = 1, . . . , N,

where U([0, 1]) is a random variable distributed uniformlyover the interval [0, 1]. Both operators are used in oursample implementation. Furthermore, we exploit the


classical fitness-proportional (roulette-wheel) selection inpassive populations (on Evolutionary Agents) additionallypreserving the best individual of each generation. Anewly sprouted deme’s population is sampled accordingto the N -dimensional Gaussian distribution centered atthe properly encoded fittest individual of the parentprocess with the diagonal covariance matrix taking values(σsproutj )2 on the diagonal. The sprout cannot beperformed in population P at level j if there exists apopulation P ′ at level j + 1 such that

|y − scalei,i+1(y)| < cj , (6)

where y is the best individual in P , y is the averagephenotype of P ′, and cj is a branch comparison constant.

Finally, it should be mentioned that a furtherutilization of the knowledge gathered during themulti-level enhanced genetic evolution is possible bymeans of the clustering technique, in which betterapproximation of attraction basins of the local minimacan be developed, allowing yet more precise applicationof local optimization methods.

2. Benchmark TestsIn order to prove HMS abilities to find the globalminimum in multi-modal cases, we performed twobenchmark tests. The test setup in both cases was asfollows. The HMS tree had three levels with EvolutionaryAgents at the root and middle levels and Local Agentsat the leaf level, which seems to be a quite standardlayout for an overall optimization. The proportional(roulette-wheel) selection, the normal mutation and thearithmetic crossover were chosen as genetic operators.First, we run the HMS against both functions 30 times tillthe global minimum was reached with assumed accuracy10−4. The choice of such a type of test influencedthe setting of the HMS stopping conditions. Namely,the global stopping condition was satisfied if a leafapproached the global minimum with the given accuracy,whereas a leaf stopping condition was satisfied if the leafapproached the global minimum or if a fixed number of itsconsecutive metaepochs were ineffective, i.e. there wasno significant variation in the leaf’s population averagefitness. To make this stopping condition applicable toactive populations, we need to adapt the notion of geneticepoch. Namely, in case of a LA population, it is simply aCA action execution sequence of length equal to the initialpopulation size. After the HMS had finished, we countedits objective calls, which set the computational budgetfor two comparative classical stochastic optimizationmethods: the simple evolutionary algorithm (SEA) andthe multi-start. The comparative methods were also run30 times.

The first benchmark was the 10-dimensional Ackleypath function with domain [−5, 5]10. It is a standard

global optimization test with one hard-to-find global0-valued minimum surrounded by numerous other localminima with greater values. This makes it similar toa few important inverse problem objectives, such as themagnetotelluric problem (see Section 3). A special featureof the Ackley function is it flatness outside the narrowattraction basin of the global minimum, which makes itstill more similar to the MT problem. The executionparameters for 10D Ackley function are summarized inTable 1. Note that the metaepoch length parameter isspecially adapted to Local Agents (see above). Similarly,the population size in this case is not constant, in oursimulations it varied between 4 and 8. The obtained HMS

Table 1. HMS execution parameters (Ackley 10D)Root Middle Leaf

Population (initial) 50 10 4Metaepoch length 2 2 2Encoding scale 4.0 2.0 1.0Mutation rate 0.2 0.05 0.01Crossover rate 0.5 0.5 0.5Mutation std. dev. 5.0 1.0 0.2Sprout std. dev. - 2.0 1.0Sprout min. dist. - 2.0 1.0

objective call means are shown in Table 2. The cost ofa local method application is included in the leaf levelcost. The averages from Table 2 allowed us to compute

Table 2. Average number of objective evaluations (Ackley 10D)Root Middle Leaf Total

1022.8 584.7 4493.6 6101.1

the predicted cost for SEA and multi-start. Taking theaverage cost of running the local method from HMSexecutions, we set the starting pool size for multi-start to70. Similarly, estimating the average cost of running theevolutionary algorithm, also on the basis of HMS runs, weset the initial SEA population size to 100. SEA geneticparameters were set according to the corresponding HMSparameters from the leaf level, i.e. the crossover rate wasset to 0.5, the mutation rate to 0.01, and the mutationstandard deviation to 0.2. Table 3 shows the averageobtained minimum values for all three methods. Neither

Table 3. Average values of computed global minimum (Ackley10D)

HMS Multi-start SEAAverage 5.27 · 10−9 3.17 2.57Best 1.96 · 10−10 1.65 1.3

SEA nor multi-start ever succeeded in reaching the actualglobal minimum, which HMS managed to do in every run.

8 M. Smołka et al.

Full statistics are shown in a concise way in Figure 3.Horizontal bars indicate minimum, mean and maximum,

HMS Multi-start SEA

0

1

2

3

4

Min

. valu

e

Fig. 3. Global minima statistics (Ackley 10D)

respectively. The width of the ’violin’ indicates thedistribution of values.

The second benchmark was the product of threereflected and vertically translated Gaussian functions overthe 4-dimensional box [−5, 5]4

f2(x) =

3∏i=1

[1− exp

(−(x−mi)

TAi(x−mi))], (7)

where m1 = (3, 2, 1, 0), m2 = (1,−3,−1, 3), m3 =(−2.5, 2, 2, 2) and

A1 =

1 0 0 00 1 0 00 0 1 00 0 0 1

,

A2 =

0.2 0 0 00 0.1 0 00 0 0.1 00 0 0 0.2

,

A3 =

1.5 0 0 00 2 0 00 0 1.5 00 0 0 1

.Benchmark f2 has three separate 0-valued global minimam1,m2 andm3 with no other local minima. The difficultyof finding the global minima is graded. m2 has thebroadest attraction basin, so it is fairly easily reachable.m1 is steeper, hence more difficult to find. And themain problem is reaching m3. Outside their attractionbasins, one can find quite large plateaus, which can posea noticeable trouble for gradient optimization methods.

Thus, the aim of this test is to check the comparedmethods’ abilities of finding all global minima.

The HMS execution parameters for benchmark f2

are gathered in Table 4. The obtained HMS objective call

Table 4. HMS execution parameters (3 Gaussians)Root Middle Leaf


means are shown in Table 5. This time to make things

Table 5. Average number of objective evaluations (3 Gaussians)Root Middle Leaf Total Weighted

6901.5 1135.1 1655.2 9691.8 3183.8

still more similar to MT computations, instead of a simplesum of different-level costs, we used a weighted cost

c = cleaf +cmiddle

3+croot

6, (8)

expressing the fact that in real problems the leafcomputations are the most expensive ones, whereas theroot calls are the cheapest. Denominators in (8) aretaken from MT simulations, see Section 4. As thecomparative methods operated at the top level of accuracy,i.e., the leaf level, based on Table 5 we set the multi-startstarter pool size to 50 and the SEA initial populationto 70. As in the Ackley 10D case, we set SEAgenetic parameters accordingly to the corresponding HMSleaf-level parameters. We stopped the SEA right afterexceeding 4000 calls. The main results of the test areshown in Table 6. HMS successfully found all three

Table 6. Statistics of finding all global minima (3 Gaussians)HMS Multi-start SEA

Fully successful runs 30/30 16/30 0/30Avg. number of mins 3 2.47 0

minima in every execution. Multi-start failed in almosthalf of the runs. Moreover, in two runs it found onlyone global minimum. SEA never succeeded, obtainingan average computed global minimum of 0.57 and abest found solution with value 0.18. As the resultsshow, the three Gaussians benchmark is a quite difficultoptimization test, despite its relatively low dimensionality.The source of the difficulty is the small volume of global


minima’s attraction basins and the presence of significantplateaus.

3. Magnetotelluric inverse problemThe solar wind produces a radiation pressure that causes acompression on the day-side and a tail on the night-sideonto the magnetosphere. Due to this interaction,hydromagnetic waves are created. When those wavesreach the ionosphere, they induce an EM field that worksas power source in magnetotellurics. Depending uponthe type of source we are dealing with, the geomagneticfluctuations range between frequencies of 10−3 − 105

Hz, which allow us to make measurements with aresolution that ranges from a few meters to hundreds ofkilometers (Vozoff, 1972)

The magnetotelluric (MT) technique is used todetermine a resistivity map of the Earth’s subsurfaceby performing electromagnetic (EM) measurements.The main difference of MT with respect to othergeophysical measurement acquisition scenarios (e.g.marine controlled electromagnetic measurements)is that MT uses natural electromagnetic radiationsources generated within the ionosphere, instead ofhuman powered antennas. Thus, acquisition of MTmeasurements is rather inexpensive, and can coverlarge areas. Applications of MT measurements includehydrocarbon (oil and gas) exploration and finding suitableregions for storage of CO2.

3.1. Forward problem. MT measurements aregoverned by the electromagnetic phenomena, whichcan be described by Maxwell’s equations. When theelectrical fields depends only upon two spatial variables(x, z), then two independent and uncoupled modes arederived from these equations, namely Transverse Elec-tric (TE) and Transverse Magnetic (TM). TE modeinvolves (Ey, Hx, Hz) field components while TM uses(Hy, Ex, Ez). We focus on the TE mode and we solve theequation for Ey .

We decouple Maxwell’s equations bypre-multiplying both sides of Faraday’s law by µ−1

and applying the curl operator. Then, after incorporatingAmpere’s law, we substitute component by componentthe result into the double curl operator on the electric fieldto obtain the equation for the y-component of the electricfield:

−∆u− k2u = f in Ω ⊂ R2, (9)

where Ω is a simply connected bounded domain withLipschitz boundary and u(x, z) = Ey(x, z), (x, z) ∈ Ω.σ, (σ > σ0 > 0 in Ω) is the electrical conductivity field,k2 = ω2µε − jµωσ, where ω 6= 0 is the wave frequency,ε, µ ∈ R, ε, µ > 0 stand for the permittivity andpermeability of the considered medium. f = −jωµJ impy ,

where J impy is a prescribed, impressed electric currentdensity radiating in the y-direction.

To obtain the corresponding variational formulation,we pre-multiply equation (9) by a test function v ∈H1

0 (Ω;C). After integrating by parts and incorporatingthe Dirichlet boundary conditions over ΓD = ∂Ω, thefollowing abstract variational formulation (suitable forFinite Element computations) is obtained:

Find u(σ) ∈ H10 (Ω;C) such that:

b(σ;u(σ), v) = F (v) ∀v ∈ H10 (Ω;C),

(10)

where

b(σ;u, v) =

∫Ω

∇v ∇u−∫

Ω

k2 v u (11)

F (v) =

∫Ω

v f. (12)

where now we assume that σ ∈ L∞(Ω) and J impy ∈L2(Ω;C). The exact solution u(σ) = Ey to problem (10)is called primal forward solution.

Let us define now the functionals Li : H10 (Ω;C) →

C associated with the receiving antennas occupying Ωi

domains respectively, i = 1, . . . ,M

Li(v) =1

meas(Ωi)

∫Ωi

v. (13)

Now, we are able to define the set of dual forwardproblems

Find G(σ) ∈ H10 (Ω;C) such that:

b(σ; v,G(σ)) = Li(v) ∀v ∈ H10 (Ω;C),

(14)

for each receiving antenna, i = 1, . . . ,M .

3.2. hp-FEM approximation. Using hp-FEM finitedimensional internal approximation Vhp ⊂ H1

0 (Ω;C) (see(Demkowicz, 2006)), we obtain the discrete versions ofboth primal and dual forward problems (10), (14)

Find uh,p(σ) ∈ Vhp such that:

b(σ;uh,p(σ), v) = F (v) ∀v ∈ Vhp,(15)

Find Gih,p(σ) ∈ Vhp such that:

b(σ; v,Gih,p(σ)) = Li(v) ∀v ∈ Vhp,(16)

for each receiving antenna, i = 1, . . . ,M .To control the discretization error by performing

grid refinements, we use the two-dimensional (2D)hp-adaptive algorithm described in (Demkowicz, 2006).It incorporates two basic components:

• Fine grid solution: giving an existing coarse meshuhp, we refine the mesh in both h and p to obtain

10 M. Smołka et al.

a finer mesh. Its solution is denoted as uh/2,p+1.The hp refined grid provides a reference solutionthat is used for comparison purposes and to decidewhat areas of the computational mesh contain largererrors.

• Optimal mesh selection: given the fine meshsolution, we use it to determine optimalmesh refinements on the coarse mesh byminimizing the projection based interpolationerror (see (Demkowicz, 2006)).

The optimization algorithm contains the followingsteps:

1. For each element in the coarse mesh, we selectbetween p and h refinements. Using theprojection-based interpolation error, we select therefinement that provides the biggest error decreaserate by comparing the fine and coarse mesh solutions.

2. We determine which elements to refine by computingthe maximum decrease rate over all elements,and selecting for refinements those that exhibit anerror decrease rate larger than a fraction of themaximum. In our case, we select 1/3, as suggestedby Demkowicz et al in (Demkowicz, 2006).

3. We determine the optimal order of approximation forthe elements to be refined: if a p-refined is selected,then we increase p by one. If h-refinement wins, thenwe determine the value of the new polynomial ordersof approximation for those new elements that yieldthe biggest descent rate.

For goal-oriented adaptivity where the mainobjective is to accurately approximate L(Ey), we employthe adjoint problem to represent the error in the quantityof interest. Thus, the corresponding refinement strategyis based on minimizing the error in the quantity ofinterest rather than an arbitrary energy norm (see (Pardoet al., 2006) for details).

3.3. Inverse problem. Our aim is to obtain theimpedance, a suitable physical magnitude to perform theinversion. To do so, the magnetic field is obtained fromMaxwell’s equations as

Hx(σ) =1

jωµ

∂Ey(σ)

∂z, (17)

and then impedance Z is computed according to:

Z = Zyx =EyHx

. (18)

The impedance at each antenna is a nonlinear functionalgi(σ) computed from two linear quantities:

gi(σ) =Li(Ey(σ))

Li(Hx(σ))= jωµLi(u(σ))

(Li(∂u(σ)

∂z

))−1

(19)for i = 1, . . . ,M , whose derivative with respect to anarbitrary variable s (in particular the conductivity σ) canbe obtained from the formula

∂gi(σ)

∂s=

1

(Li(Hx))2

(∂Li(Ey)

∂sLi(Hx) (20)

− ∂Li(Hx)

∂sLi(Ey)

),

where all above partial derivatives are distributional ones.Let us denote by diobs the measured value of impedancesat the antennas, which correspond to the quantities gi(σ),where σ ∈ L∞(Ω) is the real, unknown conductivitydistribution.

We introduce the stepwise representation of theconductivity function σ as a member of a family

D = ξ ∈ L∞(Ω); ξ(x) =∑i=1,...,n ξiχi(x),

0 < ξmini ≤ ξi ≤ ξmaxi < +∞,(21)

where χii=1,...,n are the indicator functions ofthe disjoint covering Ωii=1,...,n of domain Ω(⋃i=1,...,n Ωi = Ω,Ωi ∩ Ωj , i 6= j). This representation

is utilized by hp–FEM computing of the primary and dualGalerkin problems (15), (16). Next, we define for σ ∈ Dthe approximate impedance functions

gihp(σ) = jωµLi(uh,p(σ))

(Li(∂uh,p(σ)

∂z

))−1

.

(22)The inverse problem under consideration reads as follows:Find σ ∈ D such that for all σ ∈ D:

limh→0,p→+∞

1

2M

∑Mi=1

∣∣∣gih,p(σ)− diobs∣∣∣2

≤

limh→0,p→+∞

1

2M

∑Mi=1

∣∣∣gih,p(σ)− diobs∣∣∣2 (23)

3.4. Common error scaling. The hp-FEM adaptationis performed by solving forward primal and dual problemsfor quantities of interest Li, i = 1, . . . ,M . The relativehp–FEM error scaling is selected based on the followingProposition.

Proposition 1. Assuming gi(σ) = diobs, i = 1, . . . ,Mthe misfit function might be evaluated as the product ofsquares of norms of relative hp-FEM errors of primal anddual solutions added to the square of the norm of the ab-solute hp-FEM errors of primal solutions obtained for all


coordinates of the logging curve plus the square of the er-ror associated to the inverse problem i.e.

12M

∑Mi=1

∣∣∣gih/2,p+1(σ)− gi(σ)∣∣∣2 ≤

A ‖erel(σ)‖20,1(∑M

i=1

∥∥εirel(σ)∥∥2

0,1

)+B ‖uh,p(σ)− u(σ)‖20,1+C ‖σ − σ‖2L∞(Ω)

(24)

where erel(σ) = uh/2,p+1(σ) − uh,p(σ) is the relativehp-FEM error in solving forward primal problem (15),εirel(σ) = Gih/2,p+1(σ) −Gih,p(σ), i = 1, . . . ,M are therelative hp-FEM errors in solving forward dual problems(16) and A,B,C are positive constants.

We start with a useful Lemma which is a particularcase of the more general result obtained for the AClogging data inversion (see Proposition 9 in (Smołkaet al., 2014)).

Lemma 1. The distance between the approximated andexact quantities of interest might be evaluated as the prod-uct of squares of norms of relative hp-FEM errors of pri-mal and dual solutions added to the square of the norm ofabsolute hp-FEM errors of primal solutions obtained forall coordinates of the logging curve plus the square of theerror associated to the inverse problem i.e.∑M

i=1

∣∣Li(uh/2,p+1(σ))− Li(u(σ))∣∣2 ≤

C ′ ‖erel(σ)‖20,1(∑M

i=1


0,1

)+C ′′ ‖uh,p(σ)− u(σ)‖20,1+C ′′′ ‖σ − σ‖2L∞(Ω) .

(25)

Proof. (Proof of Proposition 1) Let us denote for thesimplicity a = uh/2,p+1(σ), b = u(σ) and by a,z , b,zto their partial derivatives with respect to the variable z.Now, we can evaluate∣∣∣gih/2,p+1(σ)− gi(σ)

∣∣∣ = jωµ

∣∣∣∣ Li(a)

Li(a,z )− Li(b)

Li(b,z )

∣∣∣∣ .(26)

The right-hand-side of the above expression is less than orequal to the following∣∣∣∣ Li(a)

Li(a,z )− Li(b)

Li(a,z )

∣∣∣∣+∣∣Li(b)∣∣ ∣∣∣∣ 1

Li(a,z )− 1

Li(b,z )

∣∣∣∣=

1

|Li(a,z )|∣∣Li(a)− Li(b)

∣∣+∣∣Li(b)∣∣|Li(a,z )| |Li(b,z )|

∣∣Li(a,z −b,z )∣∣ . (27)

Because we consider higher than zero frequencies ofa cosmic propagation, then it is possible to establisha universal constant α > 0 that bounds from below|Li(∂u∂z

)| for all solutions u to (13) and for all

i = 1, . . . ,M . Moreover, all solutions to (13) arebounded from above in H1

0 (Ω), so we have thenet universal constant β < +∞ that is greater orequal than Li(u) for all solutions to (13) and for alli = 1, . . . ,M . Next, all functionals Li

(∂u∂z

)are

uniformly Lipschitz continuous on H10 (Ω) with the

constant L1. Finally,∥∥uh/2,p+1(σ)− u(σ)

∥∥1,0

≤∥∥uh/2,p+1(σ)− u(σ)∥∥

1,0+ ‖u(σ)− u(σ)‖1,0 ≤

‖uh,p(σ)− u(σ)‖1,0 + L0 ‖σ − σ‖L∞(Ω). Summing up(26), (27), we have∣∣∣gih/2,p+1(σ)− gi(σ)

∣∣∣ ≤C1

∣∣Li(uh/2,p+1(σ))− Li(u(σ))∣∣+

C2 ‖uh,p(σ)− u(σ)‖1,0 +

C3 ‖σ − σ‖L∞(Ω) ,

(28)

where C1 = jωµα , C2 = C1

βL1

α , C3 = (C2 + 1)L0

and L0 is the Lipschitz constant of the function that mapsthe conductivity to the solution of the primal problem(13), i.e. σ → u(σ) in norms ‖·‖1,0 and ‖·‖L∞(Ω) (seeLemma 4 in (Smołka et al., 2014)). Using twice the simpleinequality (a+ b)2 ≤ 2a2 + 2b2, we obtain∣∣∣gih/2,p+1(σ)− gi(σ)

∣∣∣2 ≤C1

∣∣Li(uh/2,p+1(σ))− Li(u(σ))∣∣2 +

C2 ‖uh,p(σ)− u(σ)‖21,0 +

C3 ‖σ − σ‖2L∞(Ω) ,

(29)

for other constants C1, C2, C3. Coupling (29) with theresult of Lemma 1, we obtain:

12M

∑Mi=1

∣∣∣gih/2,p+1(σ)− gi(σ)∣∣∣2 ≤

C1C′

2M ‖erel(σ)‖20,1(∑M

i=1


0,1

)+ C1C

′′+C2

2M ‖uh,p(σ)− u(σ)‖20,1

+ C1C′′′+C3

2M ‖σ − σ‖2L∞(Ω) .

(30)

It is enough to set A = C1C′

2M , B = C1C′′+C2

2M , C =C1C

′′′+C3

2M to finish the proof.

Remark 1. The first right-hand side componentof (25) expresses the influence of the relative forwarderrors (primal and dual ones) imposed on the hp-FEMrefinement process. The second one is proportional


to the square of the FEM error, which decreases to 0during hp refinements. The third component is evaluatedfrom below by Cδ2

j , where δj is the accuracy in theinverse search domain D utilized by the j-th level HMSbranch. In order to make the hp-HMS inversion at thej-th level computationally economic, we should keepthe first and third components comparable. In otherwords, decreasing ‖erel(σ)‖0,1 and

∥∥εirel(σ)∥∥

0,1below

the quantity Ratio(j) =√

δjM

4

√CA does not improve the

accuracy of fitness evaluation.

4. Sample magnetotelluric simulations4.1. Domain description. We consider a 2D formationof the subsurface consisting of a horizontally layeredmedia with some 2D inhomogeneities. We model thesource as an infinitely long (in x and y directions)volumetric rectangular surface located at the ionosphere.This allows us to treat the electromagnetic fields as planewaves that propagate in the vertical direction towards theEarth’s surface (Vozoff, 1972).The physical problem is illustrated in Figure 4. Thecomputational domain in consideration, of 2500 × 130km. consists of air and a layered media with 2Dinhomogeneities located at the center of the domain (in thex direction). The horizontal length of the domain wherethe inhomogeneities are considered is 10 km.The physical domain is surrounded by a PerfectlyMatched Layer (PML) (Berenger, 1994) used to truncatethe computational domain, and we consider the samerelative permittivity and permeability in all materials(equal to one). The dark gray, middle gray and light grayregions in the lower half of the figure are filled with soil(or rock) deposits characterized by different conductivitiesσ1 − σ4. Receivers A1 − A7 are located at the Earth’ssurface, and are represented with the inverted t-marks ⊥ .The source, located at the ionosphere, is represented witha thin dark gray rectangle at the top of the figure.

4.2. Results. In our simulations we used thegoal-oriented hp-adaptive Finite Element Method solver,which computed both the impedance and its partialderivatives at the receivers in one run. For the simulations,the solver operated at 3 accuracy levels: 70%, 20% and3%, meaning the maximal relative FEM error percentage.The ’exact’ values diobs were computed using the samesolver with accuracy 1.2%. The HMS was executed withthe same deme-tree layout as in the case of Ackley andthree Gaussians. However, this time the computationswere much more expensive. Therefore, in this case weperformed only 5 executions of all three comparativemethods, i.e. HMS, Multi-start and SEA. The HMSexecution parameters are summarized in Table 7. EveryHMS run lasted about 32 hours. The simulations were

executed on a single node of a Linux cluster with 16Intel R© Xeon R© CPU E5620 at 2.40GHz and 16 GB oftotal memory available. Our simulation at this pointutilizes one core only, however it is possible to speed itup by linking to a parallel MUMPS solver, which deliversaround 60% efficiency up to 16 cores. It turned out that

Table 7. HMS execution parameters (MT)Root Middle Leaf


the average time of FEM computations were about: 45sfor accuracy level 70%, 1.5min for accuracy level 20%and 4.5min for accuracy level 3%. This justifies theweighted cost formula (8) applied in the three Gaussiantest and actually appropriate in this case. The obtainedHMS objective call means are shown in Table 8. After

Table 8. Average number of objective evaluations (MT)Root Middle Leaf Total Weighted398.2 629.9 148.6 1176.7 424.9

appropriate estimations, we set the multi-start initial poolsize to 100 and the initial SEA population to 40. SEAgenetic parameters were set as in the previous tests.During the computations, it appeared that the objectivefunction has several flat regions, where gradient methodsgot stuck revealing many local minima of mediocre oreven poor value. Hence, in the comparisons we emphasizethe good minima, i.e. the ones with the objective valueless than a given threshold. In Table 9, we show the totalnumber of such minima found in all runs for 2 thresholdvalues. Figure 5 shows a comparison of all minimal values

Table 9. Total number of ’good’ minima found (MT)Upper bound HMS Multi-start SEA1 · 10−10 4 2 05 · 10−10 35 21 0

obtained by the three methods.The results show that both HMS and multi-start win

by far to SEA. The advantage of HMS over multi-startis more apparent in the best minima area. Namely,HMS found twice as many minima smaller than 10−10

as multi-start and 1.7 times as many minima smaller than5 · 10−10 as multi-start: see Table 9. Moreover, Figure 5


90 km

3 km

27 km

10 km

2500 km

10 km

PML

A1 2A 5A4A3A

s1

2s s4 s2

s3

4 km 4 km 4 km 4 km12 km 12 km

7A6A

z

x

Fig. 4. Domain geometry.

HMS Multi-start SEA10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

Min

. valu

e

Fig. 5. All minima statistics (MT)

shows that multi-start is more prone to getting stuck inpoor local minima.

5. Conclusions

The proposed strategy HMS constitutes a significantextension of the standard, two-phase global optimizationstrategies composed of the global search phase followedby a couple of local convex searches. The essence ofthis extension is the application of the adaptive hierarchicgenetic search alternating with the local searches withdynamically adapted accuracy. The genetic multi-demecomponent together with the inter-processing of solutions(clustering of individuals) and fitness deterioration allowfor broad, global search together with concentrating local

searches in the basins of attraction of the deepest localminima. The additional novelty is to introduce theactive individuals (computing agents) and agent-baseddistributed governing the whole strategy. Contrarilyto traditional inversion algorithms, our hybrid strategydelivers multiple solutions in case of misfit multimodality,which enables an expert in the field to determine the bestpossible solution as well as the uncertainty level.

HMS offers an exceptionally low computationalcost, which is crucial in solving inverse parametricproblems involving partial differential equations. First,the hierarchic structure of demes decreases the numberof individuals searching with a high accuracy. Thenumber of fitness evaluations is additionally decreasedby the conditional sprouting and redundancy reduction aswell as restrictive deme stopping condition. Next, theglobal search is boosted by the concurrent identificationof separate basins of attractions by separate well fittedleaf-demes. A huge cost reduction is caused by thedynamical adaptation of the accuracy of the self-adaptivehp goal-oriented Finite Element Method applied asforward solver (see Algorithm 5). This accuracy isadapted to the inverse error at the particular level of thepopulation tree, which reduces the cost of the particularfitness evaluation for the root and branch demes.

Simulations show the main HMS advantages overthe standard global optimization methods (EvolutionaryAlgorithm and multi-start) for the Ackley and threeGaussian peaks benchmarks that exhibit typical difficultycharacteristics of the inverse parametric problems(objective multimodality and ill conditioning). Anothercomputational example shows the HMS behavior for thedifficult, real-life engineering problem of magnetotelluricdata inversion, originated by the oil deposit investigation.


Apart the simulation study of the HMS efficiency, thetheoretical verification of the common error scaling wasperformed, which is necessary for the computational costreduction shown in Proposition 1 and Remark 1.

The tests we performed was of the ”finite budget”type. We intended to provide HMS with enough resourcesto obtain satisfactory solutions, and then check the resultsof standard global optimization methods running withinthe same computational budget. In our case, the budgetis determined by the forward server calls, because incase of parametric inverse problems, their total CPUtime exceeds the cost of all other operations by not lessthan two orders of magnitude. For instance, the leastexpensive, 70%-accuracy MT solver run lasted about45 seconds, whereas the most expensive of the otheroperations, i.e. the objective deterioration (including theclustering), took a fraction of a second. Every MAglobal loop (see Subsection 1.1) featured much morethan 10 solver calls at various accuracy levels and atmost one objective deterioration request. Therefore, thecost of the other operations does not affect the methodcomparison results noticeably. On the other hand, allmethods were implemented and run in a distributedenvironment, so the wall time of each computationalrun completion depended significantly on the backgroundload of all nodes, scheduling policy, etc. Because themain goal of the paper was to analyze the algorithmicaspects of the proposed strategy, we have not refinedthe implementation and scheduling policies. It will bethe subject of our future research based on our formerresult concerning agent-based ”diffusive scheduling” indistributed environments (Grochowski et al., 2006).

AcknowledgmentThe work presented in this paper has been partiallysupported by Polish National Science Center grantno. DEC-2011/03/B/ST6/01393. The last two authorswere partially funded by the Project of the SpanishMinistry of Economy and Competitiveness with referenceMTM2013-40824-P, the BCAM ”Severo Ochoa”accreditation of excellence SEV-2013-0323, the CYTED2011 project 712RT0449, and the Basque GovernmentConsolidated Research Group Grant IT649-13 on”Mathematical Modeling, Simulation, and IndustrialApplications (M2SI)”.

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Maciej Smołka received his Ph.D. (2000) inthe area of shape optimization at the JagiellonianUniversity in Krakow. Until 2012 he worked inthe Chair of Optimization and Control Theoryat the Jagiellonian University. Currently he isa member of the Adaptive Algorithms and Sys-tems Group in the Department of Computer Sci-ence, AGH University of Science and Technol-ogy, Krakow. His current research interests in-clude stochastic modeling of computational sys-

tems, application of evolutionary and hybrid algorithms in solving in-verse problems and computational multi-agent systems.

Robert Schaefer is a Full professor at the De-partment of Computer Science, Faculty of Com-puter Science Electronics and Telecommunica-tion, AGH University of Science and Technology.Author and co-author of about 180 books, papersand conference contributions, in particular authorof the book Foundation of Genetic Global Opti-mization (Springer 2007). General chair of thePPSN 2010 Conference and Steering CommitteeMember of the PPSN Series. PC member and co-

chair of more then 100 scientific conferences in computational sciencesand artificial intelligence. Recent research areas: algorithms of oil andgas surveying, genetic and memetic algorithms, computing multi-agentsystems, adaptive algorithms of solving forward and inverse problemsfor PDEs. Former research: modeling of the blood flow in arteries, mod-eling of nonlinear flow in porous media.

Maciej Paszynski received his Ph.D. (2003) inmathematics with applications to computer sci-ence from the Jagiellonian University, Krakow,Poland and habilitation (2010) in computer sci-ence from the AGH University of Science andTechnology, Krakow, Poland. In 2003–2005,he worked as a postdoctoral fellow at the Insti-tute for Computational Engineering and Sciences(ICES) at The University of Texas at Austin. Insummer 2006 and 2007 he worked as a postdoc-

toral fellow at the Department of Petroleum and Geosystems Engineer-ing at The University of Texas at Austin. Since 2007 he is a frequentvisitor at ICES at The University of Texas at Austin, USA, at King Ab-dullah University of Science and Technology (KAUST), Saudi Arabia,at Basque Center for Applied Mathematics (BCAM), and at the Depart-ment of Mathematics, Universidad del Pais Vasco, Bilbao, Spain. Since2013 he is a Vice-Director of the Department of Computer Science, AGHUniversity, Krakow and holds a university professor position there. Hisresearch interests include parallel self-adaptive hp finite element method,parallel forward solvers, models of concurrency and isogeometric meth-ods.

David Pardo is a Research Professor at Iker-basque, the University of the Basque CountryUPV/EHU, and the Basque Center for AppliedMathematics (BCAM). He has published over120 research articles and he has given over 200presentations. In 2011, he was awarded as thebest Spanish young researcher in Applied Mathe-matics by the Spanish Society of Applied Mathe-matics (SEMA). He leads several national and in-ternational research projects, as well as research

contracts with national and international companies. He is now the PI ofthe research group on Mathematical Modeling, Simulation, and Indus-trial Applications (M2SI). His research interests include computationalelectromagnetics, petroleum-engineering applications (borehole simula-tions), adaptive finite-element and discontinuous Petrov-Galerkin meth-ods, multigrid solvers, image restoration algorithms, and multiphysicsand inverse problems.

Julen Alvarez-Aramberri completed his degreein physics at the University of Basque Country in2007. After that, he studied a M.S. in Quantita-tive Finance and a M.S. in Mathematical Mod-elization, Statistics and Computation at the Uni-versity of Basque Country. Since Sep. 2011he is a Ph.D. student within the ”Department ofApplied Mathematics, Statistics and OperationalResearch” at the University of the Basque Coun-try under the supervision of David Pardo and

within the ”Project Team MAGIQUE-3D” at the University of Pau un-der the supervision of Helene Barucq. He currently is in the last yearof his Ph.D. working at the Basque Center for Applied Mathematics(BCAM). His Ph.D is expected to be finished on February 2015. Hisresearch interest include the efficient implementation of the numericalschemes, methods, and tools. In particular, he is focused in solving thedirect and inverse problems arising in the application of the magnetotel-luric technique, which is used to retrieve information about the resistivitydistribution of the Earth’s subsurface.

an agent–oriented hierarchic strategy for solving inverse problems

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