Revista Mexicana de Física 42, Suplemento 1 (1996) 203-215

Evolving an energy dependent optieal rnodeldeseription of heavy-ion elastie seattering

K. MICIIAELIANInstituto de Física, Universidad Nacional Autónoma de México

Apartado postal 20-364, 01000 México, D.F., México

ABSTRACT.We present the application of a genetic algorithm to the problem of determining anenergy dependent optical model description of heavy-ion elastic scattering. The problem requiresa search for the global best optical model potential and its energy dependence in a very rugged 12dimensional parameter space of complex topographical features wíth many local mínima. Randomsolutions are created in the first generatíon. The fitness of a solution is related to the X2 fit ofthe calculated differential cross sections with the experimental data. Best fit solutions are evolvedthrough cross over and mutation following the biological example. This genetic algorithm approachcombined with local gradient minimization is shown to provide a global, complete and extremelyefficient search method, well adapted to complex fitness landscapes. These characteristics, com-bined with the facility of application, should make it the search method of choice for a wide varietyof problems from nuclear physics.

RESUMEN. Presentamos la aplicación de un algoritmo genético al problema de determinar una de-scripción de modelo óptico dependiente de la energía para la dispersión elástica de iones pesados.El problema requiere de la búsqueda de los mejores parámetros globales del potencial óptico enun espacio paramétrico de 12 dimensiones muy accidentado, con formas topológicas complicadas ycon muchos mínimos locales, En la primera generación se crean soluciones al azar. La idoneidad delas soluciones se relaciona con un ajuste de X2 de la sección eficaz experimental. Las soluciones másaptas se hacen evolucionar a través de su cruzamiento y mutación siguiendo el ejemplo biológico.Se muestra que esta técnica de algoritmo genético, combinado con una minimización por gradientelocal constituye un método de búsqueda global, completo y extremadamente eficiente para superfi-cies de adaptabilidad complicadas. Estas características, combinadas con su fácil aplicación, debenhacer de este método de búsqueda la alternativa natural para una gran variedad de problemas enfísica nuclear.

PACS: 25.70.Bc; 02.60.Pn; 02.70.Lq

l. INTRODUCTION

Optical model potential descriptions of heavy-ion elastic scattering are a means for de-termining a mean field cffcctive intcraction for a complieated many-body projeetile-targetseattering. A reaH,tie goal of su eh modeling is to learn about the appropriateness of effee-tive nucleon-nuclcon (N-N) interaetions within the nuclear medium by, for example, eom-paring the experimental diffcrential eross seetions with optieal model ealculations ineor-porating effeetive N-N folding-model potentials [1]. Anothcr important applieation is theuse of such potentials in gcnerating distortcd wavcs for inelastic and reaction calculations.Optical model potential determinations are eomplieated by the faet that heavy-ion in-

teraetions are, in general, strongly absorptivc. Therefore, elastie seattering is only sensitive

203

204 K. MICHAELIAN

to the details of the potential at large radii [2]. For the lighter heavy-ion systems, whereweak absorption prevails and refractive effects are observed, discrete families of equallygood potentials are found which are similar in the region of the nuclear surface but whichdiffer in the central region. These potentials reproduce the gross structures in the angulardistributions such as the location of consecutive Airy minima and maxima at the appro-priate angles [3]. At higher energies, with the appearance of far side nuclear rainbowswithin the angular region accessible to measurement, the discrete ambiguity disappears.Jt has been shown [4,5] that for light heavy-ion elastic scattering, the discrete ambiguitiesat low energies can be resolved by requiring continuity with parameters of the potentialdetermined at higher energies or with other targets.However, the ambiguity, combined with the size and complexity of the parameter search

space, makes the determination of a single best potential and its energy dependence adifficult as well as tedious and time consuming project. The traditional approach hasbeen to evaluate each incident ion energy independently by fixing one or more of thepotential parameters on a grid and doing g:adient or conjugate gradient searches on theremaining free parameters. A fina! minimization, with all parameters free, is carried outon the results. A collection of the best snch locally minimized grid points for each energyare then compared with those of other energies in the hope of observing some systematics.It has been observed that if the grid is not sufficiently fine, important potential minimamay be missed during local minimization and the search rendered incomplete [5]. Theglobal problem is aided if higher energy data are available, where the refractive regio n isprobed, but even in the best of such cases, a complete analysis of the data for a givenprojectile-target combination takes many person-months or more of dedicated work. Thereliable determination of potentials for heavy-ion elastic scattering would clearly benefitfrom an automated approach providing a global, complete and efficient search.As will be demonstrated, genetic algorithms are endowed with just such capabilities.

Following the biological examp!e, they evolve, through fitness based selection, a populationof "genetic" strings coding solutions to a specific problelll using genetic operators suchas cross over of the genes (in sexual reproductioll) and mutation. Symbolically, a geneticalgorithlll proceeds as follows:

1. generate a random initia! population of solutions S(i);2. eva!uate the fitness of S( i);3. generate a new population (second generation) ofsolutions S(i+ 1), through crossover and mutation of the best fit solutions of the previoue generation;

4. evaluate the fitness of S(i + 1);5. repeat steps 3 and 4 for a number of generations until an adequate solution isfound.

Genetic algorithms were /irst proposed by Holland in 1960's in relation to researchon cellular automata [6]. They have since been successfully applied to such diverse andcOlllplex problems as, air traffic control [71, image recognition 181,training of neural net-works [9]' and predicting the economy iJ 01. In general, it has been observed that as thecomplexity of the .ea:ch spaee iIIereases, genetic algorithms enjoy progressively greatersuecess over the more traditional so ealled "illtelligent" seareh methods such as simulatedannealing and neural networks. Although gene rally ullknown within the nuclear physicscommunity, genetie algorithllls have been sueeessfully applied to experimental event recon-

EVOLVINGAN ENERGYDEPENDENTOPTlCAL MODEL.. . 205

struction problems in high energy particle physics [11]. In nuclear physics, where problemsare normally complex because of the many-body nature of the nucleus and because of theinherent complexity of the nuclear force, genetic algorithms may provide a valuable tool.Also, their generality of application and technical simplicity make them an attractivechoice.

The following section describes the construction of a gene tic algorithm through its appli-cation to the determination of an optical model potential for 12C_12C elastic scattering. InSect. 3 we present the results of this application and compare it to those obtained throughtraditional analysis of the same experimental data seto

2. A GENETIC ALGORITlIM ApPROACII TO AN OPTICAL MODEL POTENTIAL

Our specific problem is the determination of an appropriate energy dependent opticalmodel potential for 12C_12C elastic scattering. The experimental data for the differentialcross sections as a function of center of mass scattering angle were taken from the workof Stokstad et al. [121. Nine different data sets for incident energies ranging from 71 to127 MeV were used.

The analysis of these data by Stokstad el al. concluded that no adequate Woods-Saxonform for the real part of the optical model potential could be found. However, a satisfactoryfolding-model real potential was found with a central depth of about -450 MeV [12].Recently the same data were re-analyzed by Brandan el al. [S] through a more completesearch which also allowed an energy dependence of the potential parameters. It was foundthat the data could be described well by a Woods-Saxon form, slightly energy dependent,real potential with a central depth of about -300 MeV, and a weakly energy dependentimaginary potential.

\Vithin the optical model framework, the differential cross section of a reaction is deter-mined from the solution of the Schriidinger equation for the equi'¡alent one-body problemof a particle of the given incident energy in the region of a complex attractive potential

-[V(r) + iW(r)] + Udr, Rc),

where V(r) represents the nuclear prompt elastic scattering, and where W(r) is responsiblefor all nuclear reaction processes not associated with the prompt elastic. Udr, Re) is theCoulomb potential for two uniformly charged spheres of radii, Rc = 3.17 fm. Our modelassumes nuclear potentials of Woods-Saxon form,

VV(r) = ------

1+ exp[(r - Rv)/av]

and

WW(r) = ,

1+ exp[(r - Rw)/awJ

206 K. MICHAELlAN

where Rv,w = TV,W . 121/3. \Ve also assume that the potential parameters may have anenergy dependence, and that this energy dependence, at least in lirst approximation, islinear. The six geometrical parameters can then be written,

v = VD+ VE' E, TV = Tov + TEv . E, av = aov + aEv . E,

\V = \VD + \VE' E, TW = Tow + TEw . E, aw = aow + aEw . E.

Our problem then is to lind, in an e!!icient manner, sets of the 12 potential parame-ters Vo, VE, TOV1 rEv' aOV1 aEVl Wo, WE, TOW1 TEw! aow' aEw which give good fittingdifferential cross sections to the data at all energies and angles. \Ve begin by generatingrandomly an initial population of trial solutions of the independent variables (the poten-tial parameters) in the form of a genetic codeo Choosing a binary representation for eachof the variables (because of the computational facility to manipulate bit strings) solutionsare created of the form

where for display purposes each variable is represented by only 3 bits. In the real case,each variable occupies 10 bits, implying 120 bits per solution. The number of bits pervariable determines the "coarseness" of the genetic search.Two hundred such trial solutions were chosen at random from the search space:

200 < VD < 500 -1. < VE < 2..4 < Tov < 1.2 -.001 < TEv < .002.2 < aov < 1.2 -.0005 < aEv < .0005-10. < \VD < 10. .2 < \VE < .5O. < 1"ow < 3. .001 < TEw < .003-.1 < aow < 1.9 .0005 < uEw < .0(l15

Given the upper and lower energies of the Stokstad el al. data [121 of 71 and 127 MeVrespectively, these ranges correspond approximatcly lo the following generous allowanceson the parameler values:

100. < V < 700.,0.0 < IV < 60.0,

0.3 < TV < 1.4,0.1 < TW < 2.0,

0.0 < av < 1.5,0.0 < aw < 2.0.

The main genetic program called, as a subprocess, the code PTOLEMY 1131to solve theSchriidinger e<¡uation and determine the differenlial cross seclions from the inpul potentialparameter sets (lhe lrial solutions). PTOLE;"IY was run in the mulliple input lile modewith each input file corresponding to an experimental data set of difIerent incident energy.The X2 per data point lit of the PTOLEMY calculation with the data, averaged over alldata points at all energies, provided a measure of the litness of the solution.Of the 200 initial solnlions in the lirst generation, lhe fitlest 50% were selecled for the

"gene pool" for reproduction, the otber balf were discard"d. The mating procedure starled

EVOLVING AN ENERGY DEPENDENT OPTICAL MODEL... 207

with a random selection of 100 two parent pairs from the gene pool. Each pair prod ucedtwo offspring solutions according to the following prescription which was found to beefficient at converging on fit solutions while maintaining diversity among the populationat large. \Ve discuss this prescription in more detail in Sect. 4. Each parent pair was thusgiven,

1. A 10% probability of faithful reproduction in which the offspring solutions areidentical to their parents,

parents offspring11110 11 O 11--->11110 11 O 11

1100 11 O 111---> 110 O 11 O 111(where the 120 bit solutions have been reduced to 8 bits for display purposes).

2. A 3% probability of point mutated reproduction in which one bit chosen at randomof each parent solution was flipped

parents offspring

11 Ó;O 11 O 11--->111[0;0 11 O 11. . . .1100 11 O~l;ll---> 110 O 11 0:0;11

3. An 87% próoability of cross over reproduction in which the two parent solutionswere separated at a boundary of two genes (potential parameters) chosen at ran-dom and respective portions swapped to produce new offspring

parents offspring

:-------------! - - - - - -,!I111010111:: ••••••••••••• J

~1O;;210 ti1::__ _:.J

This procedure is analogous to the biological process of crossing over of the geneswhich occurs frequently between two homologous chromosomes during meiosis toform the gametes.

The genetic algorithm was found to be rather robust in that altering this prescriptionsomewhat did not noticeably affect its performance.

In addition to the aboye reproduction prescription, we took the so called "elitist" ap-proach [14] by ensuring that always the very best individual reproduced an exact copy ofitself for the gene pool of the following generation. \Vith the above procedure, reproduc-tion is constrained to keep the total population constant from generation to generation,which is a practical necessity of the algorithm.

\Ve repeat this procedure creating a third generation by reproduction of the best fit 50%of the second generation and so on for a number of generations until little improvementis observed in the fitness from one generation to the next. The average chi squared perdata point X2 IN (of the fittest 50% of the solutions of each generation) for the fit of thecalculated differential cross section to the data of Ref. [12], is plotted as a function ofthe generation number in Fig. 1. For the early generations, the increase in average fitness(decrease in X2 IN) is exponential with generation number. The rate of average fitnessimprovement of the population decreases after generation number 4. This is a consequence

208 K. MICHAELIAN

106

105Z'-..... 104N

><103

102

O 2 4 6 8 10Generatian Number

FIGURE 1. Average (of the 50% most lit solutions) X2 per dala point of the ealculated differenlialeross seetion with 9 data sets of different energy, taken from referenee [121,as a funelion of lhegeneration number.

of the fael that the plotted X2 IN is an average over the best 100 solutions and thatapproaehing the optimal solution requires a reduetion in the seareh spaee whieh wouldimply a loss in diversity of these 100. In the biologieal world, this loss of diversity wouldbe devastating to populations in a ehanging environmeut. For this reason, on average,only near optimal individuals are found in nature. In our particular problem, we requirediversity at the beginning to prevent trapping at a local minimum and to obtain as manydistinct solutions (parameter sets) as possible. However, since our definition of fitness isfixed, we would.like results as near as possible to exact solutions. It is thus more efficientto stop the genetic evolution arter some generation at which the returns are diminishingand to use a more suitable traditional local minimization procedure applied to a numberof distinct individual solutions obtained through genetic evolution. Arter 10 generationsof genetic evolution we therefore applied PTOLEMY's internal gradient minimizer tominimize local!y the best 20 distinct solutions of the 12 potential parameters.

3. RESULTS

Three of the best parameter sets obtained after one run of genetic evolution and localminimization with PTOLEMY's internal gradient minimizer are given in Table 1. Thesevalues were obtained by fitting to al! 9 different energy data sets simultaneously. Usingthese parameters as a starting point, a local 6 parameter PTOLEMY minimization wasdone, with al! parameters free, for each energy independently. The results for parameterset 2 of Table 1 are plotted in Fig. 2. From the figure it can be observed that for the

EVOLVINGANENERGYDEPENDENTOPTICALMODEL... 209

TABLE I. The three best optica! model potential parameter sets and their resulting X2 IN (aver-aged over aH data points) obtained after 10 generations of genetic evolution and a local gradientminimization.

Se' Vo VE TOv TEV aov aEv Wo WE TOW TEw UOw aEw X'/N1 547.98 -1.596 0.374 0.002 1.006 -0.001 -4.402 0.193 1.508 -0.003 0.259 0.003 25.72 497.36 -1.842 0.314 0.002 1.077 -0.001 -3.233 0.168 1.493 -0.002 0.300 0.002 23.83 440.03 -1.980 0.220 0.003 1.191 -0.002 -2.296 0.142 1.435 -0.002 0.457 0.000 23.4

real part of the potential there are apparently two possible smooth energy dependenciesfor the depth, radius and diffuseness parameters while only one solution is found for thecorresponding imaginary parameters. To resolve the inconsistency in the real part we firstassumed that the energy dependence was determined by the line with the lesser slope.Values foC"the depth of the potential which were far from this line (at the energies 83.3,93.8 and 98.2 MeV) were fixed on the line and a local minimization was then done onthe other 5 parameters. It was found that, with only small inereases in the X2 values, allparameters then fall on only one line and give a smooth energy dependence as demon-strated by the circles in Fig. 3. The lines with the larger slope in Fig. 2 arise beca use of thediscrete ambiguities connected through rather shallow valleys. The points defining theselines represent distinct members of the various families of ambiguous potentials. The factthat the lines with the large slope do not represent a single family was established fromthe impossibility of obtaining reasonable values of X2 IN by forcing all (at each energy)real potential depths to this lineoApplying the same procedure to the first para meter set of Table 1 a second valid poten-

tial and energy dependen ce was found which gave roughly the same final X2 per data pointas the first set (of about 12 overall when each energy was minimized independently). Thispotential and its energy dependence is represented by the diamonds in Fig. 3. A similarprocedure applied to the third parameter set of Table 1 gave the squares of Fig. 3. Apply-ing a linear regression to these data for the three parameter sets we found the followingcorresponding linear relations:Parameter set 1:

V(MeV) = 376.36 - 0.4417E1ab,

rv(fm) = 0.5507 + 6.5670 x 10-4 E1ab,

av(fm) = 0.8251 + 1.4931 x 10-4 Elab,

W(MeV) = -4.838 + 0.1937 E1ab,rw(fm) = 1.525 - 2.668 x 10-3 E1ab,

aw(fm) = 0.1697 + 3.1695 x 10-3 Elab'

Parameter set 2:

V(MeV) = 304.90 - 0.3502Elab,

rv(fm) = 0.5750 + 3.9904 x 10-4 Elab,

210 K. MrCHAELIAN

80 100 120E [MeV]

60 100 120

E [MeV]

o0.6

1.1

0.9

0.6

0.7

>

'"

E 1.0.:::.

80 100 120

E [MeV]

o

60 100 120

E [MeV]

o

0.4

0.7

0.6

0.3

1.1

1.3

1.0

~E.::::. 0.5>...

E.:::. 1.2

•...

80 100 120

E [MeV]

80 100 120

E [MeV]

o

o

300

500

7.5

5.0

350

450

250

17.5

15.0

>

~ 10.0

,.......,400>"::>;

>'~ 12.~

FIGURE 2. Parameters of lhe best fitting potential for data at the 9 incident energies obtainedafter 10 generations of genetic evolution, a local PTOLEMY gradient minimization and a finalminimization for each energy independently. The Iines through the data points represent possibleenergy dependenties of the parameters (see text).

av(fm) = 0.8432 + 6.0387 x 10-4 E1ab,

W(MeV) -5.4178 + 0.1815E1ab,

Tw(fm) 1.5327 - 2.5448 x 10-3 E1ab,

aw(fm) = 0.1932 + 2.9163 x 10-3 E1ab.

Parameter sel 3:

V(MeV) = 285.67 - 0.67506E'ab,

Tv(fm) = 0.51823 + 8.0121 x 10-4 E1ab,

av(fm) = 0.8689 + 1.1186 x 10-3 E1ab,

W(MeV) = -5.6627 + 0.16945El.b,

EVOLVING AN ENERGY DEPENDENT OPTICAL MODEL... 211

600 0.8 1.2

400 0.7 1.0 00O

O O O

'> O 0000 O !=b O

8 •~ 00° •e e •'" o

¡jª@o

::li 300 O .:::. 0.6 Oc .:=. 0.8 O~ eP oCffi 000 > >> • ... O "¡}locR:J 00200 O 0.6 0.6

100 0.4 0.460 100 120 140 160 60 100 120 140 180 60 100 120 140 160

E [MeV] E [McV] E [MeV]30 1.6 0.6

26 0.7 •1.4 O

20 <> O~ 0.6> 083 E 1.3 ~

O~• <> O e'" <>::li 16 00 .:::. 00 .:::. 0.5 00 O<>g> • O •O ~ 1.2 oog »

o~¡o,o~ "10 O 0.4 oO

/l 1.1 O6 0.3

O 1.0 0.260 100 120 140 160 60 100 120 140 180 60 100 120 140 160

E [MeV] E [MeV] E [MeV]

FIGURE 3. The same as for Fig. 2 except that those parameter sets with V far from the linewith the lesser slope in Fig. 2 were forced to the line and a local minimization done OIl theremaining five parameters. The diamonds, circles ami squares represent the energy independentminimization of the !irst, second and third parameter sets of Table I. The salid circles are thevalues obtained at higher energies [15] where the Camilies oC ambiguous potentials are resolved intoa single best seto

rtv(fm) = 1.57588 - 2.6780 x 10-3 Elab,

"tv(fm) = 0.19092 + 2.6402 x 10-3 Elab'

The second set of e<¡nations are very similar to ones fonnd for the same data whenanalysed throngh an extensive grid and gradient scarch [51;

V(McV) = 386.2 - 0.868E1ab,

rv(fm) = 0.583,

"v(fm) = 0.902,

lV(MeV) = -8.19 + 0.208EI.I"

212 K. MICHAELIAN

rw(fm) = 1.833 - 5.500 x 10-3 EI•h,

aw(fm) = -0.079 + 5.700 x 10-3 EI.h'

The differences between this set and our second set may be attributed to two facts: First,the reduced real radii were fixed to the value of 0.583 fm for all energies in the gridsearch [5] while they were free to obtain an energy dependence in the present geneticsearch. Second, in the grid search, there remained an ambiguity in the value of the realdepth of the potential at the lowest two energies where two potentials seemed to fulfill therequirement of continuity [5]. The set of equations aboye were obtained from a fit to thepotentials with greater depth. In our genetic search, no such ambiguity was found and ourreal depths at these energies corresponded to the potentials with lower depth found in thegrid search. The equations of reference [5] and our second set join smoothly to scatteringdata taken at higher energies [15] where the discrete ambiguity disappears (see Fig. 3).In Fig. 4 are plotted the angular distributions of the original data of Stokstad el al. at

nine different incident energies and the best fit to these data, which determined our secondset of equations for the parameters. Our fits are both qualitatively and quantitativelysimilar to those obtained through the extensive grid search.The first and third sets of equations aboye represent other distinct families of potentials

which were hinted at, but not determined, in the analysis of rcference [5]. They fit thedata just as well as the second set in the energy region of the data analysed here but donot provide the same continuity with the data taken at higher energy (Fig. 3). Variousother good fitting families of potentials were found in the same single fUn of the geneticalgorithm. These potentials were not investigated further as their energy dependencieswere not in agreement with the data at higher energies.To obtain these best final sets of the 12 potential parameters through genetic evolution

and local gradient minimization, approximately 2500 solutions were tested which requiredless than one hour continuous CPU time on a DEC 3000 ALPHA work station. In theprevious manual grid and gradient search of this same data set [5]' as many solutionswere tested on each energy alone and the manual procedure took many person-months ofdedicated work.

4. DISCUSSION ANO CONCLUSIONS

The genetic algorithm is an "intelligent" global optimization method. It begins by globallysampling the entire solution space and continually narrows the search to smaller andsmaller regions as it learns about the fitness of the sampled regions. Information learnedabout a problem is stored in the genetic codes of the individuals that make up the genepool of a given generation.As for biological evolution, genetic diversity is found to be very important. Beginning

with a large initial population set is thus more resource efficient in finding a solutionthan many evolutive generations. It is also found that cross over is a more useful force ofeffective change than simple point mutation, which, if it occurs excessively, can depletethe' population of useful characteristics. The prescription defining the amount of cross

EVOLVING AN ENERGY DEPENDENT OPTICAL MODEL... 213

lO. 100

10-1 10-1

•~IO-2 10-2b

10-3 10-3

100 100

10-1 10-1

10-2 10 .. 2

10-3 10-3

100 100

10-1 10-1

10-2 10-2

10-3 10-3

100 100

10-1 10-1

10-2 10-2

10-3 10-3

100O 20 40

9 Cv.6O 60 100

10-1

10-2

10-3

FIGURE 4. The fit of the ealculated angular distributions obtaincd from the sccond parameter setcompared to the experimental data of Stokstad et al. 112].

over and mutation used in Sect. 2 refleets some of these coneerns. In general, the optimaldivision between eross over and mutation and the trade off between population size andthe number of generations, depends on the particular problem and its eomplexity. Sorneresults for specifie problems have been analyzed 116,14] but the general case is still beingresearehed [17].

Genetie algorithms have a number of eharaeteristies whieh endow them with consider-able advantage over traditional seareh teehniques. First, they are e!licient heeause theyoptimize the time spent h('tween looking for solntions in new regions of the seareh spaeeand exploiting the good regions aIread y fonnd. In faet, it can be shown that Darwinfitness-proportionate reproduetion assigns a mathematieally optimal alloeation of trialsto the different regions of the seareh spaee [6]. Their eflicieney is further improved by thefaet that they are implicitly paralle!. In evalnating the fitness of a genetie eode defining aparticular solution, they are also evaluating the fitness of partial solutions consisting of acontinuotls or noncontinllOUS subset of this codeo

214 K. MICHAELIAN

A second important characteristic is that genetic searches are complete in the sensethat regions of the search space are not inaccessible as is the case for searches based ona grid. Thirdly, genetic searches are global because mutation and cross over reduce thepossibility of trapping at a local minima which often happens with gradient searches incomplex spaces.Because genetic evolution is a bottom up approach, solutions to problems can be oh-

tained with little a priori knowledge of the problem domain 01' the solution space. Allthat is needed is a means of defining the fitness of a solution. This makes it more gen-eral than traditional top down approaches allowing for a multiplicity of potentially usefuluncontemplated solutions to be found.In summary, the main advantages of evolving genetic searches over other more tradi-

tional approaches are:1. Efficient -the algorithm learns about the fitness of regions of the solution spaceand Darwin fitness-proportionate reproduction assigns a mathematically optimalallocation of future trials to these regions.

2. Complete -regions of search space are not inaccessible as is the case for gridsearches.

3. Global -mutation and cross over reduce the possibility of trapping at a localminima which often happens in gradient searches.

4. Bottom up -presumed knowledge of the solution space is not required, thusremoving potentially restrictive biases.

\Ve have genetically evolved solutions, through fitness based selection, to the complexproblem of finding an optical model potential description of light heavy-ion elastic scatter-ing. It was shown that, where little a priori information is known about the solution space,this "hybrid" (genetic plus local gradient) method is more efficient and also more completethan traditional approaches. It could be applied to a host of current research topics inwhich a quick survey of all valid parameter sets wouId help with the selection of a bestpotential. Apart from the general search presented aboye for a consistent optical modeldescription of heavy-ion elastic scattering, other similar applications where this approachmight be useful are; the investigation of the "threshold anomaly" [18,19], local potentialsfor pion-nueleus scattering 120,21], and studies of exotic [22,231 and deformed [24] nuclei,such as llLi and 22Ne respectively, through heavy-ion elastic scatteriug.\Ve would like to suggest that genetically evolving solutions through fitness based se lec-

tion is a simple, general and powerful technique which could be empIoyed to solve a widevariety of complex problems in nuclear physics. Preselltly we are applying the approachto the difficult many-body problem of configuring the nuclear ground state [251.

ACKNOWLEDGMENTS

\Ve would like to thank Dr. M.E. Brandan for helpful discussions on the details of opticalmodel potential fitting ami, for a careful readiug of the manuscript. The financial sup-port of CRAY-UNAM grant number SC-003095 and use of computing facilities acquiredthrough CONACYT, graut numoer 3173E, are gratefnlly acknowledged.

EVOLVING AN ENERGY DEPENDENT OPTICAL MODEL... 215

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