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Optimization methods based on natural processes for inverse problems E. F. P. Luz, H. F. de Campos Velho and J. C. Becceneri Laboratory for Computing and Applied Mathematics - LAC Brazilian National Institute for Space Research - INPE C. Postal 515 – 12245-970 – São José dos Campos - SP BRAZIL E-mail: {eduardo.luz, haroldo, becceneri}@lac.inpe.br Keywords: metaheuristics, inverse problems, particle swarm optimization, ant colony optimization. Atmospheric pollution has worried
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  Optimization methods based on natural processes for inverse problems E. F. P. Luz, H. F. de Campos Velho and J. C. Becceneri Laboratory for Computing and Applied Mathematics - LAC Brazilian National Institute for Space Research - INPE C. Postal 515 – 12245-970 – São José dos Campos - SP BRAZIL E-mail: {eduardo.luz, haroldo, becceneri}@lac.inpe.br Keywords: metaheuristics, inverse problems, particle swarm optimization, ant colony optimization. Atmospheric pollution has worried scientists and world leaders for a long time. Today, ecological legislation in several countries is restricting the emission of pollutant gases and there is also a discussion in the international community about the emission of gases related to the greenhouse effect. One of this effort/discussion is addressed in the Kyoto protocol. Techniques based on inverse problem methodologies are been used to identify sources and sinks of pollutant gases in a very efficient way. Considering the forward mathematical model expressed by equation (1), the inverse model is represented by equation (2): ( )  Auf  =  (1) ( ) 1  Afu − =  (2) Figure bellow gives a schematic representation of direct/inverse problems. Outline for direct and inverse problems. The determination of pollution emission sources can be formulated as a non-linear optimization problem. The objective function is the square difference of the data from a mathematical model (pollutant dispersion model) and the data obtained by experimental sensor, associated with a regularization operator (Roberti, 2005; Timm et al., 2005). The location and/or the intensity of emitting sources are the unknown parameters in this inverse  problem. The inverse solution is obtained by minimizing the following functional: ( ) ( ) ( ) 21 , m ExpMod  jj j  FQCCQQ α α  =  = − + Ω  ∑  (3) where C   Exp  and C   Mod   are the observed pollutant concentration at sensor-  j  and pollutant concentration computed from the dispersion model, respectively, Ω (.) is the regularization operator, α   the regularization parameter, and Q  is a vector representing the pollutant sources/sinks inside the physical domain. The minimization of the above functional can be done by several ways (Ramos et al., 1999; Campos Velho et al., 2000). Here we are proposing  the use of two nature inspired metaheuristcs: Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO). Both based on natural behaviors observed in birds and fishes for PSO, and ants for ACO. These metaheuristcs are highly based on self-organization properties, due to some dynamical mechanisms that show some structures/patterns global emerging from the local interactions. PSO is a recent heuristic search method inspired by the collaborative behavior of biological populations, mainly  birds, as said before. For this technique, the flock of birds is represented in a n -dimensional space (Kennedy and Eberhart, 2001). The position of each bird in this space is represented by its Cartesian equivalent, and its speed for each axis at that Cartesian point. The speed is calculated by Equation (4), and the position is expressed by Equation (5). ( ) ( ) 112132 kkkk iiiii vcvcrandpbestscrandgbests + = + − + −  (4) 11 kkk iii  ssv + + = +  (5) The parameters c 1 , c 2  and c 3  express the trust for the particle (or bird) in themselves, in their past, and in the flock knowledge, respectively. In this way, the bird is able to construct a candidate solution for the problem, based on the interaction with the flock members. For the ACO, a pheromone trail is used to construct the solution, which can be represented in a graph way (Bonabeau et al., 1999; Dsrco and Stützle, 2004). The artificial ants construct their tour based on equation (6) ( )( ) ( ) ( )  [ ] ( ) k i ijijk ijilil lJ  t  pt t  α β α  β  τ η τ η  ∈       =   ∑  (6) where  p ij  represents the probability of selecting a branch, τ   is the amount of pheromone, η  is the inverse of distance and α  and  β   regulates the weight for the solution based on pheromone or the distance. Initial results for the PSO are nice enough to indicate that PSO is a good strategy for solving this optimization  problem. Now, the goal is to select the PSO parameters to get a better inverse solution. The final objective is to compute the inverse solution employing the PSO and ACO algorithms carrying out a comparison between them. REFERENCES Bonabeau, E.; Dsrco, M.; Theraulaz, G. (1999), Swarm intelligence , New York: Oxford University Press. Campos Velho, H. F.; Moraes, M. R.; Ramos, F. M.; Degrazia, D. A. (2000),  An Automatic Methodology for  Estimating Eddy Diffusivity from Experimental Data . Il Nuovo Cimento, 23 C (1), 65-84. Dsrco, M.; Stützle, T. (2004),  Ant Colony Optimization . Cambridge: The MIT Press. Kennedy, J.; Eberhart, R. C. (2001), Swarm intelligence . San Francisco: Morgan Kaufmann Publishers. Ramos, F. M.; Campos Velho, H. F.; Carvalho, J. C.; Ferreira, N. J. (1999),  Novel Approaches on Entropic  Regularization , Inverse Problems, 15 (5), 1139-1148. Roberti, D. R. (2005),  Problemas Inversos em Física da Atmosfera . Santa Maria: UFSM, 2005. 141 p. Tese (Doutorado em Física) – Centro de Ciências Naturais e Exatas, Universidade Federal de Santa Maria. Timm, A. U.; Roberti, D. R.; Degrazia, G. A. (2005) Comparação de modelos de difusão unidimensional para uma fonte pontual instantânea . Especial Micrometeorologia. Ciência e Natura – UFSM.
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