Abstract: Inverse radiative transfer problems have several relevant applications, in different areas such as engineering, environmental sciences, and medicine [1]. Some outstanding examples are parameter and function estimation for global climate models, hydrologic optics and medical tomography. When formulated implicitly inverse problems are usually written as optimization problems. Several heuristics that mimic natural behaviors have been proposed for the solution of optimization problems. In particular some of the most recent algorithms, classified within the field of swarm intelligence, are based social insects like bees, ants, etc. In the late nineties the Ant Colony System (ACS) was applied successfully for the solution of combinatorial optimization problems, and more recently it has been proposed for the solution of some specific inverse problems associated with the estimation of real parameters [2][3]. The ACS is a method that employs a metaheuristic based on the collective behaviour of ants choosing a path between the nest and the food source. Each ant marks its path with an amount of pheromone, and the most marked paths are further employed by other ants as reference. In the generation of each ant of a population, the ACS must randomly assign values to the unknowns. These values compose a candidate solution and can be viewed as a path associated to that particular ant. Each ant must be evaluated. The ACS was formerly proposed for the Traveling Salesman Problem (TSP) and other graph like problems that use visibility information in the choice of each unknown. Inverse problems associated to the reconstruction of smooth profiles may include such information in the generation of ants in order to perform a pre-selection of ants. This scheme can be viewed as a kind of pre-regularization and was proposed in a previous work [2]. Hybrid approaches coupling stochastic and deterministic methods are becoming popular. Particularly, some inverse radiative transfer problems of radiative properties estimation have been solved using Simulated Annealing or a Genetic Algorithm with the Levenberg-Marquardt (LM) method [4] [5]. These works refer to the reconstruction of optical thickness, single scattering albedo, and diffuse reflectivities in a one-dimensional plane-parallel homogeneous medium. This work shows the use of a pure ACS and a hybridization of the ACS with the LM method in a similar inverse radiative transfer problem, but considering a spatial dependent albedo. The ACS based pre-regularization scheme is applied for the albedo in reconstructions using noiseless and noisy data. The direct radiative transfer problem is modeled by the linear version of the Boltzmann equation using a discrete ordinates method combined with the finite difference method. Test case results are discussed and demonstrate the feasibility of the use of the proposed methodology.

References

1. H.F. Campos Velho, F.M. Ramos, E.S. Chalhoub, S. Stephany, J.C. Carvalho, F.L. Souza, Inverse problems in space science and technology, Proceedings of the 5th International Conference on Industrial and Applied Mathematics - ICIAM, Sidney, 2003.

2. A.J. Preto, H.F. Campos Velho, J.C. Becceneri, M. Fabbri, N.N. Arai, R.P. Souto, S. Stephany, A new regularization technique for an ant-colony based inverse solver applied to a crystal growth problem, 13th Inverse Problems in Engineering Seminar (IPES-2004), Cincinnati, 14-15 June, 2004, pp. 147-153.

3. R.P. Souto, H.F. Campos Velho, S. Stephany, S. Sandri, Reconstruction of chlorophyll concentration profile in offshore ocean water using ant colony system, First Hybrid Metaheuristics (HM-2004), Valencia, 22-23 August, 2004, pp. 19-24.

4. A.J. Silva Neto, F. J. C. P. Soeiro, Estimation of phase function of anisotropic scattering with a combination of gradient based and stochastic global optimization methods, 5th World Congress on Computational Mechanics, Vienna, 2002.

5. A.J. Silva Neto, F.J.C.P. Soeiro, Solution of implicitly formulated inverse heat transfer problems with hybrid methods, Mini - Symposium Inverse Problems from Thermal / Fluids and Solid Mechanics Application - 2nd MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, USA, 2003.