Keywords: direct and inverse problems, stabilized finite element methods, Levenberg-Marquardt algorithm, leaching of pesticides modelling.

We present two different one-dimensional models describing leaching of solutes in columns of soils: a classical linear equilibrium model for transport of non volatile solutes [4]; and a non-equilibrium sorption model to represent non-equilibrium processes during transport, which includes the chemical non-equilibrium [3], or two sites models, and the physical non-equilibrium, or two regions models [5]. The state equations given by these models are basically second order linear parabolic equations, with advective, convective and reactive terms.

The inverse problem of the parameter adjustment is a challenging task. Any numerical method we use to solve the corresponding optimal control problem implies the computation of the gradient of the cost functional. Instead of a numerical approximation of this gradient, we compute it directly by solving the corresponding adjoint problem which is again a similar second order linear parabolic equation, but backward in time.

The cost functional is ill-conditioned, so we propose a Levenberg-Marquardt algorithm [6], which combine the gradient descent and Newton methods.

The state and the adjoint state equations are both unsteady linear advection-diffusion-reaction one-dimensional equations. The main difficulty in the numerical approximation of these partial differential equations, is the accurate modelling of the interaction between advection, diffusion and reaction processes. This forces the use of small time and space steps, but if we want to fit the parameters with a low computational cost we need to work with coarse meshes, so the solution is to use stabilization techniques. Moreover, if we do not know a priori the relative weight of the diffusive, convective and reactive terms, we need to guarantee a scheme stable for any regimen to assure a right parameter adjustment.

We propose a numerical scheme based on the link cutting bubble strategy (LCB) introduced by Brezzi et al. [2] for the steady problem, adapted for the non steady case in [1] where this scheme is compared to more classical stabilization techniques. The key point of this scheme is precisely that it has been designed to work in all regimes, that is, when the diffusion term is dominant, when the advection term is dominant and when the reaction term is dominant.

The numerical experiments presented here are based on real data provided by authors in [7].

1

M.I. Asensio, B. Ayuso, G. Sangalli, "Coupling stabilized finite element methods with finite difference time integration for advection-diffusion problems", I.M.A.T.I. Technical Report PV-8, Pavia, 2006.

2

F. Brezzi, G. Hauke, L.D. Marini, G. Sangalli, "Link-cutting bubbles for the stabilization of convection-diffusion-reaction problems", Math. Models Methods Appl. Sci., 13, 445-461, 2003.
Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. doi:10.1142/S0218202503002581

3

D.R. Cameron, A. Klute, "Convective-dispersive solute transport with a combined equilibrium and kinetic adsortion model", Water Resour. Res., 13(1), 183-188, 1977. doi:10.1029/WR013i001p00183

4

M.Th. van Genuchten, R.W. Cleary, "Movement of solutes in soil: computer-simulated and laboratory results", G.H. Boltz (ed.), Soil Chemistry B: Phsysico-Chemical models. Elsevier, Amsterdam, 349-386, 1982.

5

M.Th. van Genuchten, P.J. Wierenga, "Mass transfer studies in sorbing porous media: I. Analytical solutions", Soil Sci. Soc. Am. J., 40(4), 473-480, 1976.

6

J. Nocedal, S.J. Wright, "Numerical optimization", Springer, New York, 1999. doi:10.1007/b98874

7

M.S. Rodríguez-Cruz, M.J. Sànchez-Martín, M.S. Andrades, M. Sánchez-Camazano, "Imobilization of pesticides in soils modified by cationic surfactants: effects of octadecyltrimhylammonium cation", Satured & Unsatured Zone. Integration of process knowledge into effective models, Rome, 285-290, 2004.

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