This paper describes the application of the adjoint method to the history matching problem in reservoir engineering. The history matching problem consists in adjusting a set of parameters, in this case the permeability distribution, in order to match the data obtained with the simulator to the actual production data in the reservoir. Several numerical experiments are presented which show that our method is able to reconstruct the unknown permeability distribution in a reliable and efficient way from synthetic data provided by an independent numerical forward modelling code. An efficient and flexible regularization scheme is introduced as well, which stabilizes the inversion and enables the reservoir engineer to incorporate certain types of prior information into the final result.

We study light propagation in tissues using the theory of radiative transport. In particular, we study the case in which there is both forward-peaked and large-angle scattering. Because this combination of the forward-peaked and large-angle scattering makes it difficult to solve the radiative transport equation, we present a method to construct approximations to study this problem. The delta-Eddington and Fokker-Planck approximations are special cases of this general framework. Using this approximation method, we derive two new approximations: the Fokker-Planck-Eddington approximation and the generalized Fokker-Planck-Eddington approximation. By computing the transmittance and reflectance of light by a slab we study the performance of these approximations.

We reconstruct images of the absorption and the scattering coefficients for diffuse optical tomography using five different models for light propagation in tissues: (1) the radiative transport equation, (2) the delta-Eddington approximation, (3) the Fokker-Planck approximation, (4) the Fokker-Planck-Eddington approximation and (5) the generalized Fokker-Planck-Eddington approximation. The last four models listed are approximations of the radiative transport equation that take into account forward-peaked scattering analytically. Using simulated data from the numerical solution of radiative transport equation, we solve the inverse problem for the absorption and scattering coefficients using the transport-backtransport method. Through comparison of the numerical results, we show that all of these light scattering models produce good image reconstructions. In addition, we show that these approximations afford considerable computational savings over solving the radiative transport equation. However, all of the models exhibit significant "cross-talk" between absorption and scattering coefficient images. Among the approximations, we have found that the generalized Fokker-Planck-Eddington equation produced the best image reconstructions in comparison with the image reconstructions produced by the radiative transport equation.

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a b s t r a c tKeywords: Laurent series, Inverse, Radial basis functions, Interpolation.In this paper we describe a numerical algorithm to compute the Laurent expansion of the inverse of singularly perturbed matrices. The algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices. The input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix. The matrix coefficients of the Laurent expansion of the inverse are computed using recursive analytical formulae. We show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix, but exponentially with the order of the singularity. We apply this algorithm to several matrices that arise in applications. We make special emphasis to interpolation problems with radial basis functions.