Nilson C. Roberty

Recovering the source term in a linear diffusion problem by the method of fundamental solutions

This work considers the detection of the spatial source term distribution in a multidimensional linear diffusion problem with constant (and known) thermal conductivity. This work can be physically associated with the detection of non-homogeneities in a material that are inclusion sources in a heat conduction problem. The uniqueness of the inverse problem is discussed in terms of classes of identifiable sources. Numerically, we propose to solve these inverse source problems using fundamental solution-based methods, namely an extension of the method of fundamental solutions to domain problems. Several examples are presented and the numerical reconstructions are discussed.

A SOURCE-DETECTOR METHODOLOGY FOR THE CONSTRUCTION AND SOLUTION OF THE ONE-DIMENSIONAL INVERSE TRANSPORT EQUATION

A source-detector methodology is presented for the construction of an inverse transport equation that once solved provides estimates for radiative properties and/or internally distributed sources in participating media. From the proper combination of source and detector pairs, a system of non-linear equations is assembled, taking also in consideration experimental data on the exit radiation from the medium. Test case results are also presented.

A generalized approach for atomic force microscopy image restoration with bregman distances as tikhonov regularization terms

Tikhonovs regularization approach applied to image restoration, stated in terms of ill-posed problems, has proved to be a powerful tool to solve noisy and incomplete data. This work proposes a variable norm discrepancy function as the regularization term of a Tikhonov expression, where the cross-entropy functional was derived. Our method was applied to true Atomic Force Microscopy (AFM) images obtained from biological samples, producing satisfactory results towards the most probable sample morphological aspect. These images represent a mapping of local interaction forces exerted between a reduced scaled AFM sensing tip and the biological sample surface, kept alive in aqueous or air environment.

On the identification of star-shape sources from boundary measurements using a reciprocity functional

In this article we consider the numerical problem of shape reconstruction of an unknown characteristic source inside a domain. We consider a steady-state conductivity problem modelled by the Poisson equation, where the heat source is the non-homogeneous characteristic function. A well-known result, back to 1938, by Novikov (Novikov, Sur le probleme inverse du potentiel, Dokl. Akad. Nauk 18 (1938), pp. 165–168) says that star-shaped sources can be reconstructed uniquely from the Cauchy boundary data (see also the work of Isakov e.g. (Isakov, Inverse problems for partial diferential equations, Applied Mathematical Sciences, Vol. 127, Springer-Verlag, New York, 1998). Here we consider the reciprocity functional that maps harmonic functions to their integral in the unknown characteristic support. We connect the uniqueness result with the recovery of a function from a certain knowledge of the Fourier coefficients, by taking harmonic monomials as test functions. We also establish a numerical method that consists in an algebraic non-linear system of equations, leading to an approximation of the radial function that defines the boundary of the unknown source. Simulations showing the performance of the numerical method are presented.

Absorption coefficient estimation in heterogeneous media using a domain partition consistent with divergent beams

An inverse radiative transfer problem is solved for the estimation of the absorption coefficient in a purely absorbing two-dimensional heterogeneous medium. A natural base built with a domain partition that takes into account the possibility of having divergent radiation beams originated at external sources is used in combination with a family of action by line reconstruction algorithms, built within the framework of Lebesgue measure, with Bregman distances based on a q-discrepancy functional. The domain partition and the assembly of the corresponding system of linear algebraic equations, whose unknowns are the absorption coefficients for each element of the partition, are described in detail. Results are presented for test cases using synthetic experimental data generated with a Monte Carlo simple integration technique.

Reconstruction of a combination of the absorption and scattering coefficients with a discrete ordinates method consistent with the source–detector system

In the present work a combination of the absorption and scattering coefficients in heterogeneous two-dimensional media is estimated using the source–detector methodology and a discrete ordinates method whose directions of radiation propagation are taken in a way that is consistent with the source–detector system for parallel beams of radiation. The domain partition, the solution of the direct problem, and the source–detector methodology are described. Test case results are also presented.In the present work a combination of the absorption and scattering coefficients in heterogeneous two-dimensional media is estimated using the source–detector methodology and a discrete ordinates method whose directions of radiation propagation are taken in a way that is consistent with the source–detector system for parallel beams of radiation. The domain partition, the solution of the direct problem, and the source–detector methodology are described. Test case results are also presented.

Inverse radiative transfer problems in two-dimensional participating media

In the present work inverse radiative transfer problems in two-dimensional heterogeneous participating media are considered. An implicit formulation is used in which the cost functional of the squared residues between calculated and measured exit radiation intensities is minimized. The Levenberg–Marquardt method, a gradient-based minimization algorithm, is used, and therefore at every iteration of the iterative procedure the solution of the direct problem is required. For the solution of the direct problem, which is modeled by the linearized Boltzmann equation, the discrete-ordinates method with a finite difference approximation for the spatial derivatives is used. The inverse problems considered are related to the estimation of the absorption and anisotropic scattering coefficients, as well as the estimation of source terms.

Moving Heat Source Reconstruction from the Cauchy Boundary Data

We consider the problem of reconstruction of an unknown characteristic transient thermal source inside a domain. By introducing the definition of an extended dirichlet-to-Neumann map in the time-space cylinder and the adoption of the anisotropic Sobolev-Hilbert spaces, we can treat the problem with methods similar to those used in the analysis of the stationary source reconstruction problem. Further, the finite difference scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic star-shape source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula, we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support.

A one-dimensional inverse radiative transfer problem with time-varying boundary conditions

In this work are presented the formulation and solution of two inverse radiative transfer problems in one-dimensional homogeneous participating media with time-dependent boundary conditions. In the first one an energy balance for the incoming and exit radiation allows the direct determination of the absorption coefficient. In the second inverse problem we solve the simultaneous estimation of the total extinction and scattering coefficients using the source–detector method. We first consider a more general formulation of the direct radiative transfer problem, and then for the formulation of the inverse problems we use the particular cases of “cold medium” (no internal sources), transparent boundaries (no effects associated with differences on medium and environment refractive indices), and homogeneous media (constant radiative transfer properties). Test case results are presented.

An inverse source problem for the stationary diffusion–advection–decay equation

In this work we consider a source reconstruction problem for stationary diffusion–advection–decay equation from boundary data. Making a change of variable, we obtain an equivalent modified Helmholtz source problem. Some continuation solutions in the unitary disk are then presented. We also present a variational formulation based in the reciprocity functional formulation. Test functions are plane waves. Considering characteristic sources with star-shaped support in this variational formulation, we obtain a non-linear system of integral equations that must be solved to reconstruct this class of sources. This non-linear problem is then investigated with truncated Fourier series representation and then solved by collocation method. The system of nonlinear algebraic equations which approximates the solution is solved using the Levenberg–Marquardt method. Synthetic data are produced by the finite-element method. We present some numerical reconstructions for an unidimensional and a bidimensional model.