Volume 22, Issue 3

Abstract. We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. The observational noise is assumed to be Gaussian; as a consequence the prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior, which is a constant multiplier of the prior covariance operator, can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.

Abstract. The data acquisition in computerized tomography takes a certain amount of time since the x-ray source has to be rotated around the specimen. An object that changes during the scanning causes inconsistent data sets. To avoid the motion artefacts in reconstructions, the algorithm has to take the dynamic behavior of the specimen into account. In this context, some a priori information about the movement is required. A reconstruction method is proposed that compensates for the motion with a special focus on affine deformations. It also permits the combination of reconstruction and image analysis tools to extract features of the object without motion artefacts. The algorithm is validated with a numerical example from medical imaging.

Abstract. The identification of an unknown constant coefficient in the main term of the second order differential equation -kM(|u|p-2u)+g(x)u=f(x)${ - kM(|u|^{p-2} u) + g(x) u = f(x) }$ with Dirichlet boundary condition is considered. The elliptic operator M is self-adjoint, p≥2${p \ge 2}$. The identification of k here is based on an integral boundary data. The local existence and uniqueness theorems for the inverse problem is proved in the class of the pairs involving a function u such that |u|p-2u∈W22(Ω)${|u|^{p-2} u \in W^2_2 (\Omega )}$ and a positive real number k .

Abstract. In this paper, we want to estimate the true resistivity of permeable bed using the spontaneous potential on the measuring electrode in heterogeneous formation, where the resistivity of the objective layer varies continuously as a function of position. The problem is a generalization of Peng's work for the traditional three-parameter invasion model, where the resistivity of formation is piecewise constant. The mathematical model of the problem is described by a variational problem of elliptic type, where the formation resistivity appears as a nonlinear coefficient. We study an extended problem and then apply its results to prove the existence of solutions to the parameter identification problem arising from SP logging. Finally, a sufficient condition is given to ensure the uniqueness of solutions.

Abstract. We study the inverse problem for nonlinear parabolic equation with variable thermal conductivity a(x,t)${a(x,t)}$ which is a ill-posed problem. We regularize this problem by using the truncated integral method and then obtain the error estimate between the exact solution and the regularized solution. A numerical experiment is given for illustrating our method.

Abstract. Diffuse optical tomography (DOT) is an optical imaging modality, which provides the spatial distribution of the optical parameters inside a random medium. A propagation back-propagation method named EM-like reconstruction method for stationary DOT problem has been proposed yet. This method is really time consuming. Hence the ordered-subsets (OS) technique for this reconstruction method is studied in this paper. The boundary measurements of DOT are grouped into nonoverlapping and overlapping ordered sequence of subsets with random partition, sequential partition and periodic partition, respectively. The performance of OS methods is compared with the standard EM-like reconstruction method with two-dimensional and three-dimensional numerical experiments. The numerical experiments indicate that reconstruction of nonoverlapping subsets with periodic partition, overlapping subsets with periodic partition and standard EM-like method provide very similar acceptable reconstruction results. However, reconstruction of nonoverlapping subsets with periodic partition spends a minimum of time to get proper results.

Abstract. We investigate the problem of estimating the velocity in a two-dimensional acoustic wave equation, which plays an important role in geological survey. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. To reduce the computational burden, a nonlinear multigrid method is applied to solve this inverse problem. In the multigrid inversion process, in order to make the objective functionals at different scales compatible, they are dynamically adjusted. In this way, the necessary condition of “the optimal solution should be the fixed point of multigrid inversion” can be met. The stable and fast regularized Gauss–Newton method is applied to each grid. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation, improve the inversion results, and have the anti-noise ability.