Application of the second domain derivative in inverse electromagnetic scattering

We consider the inverse scattering problem of reconstructing a perfect conductor from the far field pattern of a scattered time harmonic electromagnetic wave generated by one incident plane wave. In view of iterative regularization schemes for the severely ill-posed problem the first and the second domain derivative of the far field pattern with respect to variations of the domain are established. Charaterizations of the derivatives by boundary value problems allow for an application of second degree regularization methods to the inverse problem. A numerical implementation based on integral equations is presented and its performance is illustrated by a selection of examples

A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference to develop a derivative-free stable method easy to implement in applications where the PDE (forward) model is only accessible as a black box. The method can be derived as an approximation of the regularizing Levenberg-Marquardt (LM) scheme [14] in which the derivative of the forward operator and its adjoint are replaced with empirical covariances from an ensemble of elements from the admissible space of solutions. The resulting ensemble method consists of an update formula that is applied to each ensemble member and that has a regularization parameter selected in a similar fashion to the one in the LM scheme. Moreover, an early termination of the scheme is proposed according to a discrepancy principle-type of criterion. The proposed method can be also viewed as a regularizing version of standard Kalman approaches which are often unstable unless ad-hoc fixes, such as covariance localization, are implemented. We provide a numerical investigation of the conditions under which the proposed method inherits the regularizing properties of the LM scheme of [14]. More concretely, we study the effect of ensemble size, number of measurements, selection of initial ensemble and tunable parameters on the performance of the method. The numerical investigation is carried out with synthetic experiments on two model inverse problems: (i) identification of conductivity on a Darcy flow model and (ii) electrical impedance tomography with the complete electrode model. We further demonstrate the potential application of the method in solving shape identification problems by means of a level-set approach for the parameterization of unknown geometries

Orbital and Maxillofacial Computer Aided Surgery: Patient-Specific Finite Element Models To Predict Surgical Outcomes

This paper addresses an important issue raised for the clinical relevance of Computer-Assisted Surgical applications, namely the methodology used to automatically build patient-specific Finite Element (FE) models of anatomical structures. From this perspective, a method is proposed, based on a technique called the Mesh-Matching method, followed by a process that corrects mesh irregularities. The Mesh-Matching algorithm generates patient-specific volume meshes from an existing generic model. The mesh regularization process is based on the Jacobian matrix transform related to the FE reference element and the current element. This method for generating patient-specific FE models is first applied to Computer-Assisted maxillofacial surgery, and more precisely to the FE elastic modelling of patient facial soft tissues. For each patient, the planned bone osteotomies (mandible, maxilla, chin) are used as boundary conditions to deform the FE face model, in order to predict the aesthetic outcome of the surgery. Seven FE patient-specific models were successfully generated by our method. For one patient, the prediction of the FE model is qualitatively compared with the patient's post-operative appearance, measured from a Computer Tomography scan. Then, our methodology is applied to Computer-Assisted orbital surgery. It is, therefore, evaluated for the generation of eleven patient-specific FE poroelastic models of the orbital soft tissues. These models are used to predict the consequences of the surgical decompression of the orbit. More precisely, an average law is extrapolated from the simulations carried out for each patient model. This law links the size of the osteotomy (i.e. the surgical gesture) and the backward displacement of the eyeball (the consequence of the surgical gesture)

Inexact Bregman iteration with an application to Poisson data reconstruction

This work deals with the solution of image restoration problems by an iterative regularization method based on the Bregman iteration. Any iteration of this scheme requires to exactly compute the minimizer of a function. However, in some image reconstruction applications, it is either impossible or extremely expensive to obtain exact solutions of these subproblems. In this paper, we propose an inexact version of the iterative procedure, where the inexactness in the inner subproblem solution is controlled by a criterion that preserves the convergence of the Bregman iteration and its features in image restoration problems. In particular, the method allows to obtain accurate reconstructions also when only an overestimation of the regularization parameter is known. The introduction of the inexactness in the iterative scheme allows to address image reconstruction problems from data corrupted by Poisson noise, exploiting the recent advances about specialized algorithms for the numerical minimization of the generalized Kullback–Leibler divergence combined with a regularization term. The results of several numerical experiments enable to evaluat