Table of contents

This volume of Journal of Physics: Conference Series is dedicated to the scientific research presented during the 5th International Workshop on New Computational Methods for Inverse Problems, NCMIP 2015 (http://complement.farman.ens-cachan.fr/NCMIP_2015.html). This workshop took place at Ecole Normale Supérieure de Cachan, on May 29, 2015. The prior editions of NCMIP also took place in Cachan, France, firstly within the scope of ValueTools Conference, in May 2011, and secondly at the initiative of Institut Farman, in May 2012, May 2013 and May 2014.

The New Computational Methods for Inverse Problems (NCMIP) workshop focused on recent advances in the resolution of inverse problems. Indeed, inverse problems appear in numerous scientific areas such as geophysics, biological and medical imaging, material and structure characterization, electrical, mechanical and civil engineering, and finances. The resolution of inverse problems consists of estimating the parameters of the observed system or structure from data collected by an instrumental sensing or imaging device. Its success firstly requires the collection of relevant observation data. It also requires accurate models describing the physical interactions between the instrumental device and the observed system, as well as the intrinsic properties of the solution itself. Finally, it requires the design of robust, accurate and efficient inversion algorithms. Advanced sensor arrays and imaging devices provide high rate and high volume data; in this context, the efficient resolution of the inverse problem requires the joint development of new models and inversion methods, taking computational and implementation aspects into account. During this one-day workshop, researchers had the opportunity to bring to light and share new techniques and results in the field of inverse problems.

The topics of the workshop were: algorithms and computational aspects of inversion, Bayesian estimation, Kernel methods, learning methods, convex optimization, free discontinuity problems, metamodels, proper orthogonal decomposition, reduced models for the inversion, non-linear inverse scattering, image reconstruction and restoration, and applications (bio-medical imaging, non-destructive evaluation...).

NCMIP 2015 was a one-day workshop held in May 2015 which attracted around 70 attendees. Each of the submitted papers has been reviewed by two reviewers. There have been 15 accepted papers. In addition, three international speakers were invited to present a longer talk.

The workshop was supported by Institut Farman (ENS Cachan, CNRS) and endorsed by the following French research networks: GDR ISIS, GDR MIA, GDR MOA and GDR Ondes. The program committee acknowledges the following research laboratories: CMLA, LMT, LURPA and SATIE.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

A non-iterative topological sensitivity framework for guaranteed far field detection of a dielectric inclusion is presented. The cases of single and multiple measurements of the electric far field scattering amplitude at a fixed frequency are taken into account. The performance of the algorithm is analyzed theoretically in terms of resolution, stability, and signal-to-noise ratio.

Parameter identification from noisy data is an ill-posed inverse problem and data noise leads to poor solutions. Regularization methods are necessary to obtain stable solutions. In this paper we introduce the regularization by means of an iteratively weighted constraint and define an algorithm to compute the weights and solve the constrained problems using as prior information the given measurements. Although this approach is general, in the present work we prove the convergence in the case of least squares data fit with ℓ₂ regularization term. The data reported in the numerical experiments prove the efficiency and good quality of the results.

A stochastic pertubation of a level-set regularization method to find stable inverses for the nonlinear reconstruction of the complex refractive index in in-line phase tomography is presented under the assumption that the index is piecewise constant. The stochastic algorithm is tested on a multi-material object. It is useful to escape the local minima with decrease of the reconstruction errors localized on the boundaries.

In this paper we account for the numerical error introduced by the Finite Element approximation of the shape gradient to construct a guaranteed shape optimization method. We present a goal-oriented strategy inspired by the complementary energy principle to construct a constant-free, fully-computable a posteriori error estimator and to derive a certified upper bound of the error in the shape gradient. The resulting Adaptive Boundary Variation Algorithm (ABVA) is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion for the optimization loop. Some preliminary numerical results for the inverse identification problem of Electrical Impedance Tomography are presented.

The aim here is to study two-time-scale models and their associated parameter identification. When it is possible to consider two well-separated time scales, and when the fast component of the applied loading is periodic, a periodic time homogenization scheme, similar to what exists in space homogenization, can be used to derive a homogenized model. A parameter identification process for this latter is then proposed, consisting in homogenizing with respect to time a classical identification strategy based on the use of adjoint state formulations; it is then applied to an academic example showing the benefits of such a strategy.

In digital tomosynthesis imaging, multiple projections of an object are obtained along a small range of different incident angles in order to reconstruct a pseudo-3D representation (i.e., a set of 2D slices) of the object. In this paper we describe some mathematical models for polyenergetic digital breast tomosynthesis image reconstruction that explicitly takes into account various materials composing the object and the polyenergetic nature of the x-ray beam. A polyenergetic model helps to reduce beam hardening artifacts, but the disadvantage is that it requires solving a large-scale nonlinear ill-posed inverse problem. We formulate the image reconstruction process (i.e., the method to solve the ill-posed inverse problem) in a nonlinear least squares framework, and use a Levenberg-Marquardt scheme to solve it. Some implementation details are discussed, and numerical experiments are provided to illustrate the performance of the methods.

In this work we propose a new identification strategy based on the coupling between a probabilistic data assimilation method and a deterministic inverse problem approach using the modified Constitutive Relation Error energy functional. The idea is thus to offer efficient identification despite of highly corrupted data for time-dependent systems. In order to perform real-time identification, the modified Constitutive Relation Error is here associated to a model reduction method based on Proper Generalized Decomposition. The proposed strategy is applied to two thermal problems with identification of time-dependent boundary conditions, or material parameters.

Microwave imaging methods are useful for non-destructive inspection of dielectric targets. In this work, a numerical technique for solving the 3D Lippmann-Schwinger integral equation of the inverse scattering problem via Gauss-Newton linearization in Banach spaces is analysed. More specifically, two different approximations of the Frechet derivative are proposed in order to speed up the computation. Indeed it is well known that the computation of the Frechet derivative is generally quite expensive in three dimensional restorations. Numerical tests show that the approximations give a faster restoration without loosing accuracy.

We consider the problem of restoring blurred images affected by impulsive noise. The adopted method restores the images by solving a sequence of constrained minimization problems where the data fidelity function is the ℓ₁ norm of the residual and the constraint, chosen as the image Total Variation, is automatically adapted to improve the quality of the restored images. Although this approach is general, we report here the case of vectorial images where the blurring model involves contributions from the different image channels (cross channel blur). A computationally convenient extension of the Total Variation function to vectorial images is used and the results reported show that this approach is efficient for recovering nearly optimal images.

The inversion of Ground Penetrating Radar (GPR) data requires the development of suitable information-exploitation techniques that are able to extract as much as possible information on the unknown targets from the available measurements. An innovative singlefrequency (SF) inversion technique based on a deterministic conjugate-gradient (CG) minimization and the iterative multi-scaling approach (IMSA) is described. It is then shown how to improve the performances of the SF-IMSA-CG method by the introduction of an external frequency hopping (FH) iterative loop. On the one hand, the proposed FH-IMSA-CG method allows to exploit the intrinsic frequency diversity of wideband GPR measurements thanks to the FH strategy. On the other hand, the IMSA approach guarantees a significant reduction of the problem unknowns, providing an increased resolution within the identified regions of interest (RoIs). A numerical comparison shows the advantages of the FH-IMSA-CG over its single-frequency version. Moreover, the benefits of integrating the IMSA within the FH are verified by directly comparing the FH-IMSA-CG with its single-resolution (BARE) version (FH-BARE-CG).

Recently, Poisson noise has become of great interest in many imaging applications. When regularization strategies are used in the so-called Bayesian approach, a relevant issue is to find rules for selecting a proper value of the regularization parameter. In this work we compare three different approaches which deal with this topic. The first model aims to find the root of a discrepancy equation, while the second one estimates such parameter by adopting a constrained, approach. These two models do not always provide reliable results in presence of low counts images. The third approach presented is the inexact Bregman procedure, which allows to use an overestimation of the regularization parameter and moreover enables to obtain very promising results in case of low counts images and High Dynamic Range astronomical images.

In this paper, we propose a novel method to estimate the elbow motion, through the features extracted from electromyography (EMG) signals. The features values are normalized and then compared to identify potential relationships between the EMG signal and the kinematic information as angle and angular velocity. We propose and implement a method to select the best set of features, maximizing the distance between the features that correspond to flexion and extension movements. Finally, we test the selected features as inputs to a non-linear support vector machine in the presence of non-idealistic conditions, obtaining an accuracy of 99.79% in the motion estimation results.

We study sparse spikes deconvolution over the space of Radon measures when the input measure is a finite sum of positive Dirac masses using the BLASSO convex program. We focus on the recovery properties of the support and the amplitudes of the initial measure in the presence of noise when the minimum separation distance t of the input measure (the minimum distance between two spikes) tends to zero. We show that when ||ω||₂/λ, ||ω||₂/t^2N-1 and λ/t^2N-1 are small enough (where λ is the regularization parameter, ω the noise and N the number of spikes), which corresponds roughly to a sufficient signal-to-noise ratio and a noise level and a regularization parameter small enough with respect to the minimum separation distance, there exists a unique solution to the BLASSO program with exactly the same number of spikes as the original measure. We provide an upper bound on the error with respect to the initial measure. As a by-product, we show that the amplitudes and positions of the spikes of the solution both converge towards those of the input measure when λ and ω drop to zero faster than t^2N-1.

In 3-D microwave imaging, gradient-based optimization algorithms usually make use of the so-called stabilized version of the biconjugate gradient iterative method (BiCGStab) in order to solve multiple linear systems. We propose to use a block version of BiCGStab to jointly solve the mutiple right-hand side linear systems. Illuminations are partitioned in subgroups, which makes the method more efficient. The reconstruction process is studied on realistic simulated data and illustrates the efficiency of the method compared to BiCGStab.

A method relying on a direct semi-analytic model is proposed for reconstructing cracks from eddy current images. We consider systems featuring a uniform excitation flow which can be modelled by means of fictitious current sources distributed in the crack volume. Thanks to the relative simplicity of the model a reconstruction method based the comparison of EC images is considered. The sensitivity of different images comparison criteria is studied for 2D surface cracks, leading to a reconstruction method based on a genetic algorithm. Numerical experiments are carried out to examine the performances of the algorithm in terms of convergence and accuracy.