On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification

We present a new approach to convexification of the Tikhonov regularization using a continuation method strategy. We embed the original minimization problem into a one-parameter family of minimization problems. Both the penalty term and the minimizer of the Tikhonov functional become dependent on a continuation parameter. In this way we can independently treat two main roles of the regularization term, which are stabilization of the ill-posed problem and introduction of the a priori knowledge. For zero continuation parameter we solve a relaxed regularization problem, which stabilizes the ill-posed problem in a weaker sense. The problem is recast to the original minimization by the continuation method and so the a priori knowledge is enforced. We apply this approach in the context of topology-to-shape geometry identification, where it allows to avoid the convergence of gradient-based methods to a local minima. We present illustrative results for magnetic induction tomography which is an example of PDE constrained inverse problem

Nondestructive testing of metallic cables based on a homogenized model and global measurements

We propose a simple, quick, and cost-effective method for nondestructive eddy-current testing of metallic cables. Inclusions in the cross section of the cable are detected on the basis of certain global data: hysteresis loop measurements for different frequencies. We detect air-gap inclusions inside the cross section using a homogenized model. The problem, which can be understood as an inverse spectral problem, is posed in two dimensions. We consider its reduction to one dimension. The identifiability is studied. We obtain a uniqueness result for a single inclusion in 1D by two measurements for sufficiently low frequency. We study the sensibility of the inverse problem numerically. A study case with real data is performed to confirm the usefulness

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

We consider time series data modeled by ordinary differential equations (ODEs), widespread models in physics, chemistry, biology and science in general. The sensitivity analysis of such dynamical systems usually requires calculation of various derivatives with respect to the model parameters. We employ the adjoint state method (ASM) for efficient computation of the first and the second derivatives of likelihood functionals constrained by ODEs with respect to the parameters of the underlying ODE model. Essentially, the gradient can be computed with a cost (measured by model evaluations) that is independent of the number of the ODE model parameters and the Hessian with a linear cost in the number of the parameters instead of the quadratic one. The sensitivity analysis becomes feasible even if the parametric space is high-dimensional. The main contributions are derivation and rigorous analysis of the ASM in the statistical context, when the discrete data are coupled with the continuous ODE model. Further, we present a highly optimized implementation of the results and its benchmarks on a number of problems. The results are directly applicable in (e.g.) maximum-likelihood estimation or Bayesian sampling of ODE based statistical models, allowing for faster, more stable estimation of parameters of the underlying ODE model.Comment: 5 figure

Adjoint variable method for the study of combined active and passive magnetic shielding

For shielding applications that cannot sufficiently be shielded by only a passive shield, it is useful to combine a passive and an active shield. Indeed, the latter does the "finetuning" of the field reduction that is mainly caused by the passive shield. The design requires the optimization of the geometry of the passive shield, the position of all coils of the active shield, and the real and imaginary components of the currents (when working in the frequency domain). As there are many variables, the computational effort for the optimization becomes huge. An optimization using genetic algorithms is compared with a classical gradient optimization and with a design sensitivity approach that uses an adjoint system. Several types of active and/or passive shields with constraints are designed. For each type, the optimization was carried out by all three techniques in order to compare them concerning CPU time and accuracy. Copyright (C) 2008 Peter Sergeant et al

Heterogeneous multiscale method in eddy currents modeling

The induction of eddy currents in a conductive piece is an electromagnetic phenomenon described by Maxwell’s equations. For composite materials it is multiscale in its nature. We are looking for the macroscopic properties of the composite. We have applied the Heterogeneous Multiscale Method in order to avoid necessity of using full microscale solver

Determination of precession and dissipation parameters in the micromagnetism

The precession β and the dissipation parameter α of a ferromagnetic material can be considered microscopically space dependent. Their space distribution is difficult to obtain by direct measurements. We consider in this article an inverse problem, where we aim at recovering α and β from space measurements of the magnetization in the whole domain Ω. The evolution of the magnetization in micromagnetics is governed by the Landau-Lifshitz (LL) equation. We first study the sensitivity of the LL equation. On basis of the results we analyze the inverse problem. We support the theory by numerical examples

Homogenized eddy current model for non-destructive testing of metallic cables

Purpose - This paper aims to derive a simple and effective but still a reasonably accurate model for electromagnetic problems with hysteretic magnetic properties and/or induced currents in heterogeneous regions in 2D, meant particularly for non-destructive testing (NDT) of steel cables by eddy-currents. Design/methodology/approach - It is assumed that the diffusion of electromagnetic fields in a heterogeneous cable, which consists of many strands, can be described by the Maxwell equations with periodically oscillating coefficients. A computationally efficient model can then be derived. The idea behind this is to replace the heterogeneous material in the cross-section by a fictitious homogeneous one, whose behaviour at the macroscopic level is a good approximation of the one of the composite material. Such a homogenized model is obtained by employing the two-scale convergence. Findings - The model is validated based on experimental electromagnetic data from a steel cable (measured magnetic hysteresis loops) to show that the model is applicable for NDT of cables. The model is useful for studying NDT of cables, both for excitation at low frequency (where changes in magnetic properties are investigated) and at higher frequency (eddy current testing). It is valid for a wide range of amplitudes and frequencies. Originality/value - From the mathematical point of view the model incorporated a non-local boundary condition that has to be included in the analysis. From the engineering point of view, by solving an inverse problem based on this model and on measured hysteresis loops at several frequencies, a broader range of defects in the cable can be detected