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Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 303154, 2 pages

http://dx.doi.org/10.1155/2013/303154

## Recent Theory and Applications on Inverse Problems

^{1}Institut für Theorie Elektromagnetischer Felder, Technische Universität, 64289 Darmstadt, Germany^{2}Department of Electrical and Electronics Engineering, Dokuz Eylul University, 35160 Izmir, Turkey^{3}Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, UK^{4}German Meteorological Service, Deutscher Wetterdienst Research and Development, Head Division FE 12 (Data Assimilation), Frankfurter Straβe 135, 63067 Offenbach, Germany

Received 25 July 2013; Accepted 25 July 2013

Copyright © 2013 Fatih Yaman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The aim of this special issue is to present some recent developments in theoretical and practical areas for the solution of inverse problems which appear in physical applications. The scope of the issue not only covers a broad range of original studies but also includes two review articles on recent investigations. In addition, three papers are the collections of studies having results depending on experimental investigations with real data. It is also worthy to mention that the paper acceptance rate after the reviewing procedure is approximately 55% for this issue and in the following we give very brief descriptions of the published papers.

F. Yaman et al. contribute to the special issue with a wide range review paper on inverse problems which focus on applied sciences. In the paper, many inversion based engineering applications are introduced and mathematical investigations of some of the important ones are given in detail. The authors employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. Furthermore, an introduction with many links to the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastical tools into ensemble methods both for data assimilation and for forecasting.

D. Abbasi and M. Mortazavi study a classical problem of optimal reentry guidance with optimal control techniques, and in addition they offer a solution for the path constraints inspired by the idea of inverse problems methodology. It is stated that the final resulting algorithm seems suitable for onboard predictive guidance.

S. L. Han and T. Kinoshita apply a stochastic inverse method for the identification of the nonlinear damping moment of a ship moving at nonzero-forward speed. The workability and the accuracy of the method are verified with laboratory tests under controlled conditions. The authors consider two different types of ship rolling motions, time-dependent transient motion and frequency-dependent periodic motion, in their experimental trials. It is illustrated that the method enables the inherent nonlinearity in damping moment to be estimated, including its reliability analysis.

The second paper of S. L. Han and T. Kinoshita in our issue is related to the reconstruction of initial wave fields from the limited measurements on the specific boundaries of the fluid body for two- and three-dimensional problems. The authors adopt a stochastic method based on Bayesian formulation for the solution of the inverse problem and present illustrative simulation results.

X.-G. Lv et al. propose a fast high order total variation minimization method to restore multiplicative noisy images. The proposed method is able to preserve edges and at the same time avoid the staircase effect in the smooth regions. The authors discuss the convergence of the alternating minimization algorithm in the paper and present numerical results which show that the proposed method gives restored images in high quality.

L. Yang employs conditional nonlinear optimal perturbation techniques of a coupled Lorenz model and studied their dependence on the reference state. In the paper, a simple model is considered to demonstrate the technique which provides promising results. It is concluded that in order to meet the requirements of the realistic applications the more complex models have to be studied for improving the accuracy of weather and climate forecasts with the related results.

T. Nara aims to propose a numerical algebraic reconstruction algorithm for finding number and locations of neural currents from the knowledge of measured magnetic field data in 3D, which has an application in magnetoencephalography. It is assumed that the radial component of the magnetic field is measured on a sphere of a ball consisting of three additional spherical layers which correspond to a scalp, skull, and brain. In the inverse problem second degree algebraic equations for unknown parameters such as sources and the radial magnetic field data are obtained and solved by means of Gröbner bases. The applicability and the effectiveness of the method are supported with the numerical simulation results.

H. Chen et al. are interested in solving a large-scale optimization problem minimizing a cost functional. The main objective of the study is to make a computational investigation on the performance of the L-BFGS method and two versions of gradient descent method through a series of ideal experiments in which the open boundary conditions are inverted by assimilating the interior observations with the adjoint method. In the paper, numerical results are reported which provide comparisons of the methods for the inversion error between the inverted and prescribed open boundary conditions and the cost function normalized by its initial value as well as the gradient of the cost function with respect to the open boundary conditions.

Z.-L. Deng et al. investigate an inverse heat conduction problem on a semi-infinite domain. More precisely, the fractional diffusion equation is solved analytically in the direct problem and a mollification method with Dirichlet kernel is adopted for the solution of the inverse problem. A priori and a posteriori parameter choice rules are proposed to get the corresponding error estimate between the exact solution and its regularized approximation. Furthermore a numerical example is provided to verify the theoretical results.

F. A. A. Queiroz et al. outline the inverse problem considering the GPR assessment of concrete in microwave imaging. The aim of the inverse problem considered in the study is the inclusion reconstruction in concrete slabs. In the paper three reconstruction algorithms are implemented and tests are performed using real experimental data to assess the applicability and efficiency of these algorithms for image reconstruction.

C.-X. Li et al. discuss the detection problem using passive decomposition of the time reversal operator method. The authors derive the generalized likelihood ratio test based on Neyman-Person lemma provided that the signal components can be modeled as a linear combination of basis vectors with an unknown signal subspace. Moreover, authors present results which illustrate the performance of the time reversal operator detector and the energy detector using the real acoustic data collected in the laboratory waveguide experiment.

A. Gessese and M. Sellier report a numerical technique to identify the underlying bedrock topography from given free surface elevation data in shallow open channel flows. The technique described in the paper is based on an explicit finite difference scheme which provides the solution to the inverse problem directly. The algorithm is tested on three numerical examples, and noise data tests are performed.

D. Chapelle et al. show that the exponential convergence can be achieved with an observer strategy based on partial measurements of the solution of a wave equation. The authors adopt the observer strategy “Schur Displacement Feedback” for elasticity-like formulations to the scalar wave equation. Furthermore, numerical results are provided to illustrate the effectiveness of the proposed approach.

S. Alexandrov et al. present two solutions to design a thin annular disc of variable thickness subject to thermomechanical loading. The authors adopted two design criteria, one of which leads to a relation between optimal values of the loading parameters for each specific shape of the disc. The application of the other criterion has shown that the state of required stress and strain appears in the disc of constant thickness at certain values of the loading parameters.

Y. Chen et al. propose a strategy for reducing metallic artifacts in CT images. It is concluded in the paper that, “visual and quantitative analyses on phantom and clinical data show that the proposed correction method provides a substantial reduction of the metallic artifacts in the corrected images. The pixel-wise operations in the pre-filtering and sinogram inpainting steps are greatly accelerated by a CUDA parallelization that makes the algorithm also competitive in computation time.”

Z. Guo et al. present a method to estimate open boundary conditions by an adjoint data assimilation approach. The authors assume the open boundary conditions being position dependent and can be approximated by linear interpolation among values at certain independent points. Twin numerical experiments are performed by employing Bohai Sea model in order to verify the feasibility of the method.

#### Acknowledgment

The guest editors would like to deeply thank all the authors, the reviewers, the editorial board of the Mathematical Problems in Engineering journal, and all the staff involved in the preparation of this issue.

Fatih Yaman

Valery G. Yakhno

Roland Potthast