editorial inverse problems: theory and application to science and engineering...

EditorialInverse Problems: Theory and Application toScience and Engineering 2015

Davide La Torre,1,2 Herb Kunze,3 Franklin Mendivil,4

Manuel Ruiz Galan,5 and Rachad Zaki2

1Department of Economics, Management, and Quantitative Methods, University of Milan, 20122 Milan, Italy2Department of Applied Mathematics and Sciences, Khalifa University, P.O. Box 127788, Abu Dhabi, UAE3Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canada N1G 2W14Department of Mathematics and Statistics, Acadia University, Wolfville, NS, Canada B4P 2R65Department of Applied Mathematics, University of Granada, 18071 Granada, Spain

Correspondence should be addressed to Davide La Torre; [email protected]

Received 27 September 2015; Accepted 28 September 2015

Copyright © 2015 Davide La Torre et al.This is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is a real pleasure to announce the publication of this secondspecial issue of this journal dedicated to highlighting recentresearch, development, and applications of inverse problemsin science and engineering. This volume follows a similarspecial issue published in 2014 [1] and contains interestingand original papers.

In recent years a great deal of attention has been paid tothe problem of parameter estimation in distributed systems,that is, the determination of unknown parameters in thefunctional form of the governing model of the phenomenonunder study [2–7] from the observed data. In the mathemat-ical literature, this kind of problem is called an inverse prob-lem. According to Keller [8], “we call two problems inverseof one another if the formulation of each involves all or partof the solution of the other. Often, for historical reasons, oneof the two problems has been studied extensively for sometime, while the other one is newer and not sowell understood.In such cases, the former is called the direct problem, whilethe latter is the inverse problem.” However, there is animportant and fundamental difference between the direct andthe inverse problem: often the direct problem is well-posedwhile the corresponding inverse problem is ill-posed [9]. Inthe literature, one can find many contributions presentingad hoc methods to address ill-posed inverse problems thatinvolve the minimization of a suitable approximation erroralong with the use of some regularization techniques. Inrecent years, theory and applications of inverse problems haveundergone tremendous growth: nowadays inverse problems

can be formulated in many mathematical areas and analyzedby different theoretical and computational techniques.

In the paper “AnAdaptiveObserver-BasedAlgorithm forSolving Inverse Source Problem for the Wave Equation,” byS. Asiri et al., the authors are interested in an inverse sourceproblem for the wave equation. In particular, the objectiveof this paper is to present an alternative algorithm, based onobservers, to solve the inverse source problem for the waveequation. Observers are well known in control theory. Orig-inally designed to estimate the hidden states of dynamicalsystems given some measurements, the observers’ scope hasbeen recently extended to the estimation of some unknowns,for systems governed by partial differential equations. In thispaper, observers are used to solve an inverse source problemfor a one-dimensional wave equation. An adaptive observeris designed to estimate the state and source components for afully discretized system. The effectiveness of the algorithm isemphasized in noise-free and noisy cases and an insight intothe impact of measurements’ size and location is provided.

The aim of the article “Approximate Image Reconstruc-tion in Landscape Reflection Imaging,” by R. Regnier et al., isto single out an imaging concept based on the phenomena ofwave reflection on more or less opaque objects and registerthe reflected wave energy by a single detector. Reflectionimaging of landscape (scenery or extended objects) poses theinverse problem of reconstructing the landscape reflectivityfunction from its integrals on some particular family ofspheres. Such data acquisition is encoded in the framework

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 796094, 3 pageshttp://dx.doi.org/10.1155/2015/796094

2 Mathematical Problems in Engineering

of a Radon transform on this family of spheres. In spite ofthe existence of an exact inversion formula, the numericallandscape reflectivity function reconstitution is best obtainedvia an approximate but judiciously chosen reconstructionkernel. The authors describe the working of this reflectionimaging modality and its theoretical handling, introducean efficient and stable image reconstruction algorithm, andpresent simulation results to prove the validity of this choiceas well as to demonstrate the feasibility of this imagingprocess.

A study on a detection method for continuous mechan-ical deformations of coaxial cylindrical waveguide bound-aries using perturbation theory is discussed in the paper“Perturbation Approach to Reconstructing Deformations ina Coaxial Cylindrical Waveguide” by M. Dalarsson et al.The inner boundary of the waveguide is described as acontinuous PEC structure with deformations modeled bysuitable continuous functions. In this approach, the com-putation complexity is significantly reduced compared todiscrete conductor models studied in previous works. If themechanically deformed metallic structure is irradiated bythe microwave fields of appropriate frequencies, then theauthors can reconstruct the continuous deformation functionby means of measurements of the scattered fields at bothends. They also apply the first-order perturbation methodto the inverse problem of reconstruction of boundary defor-mations, using the dominant TEM-mode of the microwaveradiation. Different orders of Tikhonov regularization, usingthe L-curve criterion, are investigated. Using reflection data,the authors obtain reconstruction results that indicate anagreement between the reconstructed and true continuousdeformations of waveguide boundaries.

The article entitled “Nonlinear Inverse Problem for anIon-Exchange Filter Model: Numerical Recovery of Param-eters,” by B. Mukanova and N. Glazyrina, considers theproblem of identifying unknown parameters for amathemat-ical model of an ion-exchange filter via measurement at theoutlet of the filter.The proposedmathematicalmodel consistsof a material balance equation, an equation describing thekinetics of ion-exchange for the nonequilibrium case, andan equation for the ion-exchange isotherm. The materialbalance equation includes a nonlinear term that depends onthe kinetics of ion-exchange and several parameters. First, anumerical solution of the direct problem, the calculation ofthe impurities concentration at the outlet of the filter, is pro-vided. Then, the inverse problem of finding the parametersof the ion-exchange process in nonequilibrium conditionsis formulated. A method for determining the approximatevalues of these parameters from the impurities concentrationmeasured at the outlet of the filter is proposed.

Since the geological bodies where tunnels are locatedhave uncertain and complex characteristics, the inverseproblem plays an important role in geotechnical engineering.In the paper “Back Analysis of Geomechanical ParametersUsing Hybrid Algorithm Based on Difference Evolutionand Extreme Learning Machine,” by Z. Song et al., theauthors propose a new hybrid algorithm that improvesthe accuracy and speed of surrounding rock identification,construct the back analysis objective function with usage of

the displacement and stress monitoring data, and proposea hybrid algorithm. An extreme learning machine (ELM)is employed with optimal architecture trained by the differ-ence evolution (DE) arithmetic. First, a three-dimensionalnumerical simulation is used in the creation of trainingand testing samples for ELM model construction. Second,the nonlinear relationship between rock parameters anddisplacement is constructed by numerical simulation. Finally,the geophysics parameters are obtained by DE optimizationarithmetic taking into consideration themonitoring data thatincludes both displacement and pressure. This method hadbeen applied in the Fusong highway tunnel in Fusong city ofChina’s Jilin province, with a good effect obtained. It takes fulladvantage ofDE andELMand has both calculation speed andprecision in the back analysis.

In the paper “Identification of Hydraulic Conductiv-ity in Aquifer for Coupled FEM and Adaptive GeneticAlgorithm,” by X. Deng et al., the authors use hydraulicconductivity to conduct an inversion analysis according tothe measurement of head materials by combining the finiteelement method with the adaptive genetic algorithm. Thehydraulic conductivity of a natural rock mass is difficultto be determined because of the complex structure andthe significant influence of uncertain factors. The resultspresented in this paper show that themaximum relative errorof the measured and computed groundwater levels at themeasuring points is 5.3% and the average head error is 1.41%.The effective hydraulic conductivity of intensively weatheredlayer, moderately weathered layer, and fresh bedrock layerin riverbed formation tends to decline gradually and theeffective permeability coefficient in the vertical direction isthe minimum in the same aquifer.This analysis suggests thenthat the established hydraulic conductivity inversion analysismethod is effective.

Finally, the paper “JointMotionDeblurring and Superres-olution from Single Blurry Image,” by L. He et al., proposesa joint blind-deblurring and superresolution algorithm fromone single image that combines gradient and motion blurkernel priors in a coherent framework. Currently, obtainingsuperresolution from a motion blurred image remains achallenging task. The conventional approach, which prepro-cesses the blurry low resolution (LR) image with a deblurringalgorithm and employs a superresolution algorithm, has thelimitation that the high frequency texture of the image isunavoidably lost in the deblurring process, and this lossrestricts the performance of the subsequent superresolutionprocess. This paper presents a novel technique that performsmotion deblurring and superresolution jointly from one sin-gle blurry image. The basic idea is to regularize the ill-posedreconstruction problem using an edge-preserving gradientprior and a sparse kernel prior. This method derives froman inverse problem approach under an efficient optimizationscheme that alternates between blur kernel estimation andsuperresolving until convergence. Furthermore, this paperproposes a simple and efficient refinement formulation toremove artifacts and render better deblurred high resolution(HR) images. The improvements brought by the proposedcombined framework are demonstrated by the processingresults of both simulated and real-life images. Quantitative

Mathematical Problems in Engineering 3

and qualitative results on challenging examples show thatthe proposed method outperforms the existing methodsand effectively eliminates motion blur and artifacts in thesuperresolved image.

Acknowledgments

The guest editors thank all of the authors as well as all otherswho submitted papers for consideration.

Davide La TorreHerb Kunze

Franklin MendivilManuel Ruiz Galan

Rachad Zaki

References

[1] H. E. Kunze, D. La Torre, F. Mendivil, M. Ruiz Galan, and R.Zaki, “Inverse problems: theory and application to science andengineering,” Mathematical Problems in Engineering, vol. 2014,Article ID 497413, 2 pages, 2014.

[2] A. Kirsch,An Introduction to theMathematicalTheory of InverseProblems, Springer, New York, NY, USA, 2011.

[3] H. Kunze, D. La Torre, F. Mendivil, and E. R. Vrscay, Fractal-Based Methods in Analysis, Springer, New York, NY, USA, 2012.

[4] F. D. Moura Neto and A. J. da Silva Neto, An Introduction toInverse Problems with Applications, Springer, New York, NY,USA, 2013.

[5] A. Tarantola, Inverse Problem Theory and Methods for ModelParameter Estimation, SIAM, Philadelphia, Pa, USA, 2005.

[6] A. N. Tychonoff, “Solution of incorrectly formulated problemsand the regularization method,” Doklady Akademii Nauk SSSR,vol. 151, pp. 501–504, 1963.

[7] C. R.Vogel,ComputationalMethods for Inverse Problems, SIAM,Philadelphia, Pa, USA, 2002.

[8] J. B. Keller, “Inverse problems,” The American MathematicalMonthly, vol. 83, no. 2, pp. 107–118, 1976.

[9] J. Hadamard, Lectures on the Cauchy Problem in Linear PartialDifferential Equations, Yale University Press, 1923.

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