Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

Regularization methods for the solution of a nonlinear least-squares problem in tomography

In this work we study a polyenergetic and multimaterial model for the breast image reconstruction in Digital Tomosynthesis, taking into consideration the variety of the materials forming the object and the polyenergetic nature of the X-rays beam. The modelling of the problem leads to the resolution of a high-dimensional nonlinear least-squares problem that, due to its nature of inverse ill-posed problem, needs some kind of regularization. We test two main classes of methods: the Levenberg-Marquardt method (together with the Conjugate Gradient method for the computation of the descent direction) and two limited-memory BFGS-like methods (L-BFGS). We perform some experiments for different values of the regularization parameter (constant or varying at each iteration), tolerances and stop conditions. Finally, we analyse the performance of the several methods comparing relative errors, iterations number, times and the qualities of the reconstructed images.

Inverse and regular phenyl-azole ligands for luminescent Iridium(III) complexes

In the modern society, light is mostly powered by electricity which lead to a significant increase of the global energy consumption. In order to reduce it, different kinds of electric lamps have been developed over the years; it is now accepted that phosphorescence-based OLEDs offer many advantages over existing light technologies. Iridium complexes are considered excellent candidates for bright materials by virtue of the possibility to easily tune the wavelength of the emitted radiation, by appropriate modifications of the nature of the ligands. It is important to note that the synthesis of Ir(III) blue-emitting complexes is a very challenging goal, because of wide HOMO-LUMO gaps needed for produce a deep blue emission. During my thesis I planned the synthesis of two different series of new Ir(III) heteroleptic complexes, the C and the N series, using cyclometalating ligands containing an increasing number of nitrogens in inverse and regular position. I successfully performed in the synthesis of the required four ligands, i.e. 1-methyl-4-phenyl-1H-imidazole (2), 4-phenyl-1-methyl-1,2,3-triazole (3), 1-phenyl-1H-1,2,3-triazole (6) and 1-phenyl-1H-tetrazole (7), that differ in the number of nitrogens present in the heterocyclic ring and in the position of the phenyl ring. Therefore the cyclometalation of the obtained ligands to get the corresponding Ir(III)-complexes was attempted. I succeeded in the synthesis of two Ir(III)-complexes of the C series, and I carried out various attempts to set up the appropriate reaction conditions to get the remaining desired derivatives. The work is still in progress, and once all the desired complexes will be synthesized and characterized, a correlation between their structure and their emitting properties could be formulated analysing and comparing the photophysical data of the real compounds.

Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.

On the free-boundary for an elliptic inverse nonlocal problem arising in the nuclear fusion. A numerical approach

We consider a mathematical model related to the stationary regime of a plasma magnetically confined in a Stellarator device in the nuclear fusion. The mathematical problem may be reduced to an nonlinear elliptic inverse nonlocal two dimensional free{boundary problem. The nonlinear terms involving the unknown functions of the problem and its rearrangement. Our main goal is to determinate the existence and the estimate on the location and size of region where the solution is nonnegative almost everywhere (corresponding to the plasma region in the physical model)

A study of the non-uniform convergence of the inverse of a doubly-infinite matrix associated with a boundary value problem in a waveguide /

"Contract AF33(616)-6079 Project No. 9-(13-6278), Task No. 40572. Sponsored by: Aeronautical Systems Division"

Evaluation of the problem posed by in-place pollutants in Baltimore Harbor and recommendation of corrective action.

Mode of access: Internet.

I. The present status of the leprosy problem in Hawaii. II. The reaction of lepers to Moro's "percutaneous" test. III. A note upon the possibility of the mosquito acting in the transmission of leprosy.

At head of title: Treasury department. Public health and marine-hospital service of the United States.

A method of fundamental solutions for the one-dimensional inverse Stefan problem

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional inverse Stefan problem for the heat equation by extending the MFS proposed in [5] for the one-dimensional direct Stefan problem. The sources are placed outside the space domain of interest and in the time interval (-T, T). Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate and stable results can be obtained efficiently with small computational cost.

Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy–Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367–4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in Johansson et al. (2011) and show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost.

Least Squares Approach to Inverse Problems in Musculoskeletal Biomechanics

Inverse simulations of musculoskeletal models computes the internal forces such as muscle and joint reaction forces, which are hard to measure, using the more easily measured motion and external forces as input data. Because of the difficulties of measuring muscle forces and joint reactions, simulations are hard to validate. One way of reducing errors for the simulations is to ensure that the mathematical problem is well-posed. This paper presents a study of regularity aspects for an inverse simulation method, often called forward dynamics or dynamical optimization, that takes into account both measurement errors and muscle dynamics. The simulation method is explained in detail. Regularity is examined for a test problem around the optimum using the approximated quadratic problem. The results shows improved rank by including a regularization term in the objective that handles the mechanical over-determinancy. Using the 3-element Hill muscle model the chosen regularization term is the norm of the activation. To make the problem full-rank only the excitation bounds should be included in the constraints. However, this results in small negative values of the activation which indicates that muscles are pushing and not pulling. Despite this unrealistic behavior the error maybe small enough to be accepted for specific applications. These results is a starting point start for achieving better results of inverse musculoskeletal simulations from a numerical point of view.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

The problem posed by exam choice on the comparability of results in the Finnish matriculation examination

The article focuses on the upper secondary matriculation examination in Finland as a school leaving and university entrance examination. The presented research addresses the question of whether increased choice of the subject-specific examinations has the potential to undermine the comparability of examination results and to direct students’ choices not only in the examination but already beforehand at school. The authors refer to Finland’s tradition of more than 160 years of a national examination connecting the academic track of upper secondary schools with universities. The authors explain the Finnish system by describing the adoption of a course-based (vs. class- or year-based) curriculum for the three-year upper secondary education and the subsequent reforms in the matriculation examination. This increases students’ choices considerably with regard to the subject-specific exams included in the examination (a minimum of four). As a result, high-achieving students compete against each other in the more demanding subjects while the less able share the same normal distribution of grades in the less demanding subjects. As a consequence, students tend to strategic exam-planning, which in turn affects their study choices at school, often to the detriment of the more demanding subjects and, subsequently, of students’ career opportunities, endangering the traditional national objective of an all-round pre-academic upper secondary education. This contribution provides an overview of Finnish upper secondary education and of the matriculation examination (cf. Klein, 2013) while studying three separate but related issues by using data from several years of Finnish matriculation results: the relation of the matriculation examination and the curriculum; the problems of comparability vis-à-vis university entry due to the increased choice within the examination; the relations between students’ examination choices and their course selection and achievement during upper secondary school. (DIPF/Orig.)

STUDY OF A DIRECT SAMPLING METHOD FOR THE INVERSE MEDIUM SCATTERING PROBLEM

Direct sampling methods are increasingly being used to solve the inverse medium scattering problem to estimate the shape of the scattering object. A simple direct method using one incident wave and multiple measurements was proposed by Ito, Jin and Zou. In this report, we performed some analytic and numerical studies of the direct sampling method. The method was found to be effective in general. However, there are a few exceptions exposed in the investigation. Analytic solutions in different situations were studied to verify the viability of the method while numerical tests were used to validate the effectiveness of the method.

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.