We investigate the applicability of the method of maximum entropy regularization (MER) including convergence and convergence rates of regularized solutions to the specific inverse problem (SIP) of calibrating a purely time-dependent volatility function. In this context, we extend the results of [16] and [17] in some details. Due to the explicit structure of the forward operator based on a generalized Black-Scholes formula the ill-posedness character of the nonlinear inverse problem (SIP) can be verified. Numerical case studies illustrate the chances and limitations of (MER) versus Tikhonov regularization (TR) for smooth solutions and solutions with a sharp peak.

We deal with locally ill-posed nonlinear operator equations F(x) = y in L^2(0,1), where the Fréchet derivatives A = F'(x_0) of the nonlinear forward operator F are compact linear integral operators A = M ◦ J with a multiplication operator M with integrable multiplier function m and with the simple integration operator J. In particular, we give examples of nonlinear inverse problems in natural sciences and stochastic finance that can be written in such a form with linearizations that contain multiplication operators. Moreover, we consider the corresponding ill-posed linear operator equations Ax = y and their degree of ill-posedness. In particular, we discuss the fact that the noncompact multiplication operator M has only a restricted influence on this degree of ill-posedness even if m has essential zeros of various order.

In this paper, we consider the inverse problem of calibrating a generalization of the bivariate Ornstein-Uhlenbeck model introduced by Lo and Wang. Even though the generalized Black-Scholes option pricing formula still holds, option prices change in comparison to the classical Black-Scholes model. The time-dependent volatility function and the other (real-valued) parameters in the model are calibrated simultaneously from option price data and from some empirical moments of the logarithmic returns. This gives an ill-posed inverse problem, which requires a regularization approach. Applying the theory of Engl, Hanke and Neubauer concerning Tikhonov regularization we show convergence of the regularized solution to the true data and study the form of source conditions which ensure convergence rates.

In this note, we consider an inverse problem in a jump diffusion model. Using characteristic functions we prove the injectivity of the forward operator mapping the five parameters determining the model to the density function of the return distribution.

The classical test to confirm the performance of a given fibre design for use in reinforced concrete is the pull-out test. While attempts have been made to simulate the performance of such pull-out tests, in practice it has been found that there is a significant disparity between prediction and real-life performance. The high strength of steel reinforcing fibres is a consequence of the cold wire drawing process and subsequent fabrication. Residual stresses exist in cold drawn wire as a consequence of the elastic response to a non-uniform distribution of plastic strain. This also introduces a yield strength profile where yield strength varies radially through the wire. The question arises as to whether fibre design should use a starting material model that considers these properties. This thesis examines whether the tensile test, simple bend test and pull-out test provide enough information to define a starting material model that may be used for further design and simulation of such fibres. Since the details of the wire drawing process and material specification are proprietary and therefore unknown, a sensitivity study was conducted to determine which aspects of the wire drawing process have the greatest effect on the pull-out curve and the following were established as being significant: • Plastic strain due to wire drawing was shown to be the most important factor. • The bilinear curve was shown to be a suitable approximation for the stress-strain curve. • Replacing the plastic strain profile with a single value of average equivalent plastic strain is practical. The following were established as having negligible effect: • The consequences of the hooked-end forming process. • The residual stress profiles due to wire drawing provided that the above was also excluded. • The hardening law While inverse analysis demonstrated that all tests provide sufficient information to determine the required properties for this bilinear material model, the pull-out test was shown to provide more accurate approximations of the maximum pull-out force at the first and second peaks and the bend test was shown to produce more accurate approximations of the energy associated with pull-out. Good correlation with the baseline pull-out curve was found for both the isotropic and the kinematic hardening laws and it is concluded that behaviour during pull-out is insensitive to the hardening law. Sensitivity analysis and characterisation of the material model using an experimental pull-out curve demonstrated the importance of the coefficient of friction. Full characterisation using the pull-out curve therefore requires the solution to a three-variable problem: yield strength, tangent modulus and coefficient of friction. This was a suggested topic for further study. / Dissertation (MEng)--University of Pretoria, 2019. / Mechanical and Aeronautical Engineering / MEng (Mech) / Unrestricted

The purpose of this thesis is to study how to overcome difficulties one typically encounters when solving non-negative inverse problems by standard Krylov subspace methods. We first give a theoretical background to the non-negative inverse problems. Then we concentrate on selected modifications of Krylov subspace methods known to improve the solution significantly. We describe their properties, provide their implementation and propose an improvement for one of them. After that, numerical experiments are presented giving a comparison of the methods and analyzing the influence of the present parameters on the behavior of the solvers. It is clearly demonstrated, that the methods imposing nonnegativity perform better than the unconstrained methods. Moreover, our improvement leads in some cases to a certain reduction of the number of iterations and consequently to savings of the computational time while preserving a good quality of the approximation.

Nous sommes intéressés dans cette thèse par l'analyse de la diffraction directe et inverse des ondes par des couches infinies périodiques localement perturbées à une fréquence fixe. Ce problème a des connexions avec le contrôle non destructif des structures périodiques telles que des structures photoniques, des fibres optiques, des réseaux, etc. Nous analysons d'abord le problème direct et établissons certaines conditions sur l'indice de réfraction pour lesquelles il n'existe pas de modes guidés. Ce type de résultat est important car il montre les cas pour lesquels les mesures peuvent être effectuées par exemple sur une couche au dessus de la structure périodique sans perdre des informations importantes dans la partie propagative de l'onde. Nous proposons ensuite une méthode numérique pour résoudre le problème de diffraction basée sur l'utilisation de la transformée de Floquet-Bloch dans les directions de périodicité. Nous discrétisons le problème de manière uniforme dans la variable de Floquet-Bloch et utilisons une méthode spectrale dans la discrétisation spatiale. La discrétisation en espace exploite une reformulation volumétrique du problème dans une cellule (équation intégrale de Lippmann-Schwinger) et une périodisation du noyau dans la direction perpendiculaire à la périodicité. Cette dernière transformation permet d'utiliser des techniques de type FFT pour accélérer le produit matrice-vecteur dans une méthode itérative pour résoudre le système linéaire. On aboutit à un système d'équations intégrales couplées (à cause de la perturbation locale) qui peuvent être résolues en utilisant une décomposition de Jacobi. L'analyse de la convergence est faite seulement dans le cas avec absorption et la validation numérique est réalisées sur des exemples 2D. Pour le problème inverse, nous étendons l'utilisation de trois méthodes d'échantillonnage pour résoudre le problème de la reconstruction de la géométrie du défaut à partir de la connaissance de données mutistatiques associées à des ondes incidentes planes en champ proche (c.à.d incluant certains modes évanescents). Nous analysons ces méthodes pour le problème semi-discrétisée dans la variable Floquet-Bloch. Nous proposons ensuite une nouvelle méthode d'imagerie capable de visualiser directement la géométrie du défaut sans savoir ni les propriétés physiques du milieux périodique, ni les propriétés physiques du défaut. Cette méthode que l'on appelle imagerie-différentielle est basée sur l'analyse des méthodes d'échantillonnage pour un seul mode de Floquet-Bloch et la relation avec les solutions de problèmes de transmission intérieurs d'un type nouveau. Les études théoriques sont corroborées par des expérimentations numériques sur des données synthétiques. Notre analyse est faite d'abord pour l'équation d'onde scalaire où le contraste est sur le terme d'ordre inférieur de l'opérateur de Helmholtz. Nous esquissons ensuite l'extension aux cas où la le contraste est également présent dans l'opérateur principal. Nous complémentons notre travail par deux résultats sur l'analyse du problème de diffraction pour des matériaux périodiques ayant des indices négatifs. Nous établissons en premier le caractère bien posé du problème en 2D dans le cas d'un contraste est égal à -1. Nous montrons également le caractère Fredholm de la formulation Lipmann-Schwinger du problème en utilisant l'approche de T-coercivité dans le cas d'un contraste différent de -1. / We are interested in this thesis by the analysis of scattering and inverse scattering problems for locally perturbed periodic infinite layers at a fixed frequency. This problem has connexions with non destructive testings of periodic media like photonics structures, optical fibers, gratings, etc. We first analyze the forward scattering problem and establish some conditions under which there exist no guided modes. This type of conditions is important as it shows that measurements can be done on a layer above the structure without loosing substantial informations in the propagative part of the wave. We then propose a numerical method that solves the direct scattering problem based on Floquet-Bloch transform in the periodicity directions of the background media. We discretize the problem uniformly in the Floquet-Bloch variable and use a spectral method in the space variable. The discretization in space exploits a volumetric reformulation of the problem in a cell (Lippmann-Schwinger integral equation) and a periodization of the kernel in the direction orthogonal to the periodicity. The latter allows the use of FFT techniques to speed up Matrix-Vector product in an iterative to solve the linear system. One ends up with a system of coupled integral equations that can be solved using a Jacobi decomposition. The convergence analysis is done for the case with absorption and numerical validating results are conducted in 2D. For the inverse problem we extend the use of three sampling methods to solve the problem of retrieving the defect from the knowledge of mutistatic data associated with incident near field plane waves. We analyze these methods for the semi-discretized problem in the Floquet-Bloch variable. We then propose a new method capable of retrieving directly the defect without knowing either the background material properties nor the defect properties. This so-called differential-imaging functional that we propose is based on the analysis of sampling methods for a single Floquet-Bloch mode and the relation with solutions toso-called interior transmission problems. The theoretical investigations are corroborated with numerical experiments on synthetic data. Our analysis is done first for the scalar wave equation where the contrast is the lower order term of the Helmholtz operator. We then sketch the extension to the cases where the contrast is also present in the main operator. We complement our thesis with two results on the analysis of the scattering problem for periodic materials with negative indices. Weestablish the well posedness of the problem in 2D in the case of a contrast equals -1. We also show the Fredholm properties of the volume potential formulation of the problem using the T-coercivity approach in the case of a contrast different from -1.

Le caractère prédictif des simulations numériques faites pour dimensionner une structure dépend directement de la précision avec laquelle sont introduites les conditions aux limites imposées en service à la structure. Dans le cas d’une roue d’aéronef, les chargements appliqués par les différents pneumatiques qui peuvent être montés sur celle-ci lors des manœuvres au sol de l’aéronef jouent un rôle crucial dans la précision du modèle numérique. Cependant, ces chargements sont complexes et mal connus. Les charges globales appliquées à l’interface pneu-piste sont transférées à la roue au niveau de l’interface pneu-jante grâce au pneumatique, qui est un composant intermédiaire sophistiqué que ne maîtrise pas le manufacturier de roue. La détermination des conditions aux limites directement appliquées à la roue est donc particulièrement difficile et nécessite de faire des hypothèses sur le comportement en déformation du pneumatique. L’approche proposée dans le cadre de ces travaux de thèse consiste à identifier les chargements induits par le pneumatique sur la roue via la mesure de la déformée de cette dernière pour les différents cas de charge envisagés. L’objectif est de permettre au concepteur de la roue de mieux comprendre et de modéliser l’interaction entre le pneumatique et la roue afind’obtenir des simulations numériques plus prédictives. Pour ce faire, une paramétrisation objective des chargements à l’interface pneu-jante, aussi robuste et compacte que possible, est tout d’abord définie. Des mesures non-intrusives par corrélation d’images numériques (CIN) et jauges d’extensométrie, réalisées lors d’un post-doctorat antérieur à ces travaux de thèse, sont ensuite utilisées pour calibrer les paramètres recherchés. Une procédurede recalage de modèle par éléments finis est notamment mise en œuvre. Enfin, la problématique du placement optimal de capteurs est abordée afin de définir une instrumentation permettant de minimiser les incertitudes sur les paramètres à identifier. / The predictive character of numerical simulations run to design a structure critically depends on the accuracy with which the boundary conditions prescribedon the structure are introduced. For aircraft wheel manufacturers, the loadings applied to the wheel during ground maneuvering by various tires that are mounted play an important role. However, they are hardly known apart from the resultant force between the ground and the tire. Aircraft wheel manufacturers do not control the tire, which is a complex intermediate component, but must still work with it to design their products. Because tire-ground loads are transmitted to the wheel through the tire at the tire-rim interface, assumptions on the deformation behavior of the tire are to be made to determine the correct applied loading to the wheel and many experiments are often requiredto evaluate the wheel structural response. The knowledge of tire-rim interface loadings would thus provide an invaluable help to shorten aircraft wheel design processes. To tackle this challenge, an inverse identification procedure of tire-rim loadings for several cases is proposed herein. The aim is to identify the loadings applied by a tire on an aircraft wheel via the measurement of thewheel deformation and to allow the wheel manufacturer to better understand and model the interaction between the wheel and the tire. First, an objective parameterization of tire-rim loadings, which is as robust and compact as possible, is defined. Then, non-intrusive measurements (e.g., displacement fields obtained by Digital Image Correlation analyses, strains at gauge locations) obtained during a previous postdoctoral research are used to calibrate the loading parameters. A finite element model updating algorithmis used to solve this inverse problem. Last, the optimal sensor placement problem is considered to define an instrumentation that yields the most accurate estimâtes of the loading parameters.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第21212号 / 情博第665号 / 新制||情||115(附属図書館) / 京都大学大学院情報学研究科先端数理科学専攻 / (主査)教授 磯 祐介, 教授 木上 淳, 助手 藤原 宏志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

When scattering intensity is plotted against the dimensionless quantity qR, where q is the magnitude of the scattering wave vector and R is the radius of the particle, in log-log scale the scattering curve shows a power-law structure which defines characteristic crossovers. This work reveals some new relationships between the power-law structure and the particle properties. In this work, computer simulation results from T-matrix, Mie theory, and discrete dipole approximation methods are used to study the far field intensity and the internal field of the particles. Scattering by both weakly and strongly refractive particles are studied. For weakly refractive randomly oriented spheroidal particles, how the phasor cancellation-based tip volume method can be applied to predict the scattering envelope is demonstrated. The relationship between backscattering enhancement and the curvature of the weakly refractive particles is explained. In strongly-refractive particles when the phase shift parameter is high, regions with higher field amplitudes start to appear. These regions are recognized as the hot spot regions. In this work, a proper definition is given to the hot spot region. The relationships between the hot spot region and the power-law structure, between the hot spot region and the particle morphology, and between the power-law structure and the particle morphology are extensively studied for scattering by spherical particles. A new semi-quantitative phasor analysis method is introduced, and the new method is used with color-coded phasor plots to identify how different regions of the particle contribute to the scattering pattern to get an insight into the physics behind the scattering. How different regions of the particle contribute to the second crossover (SC) and the backscattering enhancement is presented. Relationships between the SC, particle size, and relative refractive index of the particle are derived. It was identified that the scattering angle at the SC depends only on the relative refractive index of the particle. How the findings of this work can be applied to solve the inverse electromagnetic scattering problem for a single non-absorbing spherical particle is also discussed.