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![The boundary ΓC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _C$$\end{document} is accessible to measures and ΓI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _I$$\end{document} is out of reach](profile/Faker-Ben-Belgacem/publication/308675460/figure/fig2/AS:960034745892865@1605901528486/The-boundary-GCdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym.gif)
The boundary ΓC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _C$$\end{document} is accessible to measures and ΓI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _I$$\end{document} is out of reach
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The variational finite element solution of Cauchy's problem, expressed in the Steklov-Poincaré framework and regularized by the Lavrentiev method, has been introduced and computationally assessed in [Inverse Problems in Science and Engineering, 18, 1063–1086 (2011)]. The present work concentrates on the numerical analysis of the semi-discrete probl...
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The outcome of data analysis depends on the quality and completeness of data. This paper considers various techniques for filling in missing precipitation data. To assess suitability of the different methods for filling in missing data, monthly precipitation data collected at six different stations was considered. The complete sets (with no missing...
Citations
... To readers interested in these subjects and in connected topics, we suggest the non-exhaustive list of publications [29,2,10,13,24,12,14,22,36,19]. References therein are also worth consulting. ...
We focus on the ill posed data completion problem and its finite element approximation, when recast via the variational duplication Kohn-Vogelius artifice and the condensation Steklov-Poincaré operators. We try to understand the useful hidden features of both exact and discrete problems. When discretized with finite elements of degree one, the discrete and exact problems behave in diametrically opposite ways. Indeed, existence of the discrete solution is always guaranteed while its uniqueness may be lost. In contrast, the solution of the exact problem may not exist, but it is unique. We show how existence of the so called "weak spurious modes", of the exact variational formulation, is source of instability and the reason why existence may fail. For the discrete problem, we find that the cause of non uniqueness is actually the occurrence of "spurious modes". We track their fading effect asymptotically when the mesh size tends to zero. In order to restore uniqueness, we recall the discrete version of the Holmgren principle, introduced in [Azaïez et al, IPSE, 18, 2011], and we discuss the effect on uniqueness of the finite element mesh, using some graph theory basic material.
... To readers interested in connected topics, we suggest the non-exhaustive list of publications and references therein [29,3,10,12,24,22,36,19,2,17]. ...
... Proof: The proof follows the same lines as in [10,Lemma 4.3]. The regularity exponent q is linked to the aperture of the largest inner angle at corners of the boundary Γ C . ...
... The purpose of the present paper is to complete the work started in [9], where the data completion had been regularized by a Lavrentiev procedure and partially approximated by a finite element method. Partially in the sense that only the unknown missing boundary data at the incomplete part of the boundary was computed as a piecewise affine finite element function. ...
... This may be compared to the finite element discretization of the Poisson boundary value problem without numerical integration. Statements in [9] on the convergence with respect to small mesh-size are listed. Section 4 proposes a full discretization of the problem. ...
... Γ I ). The proof of the following lemmas is done by a cut-off argument, see [9,Lemma 4.3], ...
... Several regularization techniques has been proposed to tackle Problem (1.1). Without being exhaustive, we may mention methods based on surface integral equations [12,23], Lavrentiev regularization [10,11], stabilized finite elements methods [15][16][17], quasi-reversibility method [13,18,26,35,36], fading regularization method [24,27], etc. ...
... Belgacem et al. [33] proposent un critère de convergence qui découle du choix du paramètre de régularisation par le principe de Morozov [164]. La solution variationnelle discrétisée par éléments finis et régularisée par la méthode de Lavrentiev est introduite dans [19] et récemment il a été montré que le cadre variationnel de Steklov-Poincaré est également adapté à l'analyse de convergence de la méthode éléments finis semi-discrète [34]. L'algorithme de Steklov-Poincaré est également applicable aux problèmes inverses de reconstruction de sources supportées par des surfaces avec la nécessité de connaître la localisation des sources [20]. ...
Les problèmes de complétion de données interviennent dans divers domaines de la physique, tels que la mécanique, l'acoustique ou la thermique. La mesure directe des conditions aux limites se heurte souvent à l'impossibilité de placer l'instrumentation adéquate. La détermination de ces données n'est alors possible que grâce à des informations complémentaires. Des mesures surabondantes sur une partie accessible de la frontière mènent à la résolution d'un problème inverse de type Cauchy. Cependant, dans certains cas, des mesures directes sur la frontière sont irréalisables, des mesures de champs plus facilement accessibles permettent de pallier ce problème. Cette thèse présente des méthodes de régularisation évanescente qui permettent de trouver, parmi toutes les solutions de l'équation d'équilibre, la solution du problème de complétion de données qui s'approche au mieux des données de type Cauchy ou de champs partiels. Ces processus itératifs ne dépendent pas d'un coefficient de régularisation et sont robustes vis à vis du bruit sur les données, qui sont recalculées et de ce fait débruitées. Nous nous intéressons, dans un premier temps, à la résolution de problèmes de Cauchy associés à l'équation d'Helmholtz. Une étude numérique complète est menée, en utilisant la méthode des solutions fondamentales en tant que méthode numérique pour discrétiser l'espace des solutions de l'équation d'Helmholtz. Des reconstructions précises attestent de l'efficacité et de la robustesse de la méthode. Nous présentons, dans un second temps, la généralisation de la méthode de régularisation évanescente aux problèmes de complétion de données à partir de mesures de champs partielles. Des simulations numériques, pour l'opérateur de Lamé, dans le cadre des éléments finis et des solutions fondamentales, montrent la capacité de la méthode à compléter et débruiter des données partielles de champs de déplacements et à identifier les conditions aux limites en tout point de la frontière. Nous retrouvons des reconstructions précises et un débruitage efficace des données lorsque l'algorithme est appliqué à des mesures réelles issues de corrélation d'images numériques. Un éventuel changement de comportement du matériau est détecté grâce à l'analyse des résidus de déplacements.
We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L 2 -boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria.
