Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid

We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated to the solution of a Laplace Neumann problem as the distance $\varepsilon>0$ between the solid and the cavity's bottom tends to zero. Denoting by $\alpha>0$ the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a non zero velocity for $\alpha <2$ (real shock case), and with null velocity for $\alpha \geqslant 2$ (smooth landing case). Our proof is based on a suitable change of variables sending to infinity the cusp singularity at the contact. More precisely, for every $\varepsilon\geqslant 0$, we transform the Laplace Neumann problem into a generalized Neumann problem set on a domain containing a horizontal strip $]0,\ell_\varepsilon[\times ]0,1[$, where $\ell_\varepsilon\to +\infty$

Reconstructing initial data using iterative observers for wave type systems. A numerical analysis

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently Proposed in Ramdani et al. (Automatica 46(10), 1616–1625, 2010). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE’s. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed)

Reconstructing initial data using observers : error analysis of the semi-discrete and fully discrete approximations

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss [15]. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and finite differences in time. The analysis is carried out for abstract Schr\"odinger and wave conservative systems with bounded observation (locally distributed).Comment: 38 pages, 1 figure

Observateurs itératifs en horizon fini. Application à la reconstruction de données initiales pour des EDP d'évolution

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss (Ramdani et al., 2010). The algorithm, which can be used for finite-dimensional and infinite-dimensional linear systems, is based on the use of two observers (also called Luenberger observers) used iteratively back and forth in time. In this paper, we first present the algorithm in a finite-dimensional context and then generalize it to Partial Differential Equations (PDE). To conclude, the method is applied to the 1D wave equation and numerical results will be given

PENGANIAYAAN BERAT SEBAGAI ALASAN PENGHALANG MEWARISI DALAM KOMPILASI HUKUM ISLAM PASAL 173 HURUF A

The purpose of this study was to explore severe persecution as the reason the barrier inherits in a compilation of Islamic law section 173 the letter A. the topic of “Severe Persecution” isan interesting topic both academic and general quarters about it can be highlighted in both the book and the legacy. Nextaya persecution heavy is the new form of inherited law Indonesia included in a compilation of Islamic law under section 173 of the lette A the view of cleric 4 of madzhab category of severe persecution is not through research. Using the yuridis normative and yuridis empiris method in the form of content analysis. After heavy mayhem is categorized as the reason for the obstacle inheriting acquired knowledge and understanding the data is then analyzed comprehensive to find the basis of his KHI’s law makes the persecution tough as an excuse for inheritable rule, the istinbath of KHI chapter 173 of the relationship between section 173 and the opinions of Indonesian scholars. The study found a result in a compilation of Islam law not given a clear, concrete understanding of what constitutes severe persecution. The priest hanafi argued that murder was not entitled to an inheritance by the one whom he killed whether intentionally or imbued, and then in this context the combination of Islamic law used the three methods of ushul fiqh in determining the renewal of the barrier, first, maslahah mursalah, second, sad dzariah, third, qiyas. And interrelated to the scholars’ opinion that a grouping of heiress in KHI chapter 173 is a legal renewal of heiress according to the code “law can change accourding to the circumstances.Penelitian ini bertujuan untuk mengeksplorasi penganiayaan berat sebagai alasan penghalang mewarisi dalam Kompilasi Hukum Islam pasal 173 huruf A. Topik mengenai “penganiayaan berat” merupakan topik menarik dikalangan akademik maupun dikalangan umum. Diskursus mengenai hal tersebut dapat disorot dari aspek hukum maupun kewarisan. Selanjutnya penganiayaan berat merupakan bentuk pembaharuan hukum kewarisan di Indonesia yang termaktub dalam Kompilasi Hukum Islam Pasal 173 huruf A sedangkan dalam pandangan Ulama 4 madzhab kategori penganiayaan berat tidak termasuk dalam pandangan para Ulama. Data penelitian ini diperoleh melalui penelitian yang menggunakan metode yuridis normatif dan yuridis empiris dengan bentuk content analysis. Setelah konsep penganiayaan berat yang dikategorikan sebagai alasan penghalang mewarisi dipelajari dan di fahami, data kemudian dianalisa secara komprehensif untuk menemukan dasar hukum KHI menjadikan penganiayaan berat sebagai alasan penghalang mewarisi, proses istinbath hukum KHI pasal 173 dan Hubungan antara pasal 173 dengan pendapat para Ulama Indonesia. Penelitian ini menemukan sebuah hasil Dalam Kompilasi Hukum Islam tidak diberikan pengertian yang jelas dan konkret tentang apa yang dimaksud dengan penganiayaan berat. Imam Hanafi berpendapat bahwasanya pembunuhan tidak berhak mendapatkan warisan dari seseorag yang ia bunuh baik dibunuh secara sengaja atau tidak sengaja, kemudian Dalam konteks ini Kompilasi Hukum Islam menggunakan tiga metode ushul fiqh dalam menentukan pembaharuan penghalang kewarisan, pertama, maslahah mursalah, kedua, sad dzariah, ketiga, qiyas. Dan Hubungan antar pasal dengan pendapat para ulama bahwasa­nya pengelompokkan penghalang waris dalam KHI pasal 173 adalah pembaharuan hukum waris sesuai dari sebuah kaidah “Hukum bisa berubah sesuai dengan keadaan tempat dan waktu

Iterative observers for inverse problems

This talk deals about the new iterative algorithm introduced by Ramdani, Tucsnak and Weiss last year for the reconstruction of the initial data of a linear system. Especially, I present the numerical analysis and some simulations of the full discretized algorithm applied to wave-type system. Then, we have shown with Kim Dang Phung that this algorithm can be used also for Maxwell’s equations

Reconstructing the potential for the 1D Schrödinger equation from boundary measurements

International audienceWe consider the inverse problem of determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the boundary. We use the Boundary Control method to recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We apply the algorithm to the situation when only the finite number of eigenvalues are known and prove the convergence of the method

Calderón cavities inverse problem as a shape-from-moments problem

International audienceIn this paper, we address a particular case of Calderón's (or conductivity) inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e. heterogeneities of infinitely high conductivities). We aim to recover the location and the shape of the cavities from the knowledge of the Dirichlet-to-Neumann (DtN) map of the problem. The proposed reconstruction method is non iterative and uses two main ingredients. First, we show how to compute so-called generalized Pólia-Szegö tensors (GPST) of the cavities from the DtN of the cavities. Secondly, we show that the obtained shape from GPST inverse problem can be transformed into a shape from moments problem, for some particular configurations. However, numerical results suggest that the reconstruction method is efficient for arbitrary geometries

Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors

International audienceWe investigate the acoustic selective focusing properties of time reversal in a two-dimensional acoustic waveguide. A far-field model of the problem is proposed in the time-harmonic case. In order to tackle the question of selective focusing, we derive an asymptotic model for small scatterers. We show that in the framework of this limit problem, approximate eigenvectors of the time reversal operator can be obtained when the number of propagating modes of the waveguide is large enough. This result provides, in particular, a mathematical justification of the selective focusing properties observed experimentally. Some numerical experiments on selective focusing are presented