The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem

The interior transmission problem appears naturally in the study of the inverse scattering problem of determining the shape of a penetrable medium from a knowledge of the time harmonic incident waves and the far field patterns of the scattered waves. We propose a variational study of this problem in the case of Maxwell's equations in an inhomogeneous anisotropic medium. Then we apply the obtained results to build an ‘extented far field’ operator and give a characterization of the medium from the knowledge of the range of this operator. We then show how the linear sampling method can be viewed as an approximation of this characterization. Copyright © 2004 John Wiley & Sons, Ltd.     

A Significant ‘False Perception’ of Wittgenstein’s Draft on Mind’s Eye

If we read the Tractatus logico-philosophicus according to the decimal numbering of its propositions, we may understand, finally, the section about the self and the limits of language and world. Proposition 5.64 follows 5.63 (not 5.634); 5.634 follows 5.633 (not 5.6331); and so on. Thus, it becomes clear that the picture of the visual field (TLP 5.6331) cannot be what scholars have always quoted and discussed, i.e. a draft of an eye inside its field of sight. Actually, Wittgenstein’s original drafts depict (to criticise it) the ordinary way of representing the visual field. Following him, the field of vision is ‘without limits’; it does not have a form that implies the existence of an eye, as far as the phenomenological experience does not have a form that implies the existence of an ‘I’. As a result, the current reproductions of the 5.6331 image must be rectified, and all references to this similitude should be radically amended.     

A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three dimensional domains

A few well-posed variational problems are constructed, whose solutions are vector potentials of the velocity field of the three-dimensional Stokes problem. A choice of the adequate boundary conditions to be imposed is performed in a systematic manner even in the case of not simply connected domains. The test functions do not have to be divergence free in the formulations given here. It is then seen that a similar approach is well suited to the treatment of the non-linear Navier-Stokes problem.     

Inverse scattering from an orthotropic medium

We consider the inverse scattering problem of determining the shape of an orthotropic homogeneous medium from a knowledge of the far field pattern corresponding to incident time harmonic plane waves. We solve the direct problem by potential theory, show the uniqueness of the inverse problem and then use a ‘simple’ method recently developed by Colton and Kirsch (1996) and Colton and Monk (1997) to solve the inverse problem. Numerical examples are given showing the practicality of our method for solving the inverse problem.     

Another approach to linearized elasticity and Korn's inequality

We describe and analyze an approach to the pure traction problem of three-dimensional linearized elasticity, whose novelty consists in considering the linearized strain tensor as the ‘primary’ unknown, instead of the displacement itself as is customary. This approach leads to a well-posed minimization problem, constrained by a weak form of the St Venant compatibility conditions. It also provides a new proof of Korn's inequality. To cite this article: P.G. Ciarlet, P. Ciarlet Jr., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation

In this paper we study a problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation. We prove that the stated problem is well-posed.     

Well-Posedness of a Problem of Propagation of Nonlinear Acoustic-Gravity Waves in the Atmosphere Generated by Surface Pressure Variations

Currently there are many international microbarograph networks for high-resolution recording of wave pressure variations on the Earth’s surface. This arouses interest in wave propagation in the atmosphere generated by atmospheric pressure variations. A full system of nonlinear hydrodynamic equations for atmospheric gases with lower boundary conditions in the form of wavelike pressure variations on the Earth’s surface is considered. Since the wave amplitudes near the Earth’s surface are small, linearized equations are used in the analysis of well-posedness of the problem. With the help of a wave energy functional method, it is shown that in the non-dissipative case the solution to the boundary value problem is uniquely determined by the variable pressure field on the Earth’s surface. The corresponding dissipative problem is well-posed if, in addition to the pressure field, appropriate conditions on the velocity and temperature on the Earth’s surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t. A good agreement between the numerical and analytical solutions is obtained. The study shows that the temperature and density can rapidly vary at the lower boundary of the boundary value problem. An example of solving the three-dimensional problem with variable pressure on the Earth’s surface taken from experimental observations is given. The developed algorithms and computer programs can be used to simulate atmospheric waves generated by pressure variations on the Earth’s surface.     

Removal of numerical instability in the solution of an inverse heat conduction problem

In this paper, we consider an inverse heat conduction problem (IHCP). A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a transformation, the ill-posed IHCP becomes a Cauchy problem. Since the solution of Cauchy problem, exists and is unique but not always stable, the ill-posed problem is closely approximated by a well-posed problem. For this new well-posed problem, the existence, uniqueness, and stability of the solution are proved.     

Numerical analysis of the equations of small strains quasistatic elastoviscoplasticity

Summary The present paper deals with the mathematical and the numerical analysis of small strains elastoviscoplasticity. By considering the problem as an evolution equation whose only unknown is the stress field, the quasistatic elastoviscoplastic evolution problem is proved to be well-posed, consistent mixed finite element approximations are introduced, and classical numerical algorithms are interpreted. In particular, augmented Lagrangian methods operating on the velocity appear as standard alternating-directions time-integrations of this stress evolution problem.     

Conditionally well-posed and generalized well-posed problems

It is proved that, for a pair of metric spaces, the operators of abstract conditionally well-posed problems admit extensions that are continuous on the original domain with respect to the ambient space. As a corollary, it is shown that an arbitrary conditionally well-posed problem can be regularized via an operator independent of the error level in the input data. Certain applications to ill-posed operator equations and variational problems are discussed.     

Well-posedness and bang-bang principle

In this paper a property P ,that may be considered a generalization for some minimum problems of the bang-bang principle, is defined. It is proved that if a minimum problem satisfies property .P and is weakly well-posed then it is strongly well-posed in the sense of Tikhonov, The results are also applied to the time-optimal control problem.     

Minimizing the Range of Order Completion Times with Multiple Job Classes

A number of important contributions have been made toward the problem of minimizing the difference in ‘customer’ treatment on a single machine. However, so far, all the research has been to solve the problem of minimizing the ‘job’ difference. That is, it is assumed implicitly that each customer places only one job, and hence minimizing the difference in customer treatment is equivalent to minimizing the difference in job treatment. In many practical situations, a customer may place an order which consists of several jobs. The jobs usually can be grouped into different classes. Thus, the range, i.e. the difference between the maximum and the minimum, of order completion times becomes an appropriate performance measure for treating customers equally. In this paper, a dynamic programming (DP) algorithm is developed to provide the optimal solution. Due to the exponential time complexity of the DP algorithm, two heuristic methods are also proposed to solve large-sized problems. Computational results for the heuristics are provided.     

Formal optimization of some reduced linear programming problems

It is often possible (and profitable) to reduce or ‘Presolve’ linear programs. In particular, there are frequently constraints which force many of the variables to be at bound. Unfortunately, the solution found by the simplex method for such reduced models is not usually ‘formally’ optimal, in the sense that nonoptimal dual values may be present when the original problem is restored. Furthermore, the restored (full) problem is now totally degenerate, and may require many iterations to achieved formal optimality. We describe an efficient ‘Postsolve’ procedure for attaining the formal optimum solution, and give computational results.     

OR in strategic land-use planning

An important historic strategic application of OR has been in the field of land-use and development plan production. Changes in Government policy and legislation have led to varying levels of interest in plan production. Three post-war cycles of ‘enthusiasm for plans’ can be identified. Whilst the first was rooted very much in the Architectural Design tradition, the second led to significant developments in OR, with far wider application. Subsequent reduced Governmental enthusiasm for ‘Development Plan production’ led to considerable atrophy of relevant skills in the planning community, including those derived from OR. However, the current ‘third period of post-war enthusiasm for planning’, reinforced by environmental concerns, has revived the need for relevant skills. It is suggested that, whilst the deficit in skills and their application remains high, there are some encouraging signs. Moreover, substantial progress in the field of ‘soft OR’ offers opportunities to both the OR and planning communities.     

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.

One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

     

On an elliptic-parabolic MEMS model with two free boundaries

We discuss an evolution free boundary problem of mixed type with two free boundaries modeling an idealized electrostatically actuated MEMS device. While the electric potential is the solution of an elliptic equation, the dynamics of the membranes’ displacement is modeled by two parabolic equations. It is shown that the model is locally well-posed in time and that solutions exist globally for small source voltages whereas non-existence holds for large voltage values. Moreover, our model possesses a steady state solution that is asymptotically stable. Finally, we show that in the vanishing aspect ratio limit, solutions of the model converge toward solutions of the associated small aspect ratio problem.     

A revenue management approach to address a truck rental problem

This paper describes an application of revenue management techniques and policies in the field of logistics and distribution. In particular, the problem of transportation operators, who offer products for hire, is considered. A product is a truck of a given capacity, which can be rented for one or several time periods, throughout a multi-period planning horizon. The logistic operator can satisfy the demand of a given product with trucks with a capacity greater than that initially required, that is an ‘upgrade’ can take place. In this context, the logistic operator has to decide whether to accept or reject a request and which type of truck should be used to address it. For this purpose, a dynamic programming (DP) formulation of the problem under consideration is devised. The ‘course of dimensionality’ leads to the necessity of introducing different mathematical programming models to represent the problem. The mathematical models we presented are an extension of the well-known approximations for the DP of traditional network capacity management analysis. Based on these models and exploiting revenue management concepts, primal and dual acceptance policies are developed and compared in a computational study.     

Process and content: two ways of using SSM

Soft systems methodology (SSM) includes several ways of gaining a rich appreciation of the problem situation addressed. ‘Analysis One’, exploration of the intervention itself, is the subject here, since it is sparsely covered in the literature. The analysis is conducted in terms of three roles: ‘client’, ‘problem solver’ and ‘problem owner’. Whoever is in the role of ‘problem solver’ is free to define a list of possible ‘problem owners’, which brings many perspectives to bear on the situation. It was realized that ‘client’ and ‘problem solver’ should themselves feature in the ‘problem owner’ list. The ‘problem’ owned by the ‘problem solver’ is that of undertaking the intervention. This led to a realization that SSM is relevant to both the content of a perceived situation (SSMc) and the process of dealing with that content (SSMp). This development is described and illustrated by work in the National Health Service. The focus of the SSM use was to define the intellectual process for a service specification project which NHS professionals would themselves carry out.     

‘Minesweeper’ and spectrum of discrete Laplacians

The article is devoted to a problem inspired by the ‘Minesweeper’ computer game. It is shown that certain configurations of open cells guarantee the existence and the uniqueness of solution. Mathematically the problem is reduced to some spectral properties of discrete differential operators. It is shown how the uniqueness can be used to create a new game which preserves the spirit of ‘Minesweeper’ but does not require a computer.