From Whitney Forms to Metamaterials: a Rigorous Homogenization Theory

A rigorous homogenization theory of metamaterials -- artificial periodic structures judiciously designed to control the propagation of electromagnetic waves -- is developed. All coarse-grained fields are unambiguously defined and effective parameters are then derived without any heuristic assumptions. The theory is an amalgamation of two concepts: Smith & Pendry's physical insight into field averaging and the mathematical framework of Whitney-Nedelec-Bossavit-Kotiuga interpolation. All coarse-grained fields are defined via Whitney forms and satisfy Maxwell's equations exactly. The new approach is illustrated with several analytical and numerical examples and agrees well with the established results (e.g. the Maxwell-Garnett formula and the zero cell-size limit) within the range of applicability of the latter. The sources of approximation error and the respective suitable error indicators are clearly identified, along with systematic routes for improving the accuracy further. The proposed approach should be applicable in areas beyond metamaterials and electromagnetic waves -- e.g. in acoustics and elasticity.Comment: 23 pages, 10 figure

A brief historical perspective of the Wiener-Hopf technique

It is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener–Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener–Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems

A Homological Approach to Belief Propagation and Bethe Approximations

We introduce a differential complex of local observables given a decomposition of a global set of random variables into subsets. Its boundary operator allows us to define a transport equation equivalent to Belief Propagation. This definition reveals a set of conserved quantities under Belief Propagation and gives new insight on the relationship of its equilibria with the critical points of Bethe free energy.Comment: 14 pages, submitted for the 2019 Geometric Science of Information colloquiu

Model Order Reduction based on Proper Generalized Decomposition for the Propagation of Uncertainties in Structural Dynamics

International audienceA priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The Proper Generalized Decomposition method is used to construct a quasi-optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. The dynamic response being highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension

Solid Modeling

To appear in the Encyclopedia of Electrical and Electronics Engineering, Ed. J. Webster, John Wiley & Sons, 1999.A solid model is a digital representation of the geometry of an existing or envisioned physical object. Solid models are used in many industries, from entertainment to health care. They play a major role in the discrete-part manufacturing industries, where precise models of parts and assemblies are created using solid modeling software or more general computer-aided design (CAD) systems. Solid modeling is an interdisciplinary field that involves a growing number of areas. Its objectives evolved from a deep understanding of the practices and requirements of the targeted application domains. Its formulation and rigor are based on mathematical foundations derived from general and algebraic topology, and from Euclidean, differential, and algebraic geometry. The computational aspects of solid modeling deal with efficient data structures and algorithms, and benefit from recent developments in the field of computational geometry. Efficient processing is essential, because the complexity of industrial models is growing faster than the performance of commercial workstations. Techniques for modeling and analyzing surfaces and for computing their intersections are important in solid modeling. This area of research, sometimes called computer aided geometric design, has strong ties with numerical analysis and differential geometry. Graphic user-interface (GUI) techniques also play a crucial role in solid modeling, since they determine the overall usability of the modeler and impace the user's productivity. There have always been strong symbiotic links and overlaps between the solid modeling community and the computer graphics community. Solid modeling interfaces are based on efficient three-dimensional (3D) graphics techniques, whereas research in 3D graphics focuses on fast or photo-realistic rendering of complex scenes, often composed of solid models, and on realistic or artistic animations of non-rigid objects. A similar symbiotic relation with computer vision is regaining popularity, as many research efforts in vision are model-based and attempt to extract 3D models from images or video sequences of existing parts or scenes. These efforts are particularly important for solid modeling, because the cost of manually designing solid models of existing objects or scenes far excees the other costs (hardware, software, maintenance, and training) associated with solid modeling. Finally, the growing complexity of solid models and the growing need for collaboration, reusability of design, and interoperability of software require expertise in distributed databases, constraint management systems, optimization techniques, object linking standards, and internet protocols. This report provides a brief overview of the solid modeling field, its fundamental technologies, and some important applications

O-Minimality and its Applications to Number Theory and Analysis

The workshop brought together researchers in the areas of o-minimal structures, analysis and number theory. The latest developments in o-minimality and their applications to number theory and analysis were presented in a series of talks; one focus, in particular, was on the Pila–Wilkie Theorem and its impact on diophantine problems

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

[Activity of Institute for Computer Applications in Science and Engineering]

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science