This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Our understanding of the "long range" electrodynamic, electrostatic, and polar interactions that dominate the organization of small objects at separations beyond an interatomic bond length is reviewed. From this basic-forces perspective, a large number of systems are described from which one can learn about these organizing forces and how to modulate them. The many practical systems that harness these nanoscale forces are then surveyed. The survey reveals not only the promise of new devices and materials, but also the possibility of designing them more effectively.

Long range interactions in nanoscale science

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.

We demonstrate that the HDG approach generates a global trace space system for the unknown that although larger in rank than the traditional static condensation system in CG, has significantly smaller bandwidth at moderate polynomial orders. We show that if one ignores set-up costs, above approximately fourth-degree polynomial expansions on triangles and quadrilaterals the HDG method can be made to be as efficient as the CG approach, making it competitive for time-dependent problems even before taking into consideration other properties of DG schemes such as their superconvergence properties and their ability to handle hp-adaptivity.

/pdf/to-cg-or-to-hdg-a-comparative-study-2suwx663kx.pdf

To CG or to HDG: A Comparative Study

Human Action Recognition (HAR) involves human activity monitoring task in different areas of medical, education, entertainment, visual surveillance, video retrieval, as well as abnormal activity identification, to name a few. Due to an increase in the usage of cameras, automated systems are in demand for the classification of such activities using computationally intelligent techniques such as Machine Learning (ML) and Deep Learning (DL). In this survey, we have discussed various ML and DL techniques for HAR for the years 2011–2019. The paper discusses the characteristics of public datasets used for HAR. It also presents a survey of various action recognition techniques along with the HAR applications namely, content-based video summarization, human–computer interaction, education, healthcare, video surveillance, abnormal activity detection, sports, and entertainment. The advantages and disadvantages of action representation, dimensionality reduction, and action analysis methods are also provided. The paper discusses challenges and future directions for HAR.

A survey on video-based Human Action Recognition: recent updates, datasets, challenges, and applications

Hybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows.

https://hal.archives-ouvertes.fr/hal-02151813/document

The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications

Analysis of variable-degree HDG methods for convection–diffusion equations. Part I: general nonconforming meshes

In this paper, we provide a projection-based analysis of the hversion of the hybridizable discontinuous Galerkin methods for convectiondiffusion equations on semimatching nonconforming meshes made of simplexes; the degrees of the piecewise polynomials are allowed to vary from element to element. We show that, for approximations of degree k on all elements, the order of convergence of the error in the diffusive flux is k+ 1 and that that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k > 0. We also show that, for the variable-degree case, the projection of the error in the scalar variable is h-times the projection of the error in the vector variable, provided a simple condition is satisfied for the choice of the degree of the approximation on the elements with hanging nodes. These results hold for any (bounded) irregularity index of the nonconformity of the mesh. Moreover, our analysis can be extended to hypercubes.

Analysis of variable-degree HDG methods for convection-diffusion equations. Part II: Semimatching nonconforming meshes

https://hal.archives-ouvertes.fr/hal-00644805/document

Yanlai Chen

Papers

Analysis of variable-degree HDG methods for convection–diffusion equations. Part I: general nonconforming meshes

Analysis of variable-degree HDG methods for convection-diffusion equations. Part II: Semimatching nonconforming meshes

Certified reduced basis method for electromagnetic scattering and radar cross section estimation

Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension

A new discontinuous Galerkin method, conserving the discrete H2-norm, for third-order linear equations in one space dimension