Higher order Hochschild cohomology of schemes

We show that Higher Hochschild complex associated to a connected pointed simplicial set commutes with localization of commutative algebras over a field of characteristic zero. Then, we define in two ways higher order Hochschild cohomology of schemes over a field of characteristic zero. Originally, we can take the hyperext functor of the sheaf associated to Higher Hochschild presheaf. We obtain a Hodge decomposition for higher order Hochschild cohomology of smooth algebraic varieties over a field of characteristic zero which generalizes Pirashvili's Hodge decomposition. We can also define the Higher Hochschild cohomology of order d of a separated scheme by taking the ext functor of its structure presheaf over the Higher Hochschild presheaf of order d -- 1. These two definitions are really close to those of Swanfor classical Hochschild cohomology, but our tools are model categories and derived functors. We also generalize the equivalence of Swan's definitions to any separated schemes over a field

Peri-urbanisation, Social Heterogeneity and Ecological Simplification

Peri-urban development pressure on and near Australian coastlines is resulting in the conversion of agricultural land for rural-residential use. The impact of larger and more diverse human populations upon the ecological assets remaining in agricultural landscapes has consequently become a policy concern. This paper contributes to these policy debates by integrating the results of parallel social and ecological research projects commissioned to improve natural resource management in peri-urbanising regions. The research was undertaken in the case study region of South East Queensland, the region supporting Australia’s most rapid population growth. Our results indicate that both social and ecological communities cross a fragmentation threshold due to peri-urban development whereby they become ecologically simple and socially heterogeneous in a coupling that cedes a poor diagnosis for biodiversity retention.stored soil water, dryland grain cropping, extension, social systems, RD&E, differentiation

Combining Analytic Preconditioner and Fast Multipole Method for the 3-D Helmholtz Equation

International audienceThe paper presents a detailed numerical study of an iterative solution to 3-D sound-hard acoustic scattering problems at high frequency considering the Combined Field Integral Equation (CFIE). We propose a combination of an OSRC preconditioning technique and a Fast Multipole Method which leads to a fast and efficient algorithm independently of both a frequency increase and a mesh refinement. The OSRC-preconditioned CFIE exhibits very interesting spectral properties even for trapping domains. Moreover, this analytic preconditioner shows highly-desirable advantages: sparse structure, ease of implementation and low additional computational cost. We first investigate the numerical behavior of the eigenvalues of the related integral operators, CFIE and OSRC-preconditioned CFIE, in order to illustrate the influence of the proposed preconditioner. We then apply the resolution algorithm to various and significant test-cases using a GMRES solver. The OSRC-preconditioning technique is combined to a Fast Multipole Method in order to deal with high-frequency 3-D cases. This variety of tests validates the effectiveness of the method and fully justifies the interest of such a combination

Eddy current interaction between a probe coil and a conducting plate

International audienceConsider a coil above a conducting plate. The interaction between the probe-coil and the plate is modeled by a quasi-static approximation of Maxwell's equations: the eddy current model. The associated electromagnetic transmission boundary-value problem can be solved by the integral equations method. However, the discretization of integral operators gives dense, complex and ill-conditioned linear systems. We present here a method to compute the reaction field and the coil impedance variation by solving only surface partial differential equations

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

International audienceThe fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles

An introduction to operator preconditioning for the fast iterative integral equation solution of time-harmonic scattering problems

International audienceThe aim of this paper is to provide an introduction to the improved iterative Krylov solution of boundary integral equations for time-harmonic scattering problems arising in acoustics, electromagnetism and elasticity. From the point of view of computational methods, considering large frequencies is a challenging issue in engineering since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. More specifically, we explain the problematic and some partial solutions through analytical preconditioning for high-frequency acoustic scattering problems and the introduction of new combined field integral equations. We complete the paper with some recent extensions to the case of electromagnetic and elastic waves equations

Integral Equations and Iterative Schemes for Acoustic Scattering Problems

International audienceThe aim of this chapter is to provide an introduction to the iterative Krylov solution of integral equations for time-harmonic acoustic scattering. From the point of view of computational methods, considering large frequencies in acoustics is challenging since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. Most specifically, we develop the problematic and explain some partial solutions through analytical preconditioning for high frequency scattering and the introduction of new combined field integral equations