Cost-efficient design and production of flexible and re-usable near real-time tactical human-machine interfaces

International audienceMaking complex systems accessible to human operators supposes to design HMIs that provide the operator with means to manage the complexity in an efficient manner. This is particularly true in the aeronautics domain for tactical HMIs where complexity is present in many dimensions. Current technical requirements, such as being able to display thousands of objects updated on the basis of time intervals inferior to half a second, coupled with economic requirements such as manning and cost reductions, make this issue even more crucial. We present our approach to the design and production of near real-time tactical HMIs, that enables us to devise HMIs that meet such requirements while being flexible enough to be re- used in a wide variety of contexts and produced at a reasonable cost

A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions

Stents are medical devices designed to modify blood flow in aneurysm sacs, in order to prevent their rupture. Some of them can be considered as a locally periodic rough boundary. In order to approximate blood flow in arteries and vessels of the cardio-vascular system containing stents, we use multi-scale techniques to construct boundary layers and wall laws. Simplifying the flow we turn to consider a 2-dimensional Poisson problem that conserves essential features related to the rough boundary. Then, we investigate convergence of boundary layer approximations and the corresponding wall laws in the case of Neumann type boundary conditions at the inlet and outlet parts of the domain. The difficulty comes from the fact that correctors, for the boundary layers near the rough surface, may introduce error terms on the other portions of the boundary. In order to correct these spurious oscillations, we introduce a vertical boundary layer. Trough a careful study of its behavior, we prove rigorously decay estimates. We then construct complete boundary layers that respect the macroscopic boundary conditions. We also derive error estimates in terms of the roughness size epsilon either for the full boundary layer approximation and for the corresponding averaged wall law.Comment: Dedicated to Professor Giovanni Paolo Galdi 60' Birthda

Boundary regularity for the Poisson equation in reifenberg-flat domains

This paper is devoted to the investigation of the boundary regularity for the Poisson equation {{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $\alpha\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally H\"older continuous provided that $\Omega$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem

The Influence of Quadrature Errors on Isogeometric Mortar Methods

Mortar methods have recently been shown to be well suited for isogeometric analysis. We review the recent mathematical analysis and then investigate the variational crime introduced by quadrature formulas for the coupling integrals. Motivated by finite element observations, we consider a quadrature rule purely based on the slave mesh as well as a method using quadrature rules based on the slave mesh and on the master mesh, resulting in a non-symmetric saddle point problem. While in the first case reduced convergence rates can be observed, in the second case the influence of the variational crime is less significant

Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers

The rotated multipliers method is performed in the case of the boundary stabilization by means of a(linear or non-linear) Neumann feedback. this method leads to new geometrical cases concerning the "active" part of the boundary where the feedback is apllied. Due to mixed boundary conditions, these cases generate singularities. Under a simple geometrical conditon concerning the orientation of boundary, we obtain a stabilization result in both cases.Comment: 17 pages, 9 figure

Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients

This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK

Bayesian inverse problems for recovering coefficients of two scale elliptic equations

We consider the Bayesian inverse homogenization problem of recovering the locally periodic two scale coefficient of a two scale elliptic equation, given limited noisy information on the solution. We consider both the uniform and the Gaussian prior probability measures. We use the two scale homogenized equation whose solution contains the solution of the homogenized equation which describes the macroscopic behaviour, and the corrector which encodes the microscopic behaviour. We approximate the posterior probability by a probability measure determined by the solution of the two scale homogenized equation. We show that the Hellinger distance of these measures converges to zero when the microscale converges to zero, and establish an explicit convergence rate when the solution of the two scale homogenized equation is sufficiently regular. Sampling the posterior measure by Markov Chain Monte Carlo (MCMC) method, instead of solving the two scale equation using fine mesh for each proposal with extremely high cost, we can solve the macroscopic two scale homogenized equation. Although this equation is posed in a high dimensional tensorized domain, it can be solved with essentially optimal complexity by the sparse tensor product finite element method, which reduces the computational complexity of the MCMC sampling method substantially. We show numerically that observations on the macrosopic behaviour alone are not sufficient to infer the microstructure. We need also observations on the corrector. Solving the two scale homogenized equation, we get both the solution to the homogenized equation and the corrector. Thus our method is particularly suitable for sampling the posterior measure of two scale coefficients

Second-order L2L^2-regularity in nonlinear elliptic problems

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the existence and square integrability of the weak derivatives of the nonlinear expression of the gradient under the divergence operator. This provides a nonlinear counterpart of the classical $L^2$-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are established. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required. If the domain is convex, no regularity of its boundary is needed at all

Maximal regularity for non-autonomous equations with measurable dependence on time

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication in Potential Analysi

Sparse initial data indentification for parabolic pde and its finite element approximations

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.The first author was supported by Spanish Ministerio de Economía y Competitividad under project MTM2011-22711. The third author was supported by the Advanced Grant NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the BERC 2014-2017 program of the Basque Government, the MTM2011-29306 and SEV-2013-0323 Grants of the MINECO, the CIMI-Toulouse Excellence Chair in PDEs, Control and Numerics and a Humboldt Award at the University of Erlangen-Nürnberg