Abstract. In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtaining improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the setting and convergence analysis of the data assimilation algorithm.MSC Subject Classifications: 35Q35, 76B03, 86A10.

In this paper we formulate and analyze the Multi-Configuration Time-Dependent Hartree-Fock (MCTDHF) equations for molecular systems with pairwise interaction. This is an approximation of the N -particle timedependent Schrödinger equation which involves (time-dependent) linear combination of (time-dependent) Slater determinants. The mono-electronic wavefunctions satisfy nonlinear Schrödinger-type equations coupled to a linear system of ordinary differential equations equations for the expansion coefficients. The invertibility of the one-body density matrix (full-rank hypothesis) plays a crucial rôle in the analysis. Under the full-rank assumption a fiber bundle structure shows up and produces unitary equivalence between different useful representations of the approximation. We establish existence and uniqueness of maximal solutions to the Cauchy problem in the energy space as long as the density matrix is not singular for a large class of interactions (including Coulomb potential). A sufficient condition in terms of the energy of the initial data ensuring the global-in-time invertibility is provided (first result in this direction). Regularizing the density matrix breaks down energy conservation. However a global well-posedness for this system in L 2 is obtained with Strichartz estimates. Eventually solutions to this regularized system are shown to converge to the original one on the time interval when the density matrix is invertible.

a b s t r a c tThe multiconfiguration time-dependent Hartree-Fock (MCTDHF for short) system is an approximation of the linear many-particle Schrödinger equation with a binary interaction potential by nonlinear "one-particle" equations. MCTDHF methods are widely used for numerical calculations of the dynamics of few-electron systems in quantum physics and quantum chemistry, but the time-dependent case still poses serious open problems for the analysis, e.g. in the sense that global-in-time existence of solutions is not proved yet. In this letter we present the first result ever where global existence is proved under a condition on the initial datum that it has to be somewhat close to the "ground state".

This paper presents an efficient cyberphysical platform for the smart management of smart territories. It is efficient because it facilitates the implementation of data acquisition and data management methods, as well as data representation and dashboard configuration. The platform allows for the use of any type of data source, ranging from the measurements of a multi-functional IoT sensing devices to relational and non-relational databases. It is also smart because it incorporates a complete artificial intelligence suit for data analysis; it includes techniques for data classification, clustering, forecasting, optimization, visualization, etc. It is also compatible with the edge computing concept, allowing for the distribution of intelligence and the use of intelligent sensors. The concept of smart cities is evolving and adapting to new applications; the trend to create intelligent neighbourhoods, districts or territories is becoming increasingly popular, as opposed to the previous approach of managing an entire megacity. In this paper, the platform is presented, and its architecture and functionalities are described. Moreover, its operation has been validated in a case study where the bike renting service of Paris—Vélib’ Métropole has been managed. This platform could enable smart territories to develop adapted knowledge management systems, adapt them to new requirements and to use multiple types of data, and execute efficient computational and artificial intelligence algorithms. The platform optimizes the decisions taken by human experts through explainable artificial intelligence models that obtain data from IoT sensors, databases, the Internet, etc. The global intelligence of the platform could potentially coordinate its decision-making processes with intelligent nodes installed in the edge, which would use the most advanced data processing techniques.