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Nonlinear Reduced Modeling of Dynamical Systems Using Kernel Methods and Low-Rank Approximation
No full textInternational audienceReduced modeling of a computationally demanding dynamical system aims at approximating its trajectories, while optimizing the trade-off between accuracy and computational complexity. In this work, we propose to achieve such an approximation by first embedding the trajectories in a reproducing kernel Hilbert space (RKHS), which has interesting approximation and calculation capabilities, and then solving the associated reduced model problem. More specifically, we propose a new efficient algorithm for data-driven reduced modeling of nonlinear dynamics based on linear approximations in a RKHS. This algorithm takes advantage of the closed-form solution of a low-rank constraint optimization problem while exploiting advantageously kernel-based computations. Reduced modeling with this algorithm reveals a gain in approximation accuracy, as shown by numerical simulations, and in complexity with respect to existing approaches
Spectral signature of high-order photon processes mediated by Cooper-pair pairing
No full text12 pages, 10 figuresInternational audienceInducing interactions between individual photons is essential for applications in photonic quantum information processing and fundamental research on many-body photon states. A field that is well suited to combine strong interactions and low losses is microwave quantum optics with superconducting circuits. Photons are typically stored in an circuit, and interactions appear when the circuit is shunted by a Josephson tunnel junction. Importantly, the zero-point fluctuations of the superconducting phase across the junction control the strength and order of the induced interactions. Superconducting circuits have almost exclusively operated in the regime where phase fluctuations are smaller than unity, and two-photon interactions, known as the Kerr effect, dominate. In this experiment, we shunt a high-impedance oscillator by a dipole that only allows pairs of Cooper pairs to tunnel. Phase fluctuations, which are effectively doubled by this pairing, reach the value of 3.4. In this regime of extreme fluctuations, we observe transition frequencies that shift non-monotonically as we climb the anharmonic ladder. From this spectroscopic measurement, we extract two-, three- and four-photon interaction energies of comparable amplitude, and all exceeding the photon loss rate. This work explores a new regime of high-order photon interactions in microwave quantum optics, with applications ranging from multi-photon quantum logic to the study of highly correlated microwave radiation
A refined extreme quantiles estimator for Weibull tail-distributions
No full textInternational audienceWe address the estimation of extreme quantiles of Weibull tail-distributions. Since such quantiles are asymptotically larger than the sample maximum, their estimation requires extrapolation methods. In the case of Weibull tail-distributions, classical extreme-value estimators are numerically outperformed by estimators dedicated to this set of light-tailed distributions. The latter estimators of extreme quantiles are based on two key quantities: an order statistic to estimate an intermediate quantile and an estimator of the Weibull tail-coefficient used to extrapolate. The common practice is to select the same intermediate sequence for both estimators. We show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined estimator. This analysis is supported by an asymptotic normality result associated with the refined estimator. A data-driven method is introduced for the practical selection of the intermediate sequences and our approach is compared to three estimators of extreme quantiles dedicated to Weibull tail-distributions on simulated data. An illustration on a real data set of daily wind measures is also provided
Improving sampling by modifying the effective diffusion
No full text33 pages, 7 figuresInternational audienceMarkov chain Monte Carlo samplers based on discretizations of (overdamped) Langevin dynamics are commonly used in the Bayesian inference and computational statistical physics literature to estimate high-dimensional integrals. One can introduce a non-constant diffusion matrix to precondition these dynamics, and recent works have optimized it in order to improve the rate of convergence to stationarity by overcoming entropic and energy barriers. However, the introduced methodologies to compute these optimal diffusions are generally not suited to high-dimensional settings, as they rely on costly optimization procedures. In this work, we propose to optimize over a class of diffusion matrices, based on one-dimensional collective variables (CVs), to help the dynamics explore the latent space defined by the CV. The form of the diffusion matrix is chosen in order to obtain an efficient effective diffusion in the latent space. We describe how this class of diffusion matrices can be constructed and learned during the simulation. We provide implementations of the Metropolis–Adjusted Langevin Algorithm and Riemann Manifold (Generalized) Hamiltonian Monte Carlo algorithms, and discuss numerical optimizations in the case when the CV depends only on a few degrees of freedom of the system. We illustrate the efficiency gains by computing mean transition durations between two metastable states of a dimer in a solvent
Well-posedness of nonlocal macroscopic models of multi-population pedestrian flows for domain shape optimization
No full textInternational audienceWe consider a class of multi-population pedestrian models consisting in a system of nonlocal conservation laws coupled in the nonlocal components and describing several groups of pedestrians moving towards their respective targets while trying to avoid each other and the obstacles limiting the walking domain. Specifically, the nonlocal operators account for interactions occurring at the microscopic level as a reaction to the presence of other individuals or obstacles along the preferred path. In particular, the presence of obstacles is implemented in the nonlocal terms of the equations and not as classical boundary conditions. This allows to rewrite domain shape optimization problems as PDE-constrained problems.In this paper, we investigate the well-posedness of such optimization problems by proving the stability of solutions with respect to the positions and shapes of the obstacles. A differentiability result in the linear case is also provided. These properties are illustrated with a numerical example
Multiple-Impact Modeling in Multibody Systems
No full textInternational audienceThis chapter deals with the modeling of simultaneous collisions in rigid multibody systems, i.e., mechanical systems made of rigid bodies connected by unilateral constraints
Exponentially fast selection of sectors for quantum trajectories beyond non demolition measurements
No full textInternational audienceWe show that, in long time, quantum trajectories select an invariant subspace of the Hilbert space of the system being indirectly measured. This selection is shown to be exponentially fast in an almost sure sense and in average. This result generalizes a known result for non demolition measurements to arbitrary repeated indirect measurements. Our proofs are based on the introduction of a deformation of the original instrument to an equivalent one with a unique invariant state
Construction of birational trilinear volumes via tensor rank criteria
No full textInternational audienceWe provide effective methods to construct and manipulate trilinear birational maps by establishing a novel connection between birationality and tensor rank. These yield four families of nonlinear birational transformations between 3D spaces that can be operated with enough flexibility for applications in computer-aided geometric design. More precisely, we describe the geometric constraints on the defining control points of the map that are necessaryfor birationality, and present constructions for such configurations. For adequately constrained control points, we prove that birationality is achieved if and only if a certain tensor has rank one. As a corollary, we prove that the locus of weights that ensure birationality is . Additionally, we provide formulas for the inverse as well as the explicit defining equations of the irreducible components of the base loci.Finally, we introduce a notion of ``distance to birationality'' for trilinear rational maps, and explain how to continuously deform birational maps
Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations
No full textInternational audienceWe analyze the conforming approximation of the time-harmonic Maxwell's equations using Nédélec (edge) finite elements. We prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded by the bestapproximation error times a constant that tends to one as the mesh is refined and/or the polynomial degree is increased. Moreover, under the same conditions on the mesh and/or the polynomial degree, we establish discrete inf-sup stability with a constant that corresponds to the continuous constant up to a factor of two at most. Our proofs apply under minimal regularity assumptions on the exact solution, so that general domains, material coefficients, and right-hand sides are allowed
Numerical simulation of tokamak plasma equilibrium evolution
No full textInternational audienceThis paper focuses on the numerical methods recently developed in order to simulate the time evolution of a tokamak plasma equilibrium at the resistive diffusion time scale. Starting from the method proposed by Heumann in 2021 for the coupling of magnetic equilibrium and cur-rent diffusion, we introduce a new space discretization for the poloidal flux using coupled C0 and C1 finite elements. This, together with the use of cubic spline functions to represent the poloidal current function in the resistive diffusion equation, enables to restrain numerical oscillations which can occur with the original method. In order to compute consistently the plasma resistivity and the non-inductive bootstrap current terms needed in the resistive diffusion equation we add to the model an evolution equation for electron temperature in the plasma. It is also used to evolve the pressure term in the simulation. These numerical methods are implemented in the plasma equi-librium code NICE. A vertical displacement event is simulated and comparison with experimental results from the WEST tokamak are used to validate the simulation. The code is also coupled to a magnetic feedback controller making it possible to simulate a prescribed plasma scenario. The results for an X-point formation scenario in the WEST tokamak are presented as an illustration of the efficiency of the developed numerical methods.Cet article se concentre sur les méthodes numériques récemment développées dans le but de simuler l’évolution temporelle de l’équilibre du plasma dans un tokamak à l’échelle de temps de la diffusion résistive. Avec comme point de départ la formulation proposée par Heumann en 2021 pour le couplage de l’équilibre magnétique avec la diffusion du courant, nous introduisons une nouvelle discrétisation spatiale pour le flux poloïdal utilisant un couplage d’élément finis de classes C0 et C1. Ceci ajouté à l’utilisation de splines cubiques pour approcher la fonction de courant poloïdal dans l’équation de diffusion résistive nous permet de restreindre les oscillations numériques qui peuvent apparaitre dans la méthode originelle. Afin de calculer de manière consistente la résistivité du plasma et les courants non-inductifs dits de bootstrap qui sont nécessaires à la résolution de l’équation de diffusion résistive, nous ajoutons au modèle une équation d’évolution pour la température électronique dans le plasma. Ceci permet également de faire évoluer le terme de pression dans la simulation. Ces méthodes numériques sont implémentées dans le code d’équilibre NICE. Une instabilité verticale est simulée et comparée à des résultats expérimentaux d’une décharge du tokamak WEST. Le code est également couplé à un contrôleur magnétique rendant possible la simulation d’un scénario prescrit. Les résultats pour un scénario de formation du point X dans le tokamak WEST sont présentés et illustrent l’efficacité des méthodes numériques développées dans ce travail