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Ferromagnetic particles are incorporated in a thin soft elastic matrix. A lamella, made of this smart material, is studied experimentally and modeled. We show herein that thin films can be actuated using an external magnetic field applied through the system. The system is found to be switchable since subcritical pitchfork bifurcation is discovered in the beam shape when the magnetic field orientation is modified. Strong magnetoelastic effects can be obtained depending on both field strength and orientation. Our results provide versatile ways to contribute to many applications from the microfabrication of actuators to soft robotics. As an example, we created a small synthetic octopus piloted by an external magnetic field.

In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics, and porous media flow in particular, the uncertain permeability of the material is modelled as a random field. These random fields can be highly anisotropic. Efficient solvers, such as the Multiple Semi-Coarsened Multigrid (MSG) method, see [15, 16, 17], are required to compute solutions for various realisations of the uncertain material. The MSG method is an extension of the classic Multigrid method, that uses additional coarse grids that are coarsened in only a single coordinate direction. In this sense, it closely resembles the extension of Multilevel Monte Carlo (MLMC) [6] to Multi-Index Monte Carlo (MIMC) [9]. We present an unbiased MIMC method that reuses the MSG coarse solutions, similar to the work in [11]. Our formulation of the estimator can be interpreted as the problem of learning the unknown distribution of the number of samples across all indices, and unifies the previous work on adaptive MIMC [19] and unbiased estimation [18]. We analyse the cost of this new estimator theoretically and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over unbiased MIMC without sample reuse.

Studies of model microswimmers have significantly contributed to the understanding of the principles of self-propulsion we have today. However, only a small number of microswimmer types have been amenable to analytic modeling, and further development of such approaches is necessary to identify the key features of these active systems. Here we present a general perturbative calculation scheme for swimmers composed of beads interacting by harmonic potentials, driven by an arbitrary force protocol. The hydrodynamic interactions are treated using the Oseen and Rotne-Pragner approximations. We validate our approach by using 3 bead assemblies and comparing the results with the numerically obtained full-solutions of the governing equations of motion, as well as with the existing analytic models for a linear and a triangular swimmer geometries. While recovering the relation between the force and swimming velocity, our detailed analysis and the controlled level of approximation allow us to find qualitative differences already in the far field flow of the devices. Consequently, we are able to identify a behavior of the swimmer that is richer than predicted in previous models. Given its generality, the framework can be applied to any swimmer geometry, driving protocol and beads interactions, as well as in many swimmers problems., Comment: 19 pages, 6 figures

Small objects floating on a fluid have a tendency to aggregate due to capillary forces. This effect has been used, with the help of a magnetic induction field, to assemble submillimeter metallic spheres into a variety of structures, whose shape and size can be tuned. Under time-varying fields, these assemblies can propel themselves due to a breaking of time reversal symmetry in their adopted shapes. In this article, we study the influence of an in-plane rotation of the magnetic field on these structures. Various rotational modes have been observed with different underlying mechanisms. The magnetic properties of the particles cause them to rotate individually. Dipole-dipole interactions in the assembly can cause the whole structure to align with the field. Finally, non-reciprocal deformations can power the rotation of the assembly. Symmetry plays an important role in the dynamics, as well as the frequency and amplitude of the applied field. Understanding the interplay of these effects is essential, both to explain previous observations and to develop new functions for these assemblies., Comment: 10 pages, 9 figures

Kinetic equations model distributions of particles in position-velocity phase space. Often, one is interested in studying the long-time behavior of particles in high-collisional regimes in which an approximate (advection)-diffusion model holds. In this paper we consider the diffusive scaling. Classical particle-based techniques suffer from a strict time-step restriction in this limit, to maintain stability. Asymptotic-preserving schemes avoid this problem, but introduce an additional time discretization error, possibly resulting in an unacceptably large bias for larger time steps. Here, we present and analyze a multilevel Monte Carlo scheme that reduces this bias by combining estimates using a hierarchy of different time step sizes. We demonstrate how to correlate trajectories from this scheme, using different time steps. We also present a strategy for selecting the levels in the multilevel scheme. Our approach significantly reduces the computation required to perform accurate simulations of the considered kinetic equations, compared to classical Monte Carlo approaches., Comment: 41 page article (incl. 6 pages of appendix), 11 figures, 8 tables, 3 pages of additional supplementary material, updated title in metadata. A final peer-reviewed version of this article has been published in Numerische Mathematik

Practical structural engineering problems are often characterized by significant uncertainties. Historically, one of the prevalent methods to account for this uncertainty has been the standard Monte Carlo (MC) method. Recently, improved sampling methods have been proposed, based on the idea of variance reduction by employing a hierarchy of mesh refinements. We combine an h- and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) and Multilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicability of these novel combination methods on three structural engineering problems, for which the uncertainty resides in the Young's modulus: the static response of a cantilever beam with elastic material behavior, its static response with elastoplastic behavior, and its dynamic response with elastic behavior. The uncertainty is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a gamma random field. This random field results from a truncated Karhunen-Lo\`eve (KL) expansion. In this paper, we compare the computational costs of these Monte Carlo methods. We demonstrate that MQLMC and MLMC have a significant speedup with respect to MC, regardless of the mesh refinement hierarchy used. We empirically demonstrate that the MLQMC cost is optimally proportional to 1/epsilon under certain conditions, where epsilon is the tolerance on the root-mean-square error (RMSE). In addition, we show that, when the uncertainty is modeled as a random field, the multilevel methods combined with p-refinement have a significant lower computation cost than their counterparts based on h-refinement. We also illustrate the effect the uncertainty models have on the uncertainty bounds in the solutions., Comment: arXiv admin note: substantial text overlap with arXiv:1808.10680

A walking droplet possesses fascinating properties due to its peculiar wave/particle interaction. The self-propelling motion of such a droplet is driven by the Faraday instability triggered around the droplet at each impact. We studied in this article how such a droplet behaves in an annular cavity where a periodic pattern is placed underneath the liquid-air interface, altering the Faraday instability. We show that, while the annulus ensures a circular motion of the droplet, the periodic pattern affects the global droplet motion. Similarly to electromagnetic waves in photonic crystals, the average droplet speed nearly vanishes when the pattern has a characteristic length close to half the Faraday wavelength. This effect opens ways to design guides, reflectors, lattices and metamaterials for such macroscopic particles., Comment: 6 pages, 5 figures, 1 movie. arXiv admin note: text overlap with arXiv:1504.00484

For regulatory and interpretability reasons, logistic regression is still widely used. To improve prediction accuracy and interpretability, a preprocessing step quantizing both continuous and categorical data is usually performed: continuous features are discretized and, if numerous, levels of categorical features are grouped. An even better predictive accuracy can be reached by embedding this quantization estimation step directly into the predictive estimation step itself. But doing so, the predictive loss has to be optimized on a huge set. To overcome this difficulty, we introduce a specific two-step optimization strategy: first, the optimization problem is relaxed by approximating discontinuous quantization functions by smooth functions; second, the resulting relaxed optimization problem is solved via a particular neural network. The good performances of this approach, which we call glmdisc, are illustrated on simulated and real data from the UCI library and Cr\'edit Agricole Consumer Finance (a major European historic player in the consumer credit market)., Comment: 9 pages, 2 figures, 3 tables

The granting process of all credit institutions rejects applicants who seem risky regarding the repayment of their debt. A credit score is calculated and associated with a cut-off value beneath which an applicant is rejected. Developing a new score implies having a learning dataset in which the response variable good/bad borrower is known, so that rejects are de facto excluded from the learning process. We first introduce the context and some useful notations. Then we formalize if this particular sampling has consequences on the score's relevance. Finally, we elaborate on methods that use not-financed clients' characteristics and conclude that none of these methods are satisfactory in practice using data from Cr\'edit Agricole Consumer Finance. ----- Un syst\`eme d'octroi de cr\'edit peut refuser des demandes de pr\^et jug\'ees trop risqu\'ees. Au sein de ce syst\`eme, le score de cr\'edit fournit une valeur mesurant un risque de d\'efaut, valeur qui est compar\'ee \`a un seuil d'acceptabilit\'e. Ce score est construit exclusivement sur des donn\'ees de clients financ\'es, contenant en particulier l'information `bon ou mauvais payeur', alors qu'il est par la suite appliqu\'e \`a l'ensemble des demandes. Un tel score est-il statistiquement pertinent ? Dans cette note, nous pr\'ecisons et formalisons cette question et \'etudions l'effet de l'absence des non-financ\'es sur les scores \'elabor\'es. Nous pr\'esentons ensuite des m\'ethodes pour r\'eint\'egrer les non-financ\'es et concluons sur leur inefficacit\'e en pratique, \`a partir de donn\'ees issues de Cr\'edit Agricole Consumer Finance., Comment: In French, 6 pages, 1 table, 1 figure; long abstract for the 49th 'Journ\'ees de Statistiques' conference, May 2017, Avignon, France

We propose a multilevel Monte Carlo method for a particle-based asymptotic-preserving scheme for kinetic equations. Kinetic equations model transport and collision of particles in a position-velocity phase-space. With a diffusive scaling, the kinetic equation converges to an advection-diffusion equation in the limit of zero mean free path. Classical particle-based techniques suffer from a strict time-step restriction to maintain stability in this limit. Asymptotic-preserving schemes provide a solution to this time step restriction, but introduce a first-order error in the time step size. We demonstrate how the multilevel Monte Carlo method can be used as a bias reduction technique to perform accurate simulations in the diffusive regime, while leveraging the reduced simulation cost given by the asymptotic-preserving scheme. We describe how to achieve the necessary correlation between simulation paths at different levels and demonstrate the potential of the approach via numerical experiments., Comment: 20 pages, 7 figures, published in Monte Carlo and Quasi-Monte Carlo Methods 2018, correction of minor typographical errors

A system of ferromagnetic particles trapped at a liquid-liquid interface and subjected to a set of magnetic fields (magnetocapillary swimmers) is studied numerically using a hybrid method combining the pseudopotential lattice Boltzmann method and the discrete element method. After investigating the equilibrium properties of a single, two and three particles at the interface, we demonstrate a controlled motion of the swimmer formed by three particles. It shows a sharp dependence of the average center-of-mass speed on the frequency of the time-dependent external magnetic field. Inspired by experiments on magnetocapillary microswimmers, we interpret the obtained maxima of the swimmer speed by the optimal frequency centered around the characteristic relaxation time of a spherical particle. It is also shown that the frequency corresponding to the maximum speed grows and the maximum average speed decreases with increasing inter-particle distances at moderate swimmer sizes. The findings of our lattice Boltzmann simulations are supported by bead-spring model calculations., Comment: 10 pages, 11 figures

Practical structural engineering problems often exhibit a significant degree of uncertainty in the material properties being used, the dimensions of the modeled structures, etc. In this paper, we consider a cantilever beam and a beam clamped at both ends, both subjected to a static and a dynamic load. The material uncertainty resides in the Young's modulus, which is modeled by means of one random variable, sampled from a univariate Gamma distribution, or with multiple random variables, sampled from a Gamma random field. Three different responses are considered: the static elastic, the dynamic elastic and the static elastoplastic response. In the first two cases, we simulate the spatial displacement of a concrete beam and its frequency response in the elastic domain. The third case simulates the spatial displacement of a steel beam in the elastoplastic domain. In order to compute the statistical quantities of the static deflection and frequency response function, Multilevel Monte Carlo (MLMC) is combined with a Finite Element solver. In this paper, the computational costs and run times of the MLMC method are compared with those of the classical Monte Carlo method, demonstrating a significant speedup of up to several orders of magnitude for the studied cases.

We present a wave-memory driven system that exhibits intermittent switching between two propulsion modes in free space. The model is based on a point-like particle emitting periodically cylindrical standing waves. Submitted to a force related to the local wavefield gradient, the particle is propelled, while the wave field stores positional information on the particle trajectory. For long memory, the linear motion is unstable and we observe erratic switches between two propulsive modes : linear motion and diffusive motion. We show that the bimodal propulsion and the stochastic aspect of the dynamics at long time are generated by a Shil'nikov chaos. The memory of the system controls the fraction of time spent in each phase. The resulting bimodal dynamics shows analogies with intermittent search strategies usually observed in living systems of much higher complexity., Comment: 14 pages, 4 figures