Examinando por Autor "Zuazua, Enrique"
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Ítem Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian(Walter de Gruyter GmbH, 2017) Biccari, Umberto; Warma, Mahamadi; Zuazua, EnriqueIn [1], for 1 < p < 1Ítem Averaged dynamics and control for heat equations with random diffusion(Elsevier B.V., 2021-10-30) Bárcena Petisco, Jon Asier; Zuazua, EnriqueThis paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the [Formula presented]-fractional Laplacian, which is well-known to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime.Ítem Constrained control of gene-flow models(European Mathematical Society Publishing House, 2023) Mazari, Idriss; Ruiz Balet, Domènec ; Zuazua, EnriqueIn ecology and population dynamics, gene flow refers to the transfer of a trait (e.g. genetic material) from one population to another. This phenomenon is of great relevance in studying the spread of diseases or the evolution of social features, such as languages. From the mathematical point of view, gene flow is modeled using bistable reaction-diffusion equations. The unknown is the proportion p of the population that possesses a certain trait, within an overall population N. In such models, gene flow is taken into account by assuming that the population density N depends either on p (if the trait corresponds to fitter individuals) or on the location x (if some zones in the domain can carry more individuals). Recent applications stemming from mosquito-borne-disease control problems or from the study of bilingualism have called for the investigation of the controllability properties of these models. At the mathematical level, this corresponds to boundary control problems and, since we are working with proportions, the control u has to satisfy the constraint 0 6 u 6 1. In this article, we provide a thorough analysis of the influence of the gene-flow effect on boundary controllability properties. We prove that, when the population density N only depends on the trait proportion p, the geometry of the domain is the only criterion that has to be considered. We then tackle the case of population densities N varying in x. We first prove that, when N varies slowly in x and when the domain is narrow enough, controllability always holds. This result is proved using a robust domain perturbation method. We then consider the case of sharp fluctuations in N: we first give examples that prove that controllability may fail. Conversely, we give examples of heterogeneities N such that controllability will always be guaranteed: in other words the controllability properties of the equation are very strongly influenced by the variations of N. All negative controllability results are proved by showing the existence of nontrivial stationary states, which act as barriers. The existence of such solutions and the methods of proof are of independent interest. Our article is completed by several numerical experiments that confirm our analysis.Ítem Control and numerical approximation of fractional diffusion equations(Elsevier B.V., 2022) Biccari, Umberto; Warma, Mahamadi; Zuazua, EnriqueThe aim of this chapter is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain Ω. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω⊂Ω. Secondly, we move our attention to the exterior control problem, in which the control region O⊂Ωc is located outside Ω. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.Ítem Dynamics and control for multi-agent networked systems: a finite-difference approach(World Scientific Publishing Co. Pte Ltd, 2019-04) Biccari, Umberto; Ko, Dongnam; Zuazua, EnriqueWe analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time horizon and the size of the controls.Ítem FedADMM-InSa: an inexact and self-adaptive ADMM for federated learning(Elsevier Ltd, 2025-01) Song, Yongcun; Wang, Ziqi; Zuazua, EnriqueFederated learning (FL) is a promising framework for learning from distributed data while maintaining privacy. The development of efficient FL algorithms encounters various challenges, including heterogeneous data and systems, limited communication capacities, and constrained local computational resources. Recently developed FedADMM methods show great resilience to both data and system heterogeneity. However, they still suffer from performance deterioration if the hyperparameters are not carefully tuned. To address this issue, we propose an inexact and self-adaptive FedADMM algorithm, termed FedADMM-InSa. First, we design an inexactness criterion for the clients’ local updates to eliminate the need for empirically setting the local training accuracy. This inexactness criterion can be assessed by each client independently based on its unique condition, thereby reducing the local computational cost and mitigating the undesirable straggle effect. The convergence of the resulting inexact ADMM is proved under the assumption of strongly convex loss functions. Additionally, we present a self-adaptive scheme that dynamically adjusts each client's penalty parameter, enhancing algorithm robustness by mitigating the need for empirical penalty parameter choices for each client. Extensive numerical experiments on both synthetic and real-world datasets have been conducted. As validated by some tests, our FedADMM-InSa algorithm improves model accuracy by 7.8% while reducing clients’ local workloads by 55.7% compared to benchmark algorithms.Ítem Interplay between depth and width for interpolation in neural ODEs(Elsevier Ltd, 2024-12) Álvarez López, Antonio; Slimane, Arselane Hadj; Zuazua, EnriqueNeural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin ɛ>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op−1+(1+p)−1ɛ−d for measures. In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.Ítem Local elliptic regularity for the Dirichlet fractional Laplacian(Walter de Gruyter GmbH, 2017) Biccari, Umberto; Warma, Mahamadi; Zuazua, EnriqueWe prove the Wloc2s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of RN. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.Ítem Local regularity for fractional heat equations(Springer International Publishing, 2018) Biccari, Umberto; Warma, Mahamadi; Zuazua, EnriqueWe prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set Ω ⊂ ℝN. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Ítem Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave(Institute of Physics, 2023-11-04) Coclite, Giuseppe Maria; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas; Zuazua, EnriqueWe study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the ‘N-wave’ entropy solution of the Burgers equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.Ítem On the controllability of entropy solutions of scalar conservation laws at a junction via Lyapunov methods(Springer, 2023-01-04) De Nitti, Nicola ; Zuazua, EnriqueIn this note, we prove a controllability result for entropy solutions of scalar conservation laws on a star-shaped graph. Using a Lyapunov-type approach, we show that, under a monotonicity assumption on the flux, if u and v are two entropy solutions corresponding to different initial data and same in-flux boundary data (at the exterior nodes of the star-shaped graph), then u ≡ v for a sufficiently large time. In order words, we can drive u to the target profile v in a sufficiently large control time by inputting the trace of v at the exterior nodes as in-flux boundary data for u. This result can also be shown to hold on tree-shaped networks by an inductive argument. We illustrate the result with some numerical simulations.Ítem Optimal design of sensors via geometric criteria(Springer, 2023-05-23) Ftouhi, Ilias; Zuazua, EnriqueWe consider a convex set Ω and look for the optimal convex sensor ω⊂ Ω of a given measure that minimizes the maximal distance to the points of Ω. This problem can be written as follows inf{dH(ω,Ω)||ω|=candω⊂Ω}, where c∈ (0 , | Ω |) , dH being the Hausdorff distance. We show that the parametrization via the support functions allows us to formulate the geometric optimal shape design problem as an analytic one. By proving a judicious equivalence result, the shape optimization problem is approximated by a simpler minimization problem of a quadratic function under linear constraints. We then present some numerical results and qualitative properties of the optimal sensors and exhibit an unexpected symmetry breaking phenomenon.Ítem Propagation of one- and two-dimensional discrete waves under finite difference approximation(Springer, 2020-12) Biccari, Umberto; Marica, Aurora; Zuazua, EnriqueWe analyze the propagation properties of the numerical versions of one- and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by micro-local tools. We consider uniform and non-uniform numerical grids as well as constant and variable coefficients. The energy of continuous and semi-discrete high-frequency solutions propagates along bi-characteristic rays, but their dynamics are different in the continuous and the semi-discrete setting, because of the nature of the corresponding Hamiltonians. One of the main objectives of this paper is to illustrate through accurate numerical simulations that, in agreement with micro-local theory, numerical high-frequency solutions can bend in an unexpected manner, as a result of the accumulation of the local effects introduced by the heterogeneity of the numerical grid. These effects are enhanced in the multi-dimensional case where the interaction and combination of such behaviors in the various space directions may produce, for instance, the rodeo effect, i.e., waves that are trapped by the numerical grid in closed loops, without ever getting to the exterior boundary. Our analysis allows to explain all such pathological behaviors. Moreover, the discussion in this paper also contributes to the existing theory about the necessity of filtering high-frequency numerical components when dealing with control and inversion problems for waves, which is based very much on the theory of rays and, in particular, on the fact that they can be observed when reaching the exterior boundary of the domain, a key property that can be lost through numerical discretization.Ítem Reachable set for Hamilton–Jacobi equations with non-smooth Hamiltonian and scalar conservation laws(Elsevier Ltd, 2023-02) Esteve Yagüe, Carlos; Zuazua, EnriqueWe give a full characterization of the range of the operator which associates, to any initial condition, the viscosity solution at time T of a Hamilton–Jacobi equation with convex Hamiltonian. Our main motivation is to be able to treat the case of convex Hamiltonians with no further regularity assumptions. We give special attention to the case H(p)=|p|, for which we provide a rather geometrical description of the range of the viscosity operator by means of an interior ball condition on the sublevel sets. From our characterization of the reachable set, we are able to deduce further results concerning, for instance, sharp regularity estimates for the reachable functions, as well as structural properties of the reachable set. The results are finally adapted to the case of scalar conservation laws in dimension one.Ítem A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation(Institute of Physics, 2023-09) Biccari, Umberto; Song, Yongcun; Yuan, Xiaoming; Zuazua, EnriqueWe consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion–advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.Ítem Uniform Turnpike property and singular limits(Springer Science and Business Media B.V., 2024-04) Hernández Salinas, Martín Sebastián; Zuazua, EnriqueMotivated by singular limits for long-time optimal control problems, we investigate a class of parameter-dependent parabolic equations. First, we prove a turnpike result, uniform with respect to the parameters within a suitable regularity class and under appropriate bounds. The main ingredient of our proof is the justification of the uniform exponential stabilization of the corresponding Riccati equations, which is derived from the uniform null control properties of the model. Then, we focus on a heat equation with rapidly oscillating coefficients. In the one-dimensional setting, we obtain a uniform turnpike property with respect to the highly oscillatory heterogeneous medium. Afterward, we establish the homogenization of the turnpike property. Finally, our results are validated by numerical experiments