arXiv subjects

Carlos Esteve

Publications and source records attributed to Carlos Esteve.

7 recordsLinked to original sources

arXiv2021

The turnpike property and the long-time behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems

We analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon $T$ tends to infinity, the value function asymptotically behaves as $W(x) + c\, T + λ$, and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we showed that linear-quadratic optimal control problems with constrained control enjoy a turnpike property, also particularly when the steady optimum may saturate the control constraints.

math.AP↗

arXiv2021

Large-time asymptotics in deep learning

We consider the neural ODE perspective of supervised learning and study the impact of the final time $T$ (which may indicate the depth of a corresponding ResNet) in training. For the classical $L^2$--regularized empirical risk minimization problem, whenever the neural ODE dynamics are homogeneous with respect to the parameters, we show that the training error is at most of the order $\mathcal{O}\left(\frac{1}{T}\right)$. Furthermore, if the loss inducing the empirical risk attains its minimum, the optimal parameters converge to minimal $L^2$--norm parameters which interpolate the dataset. By a natural scaling between $T$ and the regularization hyperparameter $λ$ we obtain the same results when $λ\searrow0$ and $T$ is fixed. This allows us to stipulate generalization properties in the overparametrized regime, now seen from the large depth, neural ODE perspective. To enhance the polynomial decay, inspired by turnpike theory in optimal control, we propose a learning problem with an additional integral regularization term of the neural ODE trajectory over $[0,T]$. In the setting of $\ell^p$--distance losses, we prove that both the training error and the optimal parameters are at most of the order $\mathcal{O}\left(e^{-μt}\right)$ in any $t\in[0,T]$. The aforementioned stability estimates are also shown for continuous space-time neural networks, taking the form of nonlinear integro-differential equations. By using a time-dependent moving grid for discretizing the spatial variable, we demonstrate that these equations provide a framework for addressing ResNets with variable widths.

math.OC↗

arXiv2020

The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes

We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function $u_T$ and a time horizon $T>0$, we aim to construct all the initial conditions for which the viscosity solution coincides with $u_T$ at time $T$. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of $u_T$, is the one obtained by reversing the direction of time in the equation, considering $u_T$ as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when $u_T$ is reachable, we construct the set of all initial conditions for which the solution coincides with $u_T$ at time $T$. Note that in general, such initial conditions are not unique. Finally, for the case when the target $u_T$ is not necessarily reachable, we study the projection of $u_T$ on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of $u_T$, i.e. the smallest reachable target bounded from below by $u_T$.

math.AP↗

arXiv2019

Single-point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equation in domains with non-constant curvature

We consider the diffusive Hamilton-Jacobi equation $u_t - Δu = |\nabla u|^p$ in a bounded planar domain with zero Dirichlet boundary condition. It is known that, for $p>2$, the solutions to this problem can exhibit gradient blow-up (GBU) at the boundary. In this paper we study the possibility of the GBU set being reduced to a single point. In a previous work [Y.-X. Li, Ph. Souplet, 2009], it was shown that single point GBU solutions can be constructed in very particular domains, i.e.~locally flat domains and disks. Here, we prove the existence of single point GBU solutions in a large class of domains, for which the curvature of the boundary may be nonconstant near the GBU point. Our strategy is to use a boundary-fitted curvilinear coordinate system, combined with suitable auxiliary functions and appropriate monotonicity properties of the solution. The derivation and analysis of the parabolic equations satisfied by the auxiliary functions necessitate long and technical calculations involving boundary-fitted coordinates.

math.AP↗

arXiv2019

The evolution problem associated with eigenvalues of the Hessian

In this paper we study the evolution problem \[ \left\lbrace\begin{array}{ll} u_t (x,t)- λ_j(D^2 u(x,t)) = 0, & \text{in } Ω\times (0,+\infty), \\ u(x,t) = g(x,t), & \text{on } \partial Ω\times (0,+\infty), \\ u(x,0) = u_0(x), & \text{in } Ω, \end{array}\right. \] where $Ω$ is a bounded domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its boundary) and $λ_j(D^2 u)$ stands for the $j-$st eigenvalue of the Hessian matrix $D^2u$. We assume that $u_0 $ and $g$ are continuous functions with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial Ω$. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, $g(x,t) =g(x)$, we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as $t\to \infty$. For $j=1$ the limit profile is just the convex envelope inside $Ω$ of the boundary datum $g$, while for $j=N$ it is the concave envelope. We obtain this result with two different techniques: with PDE tools and and with game theoretical arguments. Moreover, in some special cases (for affine boundary data) we can show that solutions coincide with the stationary solution in finite time (that depends only on $Ω$ and not on the initial condition $u_0$).

math.AP↗

arXiv2017

Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity

We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, based on a parabolic equation with singular nonlinearity. We study the touchdown or quenching phenomenon. Recently, the question whether or not touchdown can occur at zero points of the permittivity profile, which had long remained open, was answered negatively by Guo and Souplet for the case of interior points, and we then showed that touchdown can actually be ruled out in subregions of the domain where the permittivity is positive but suitably small. The goal of this paper is to further investigate the touchdown localization problem and to show that, in one space dimension, one can obtain quite quantitative conditions. Namely, for large classes of typical, one-bump and two-bump permittivity profiles, we find good lower estimates of the ratio between f and its maximum, below which no touchdown occurs outside of the bumps. The ratio is rigorously obtained as the solution of a suitable finite-dimensional optimization problem (with either three or four parameters), which is then numerically estimated. Rather surprisingly, it turns out that the values of the ratio are not "small" but actually up to the order 0.3, which could hence be quite appropriate for robust use in practical MEMS design. The main tool for the reduction to the finite-dimensional optimization problem is a quantitative type I, temporal touchdown estimate. The latter is proved by maximum principle arguments, applied to a multi-parameter family of refined, nonlinear auxiliary functions with cut-off.

math.AP↗

arXiv2017

No touchdown at points of small permittivity and nontrivial touchdown sets for the MEMS problem

We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, involving a parabolic equation with singular nonlinearity. We study the touchdown, or quenching, phenomenon. Recently, the question whether or not touchdown can occur at zero points of the premittivity profile f, which had long remained open, was answered negatively for the case of interior points. The first aim of this article is to go further by considering the same question at points of positive but small permittivity. We show that, in any bounded domain, touchdown cannot occur at an interior point where the permittivity profile is suitably small. We also obtain a similar result in the boundary case, under a smallness assumption on f in a neighborhood of the boundary. This allows in particular to construct f producing touchdown sets concentrated near any given sphere. Our next aim is to obtain more information on the structure and properties of the touchdown set. In particular, we show that the touchdown set need not in general be localized near the maximum points of the premittivity profile f. In the radial case in a ball, we show the existence of M-shaped profiles for which the touchdown set is located far away from the maximum points of f and we even obtain strictly convex f for which touchdown occurs at the unique minimum point of f. These results show that some kind of smallness condition as above cannot be avoided in order to rule out touchdown at a point. On the other hand, we construct profiles f producing more complex behaviors: in any bounded domain the touchdown set may be concentrated near two arbitrarily given points, or two arbitrarily given (n-1)-dimensional spheres in a ball. These examples are obtained as a consequence of stability results for the touchdown time and touchdown set under small perturbations of the permittivity profile.

math.AP↗