arXiv subjects

Carlos Esteve-Yagüe

Publications and source records attributed to Carlos Esteve-Yagüe.

13 recordsLinked to original sources

arXiv2026

When does least-squares residual minimization solve a differential equation?

Least-squares residual minimization approximates a differential equation through an optimization problem. This paper investigates the conditions required to transition from stationary points and minimizing sequences of the residual functional to the exact solution. Starting from the classical orthogonality relation between the residual and the range of the linearized operator, we study how boundary conditions and parameterized variations determine what can be inferred about the residual at a stationary point. Smooth spurious critical points can occur even for well-posed problems; for linear equations, a compatibility condition yields an exact projection characterization. For a scalar conservation law with transonic boundary data, every minimizing sequence converges to a continuous function that does not solve the equation, even when a stationary entropy shock exists. We give conditions under which approaching this limit forces the parameter norms to diverge. A Hamilton--Jacobi example shows that a function can have zero residual without being the viscosity solution. Favorable results for a viscous model and uniformly convex Monge--Ampère critical points identify conditions that restore the corresponding implications. For piecewise-smooth trial families, the regularity needed to define the residual functional does not by itself justify the sampled gradient used in training. Moving residual jumps can contribute terms absent from that gradient; an explicit example has identically vanishing sampled gradients and a nonzero continuous derivative. Together, these results clarify what stationarity and sampled gradients imply about the residual, and under which conditions minimizing sequences lead to the intended solution.

math.NA↗

arXiv2026

Solving Hamilton-Jacobi equations by residual minimization of monotone finite-difference discretizations

We introduce a method for solving Hamilton--Jacobi equations, both inviscid and viscous, by minimizing the squared residuals of monotone finite-difference discretizations on grids of varying resolution. The method is designed to leverage neural networks and modern GPUs to solve these equations in higher dimensions; consequently, the setting for our analysis is the minimization of the residual functionals via gradient-based optimization. We establish a well-posedness theory for this approach: any critical point of the finite-difference loss solves the monotone scheme together with the prescribed Dirichlet boundary conditions, and the error of an approximation is controlled by its residual. We then derive the rate of convergence of the gradient flow that minimizes the residual in several settings, noting that the rate may depend on the grid resolution and the domain dimension. Building on this foundation, we propose a multi-level training algorithm that exploits the faster convergence available on the coarser grids of the discretization. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee convergence to the unique viscosity solution as the grid is refined. We illustrate the approach on eikonal equations, level-set problems, and a Hamilton--Jacobi--Isaacs equation arising from a stochastic differential game, in dimensions up to eight.

math.NA↗

arXiv2026

Sample Complexity Analysis of Multi-Target Detection via Markovian and Hard-Core Multi-Reference Alignment

Motivated by single-particle cryo-electron microscopy, we study the sample complexity of the multi-target detection (MTD) problem, in which an unknown signal appears multiple times at unknown locations within a long, noisy observation. We propose a patching scheme that reduces MTD to a non-i.i.d. multi-reference alignment (MRA) model. In the one-dimensional setting, the latent group elements form a Markov chain, and we show that the convergence rate of any estimator matches that of the corresponding i.i.d. MRA model, up to a logarithmic factor in the number of patches. Moreover, for estimators based on empirical averaging, such as the method of moments, the convergence rates are identical in both settings. We further establish an analogous result in two dimensions, where the latent structure arises from an exponentially mixing random field generated by a hard-core placement model. As a consequence, if the signal in the corresponding i.i.d. MRA model is determined by moments up to order $n_{\min}$, then in the low-SNR regime the number of patches required to estimate the signal in the MTD model scales as $σ^{2n_{\min}}$, where $σ^2$ denotes the noise variance.

eess.SP↗

arXiv2025

FedMAP: Personalised Federated Learning for Real Large-Scale Healthcare Systems

Federated learning (FL) promises to enable collaborative machine learning across healthcare sites whilst preserving data privacy. Practical deployment remains limited by statistical heterogeneity arising from differences in patient demographics, treatments, and outcomes, and infrastructure constraints. We introduce FedMAP, a personalised FL (PFL) framework that addresses heterogeneity through local Maximum a Posteriori (MAP) estimation with Input Convex Neural Network priors. These priors represent global knowledge gathered from other sites that guides the model while adapting to local data, and we provide a formal proof of convergence. Unlike many PFL methods that rely on fixed regularisation, FedMAP's prior adaptively learns patterns that capture complex inter-site relationships. We demonstrate improved performance compared to local training, FedAvg, and several PFL methods across three large-scale clinical datasets: 10-year cardiovascular risk prediction (CPRD, 387 general practitioner practices, 258,688 patients), iron deficiency detection (INTERVAL, 4 donor centres, 31,949 blood donors), and mortality prediction (eICU, 150 hospitals, 44,842 patients). FedMAP incorporates a three-tier design that enables participation across healthcare sites with varying infrastructure and technical capabilities, from full federated training to inference-only deployment. Geographical analysis reveals substantial equity improvements, with underperforming regions achieving up to 14.3% performance gains. This framework provides the first practical pathway for large-scale healthcare FL deployment, which ensures clinical sites at all scales can benefit, equity is enhanced, and privacy is retained.

cs.LG↗

arXiv2024

Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks

We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi (HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based method to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's critical points and derive conditions that guarantee that any critical point approximates the sought viscosity solution. The use of a deep artificial neural network on a finite difference scheme lifts the restriction of conventional finite difference methods that rely on computing functions on a fixed grid. This feature makes it possible to solve HJ equations posed in higher dimensions where conventional methods are infeasible. We demonstrate the efficacy of our algorithm through numerical studies on various canonical HJ equations across different dimensions, showcasing its potential and versatility.

math.NA↗

arXiv2023

Riemannian geometry for efficient analysis of protein dynamics data

An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich structure to account for a wide range of geometries that can be modelled after an energy landscape. Second, many standard data analysis tools initially developed for data in Euclidean space can also be generalised to data on a Riemannian manifold. In the context of protein dynamics, a conceptual challenge comes from the lack of a suitable smooth manifold and the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a novel local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold of point clouds modulo rigid body group actions and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. It performs exceptionally well on coarse-grained molecular dynamics simulated data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

q-bio.BM↗

arXiv2023

Usable boundary for visibility-based surveillance-evasion games

We consider a surveillance-evasion game in an environment with obstacles. In such an environment, a mobile pursuer seeks to maintain the visibility with a mobile evader, who tries to get occluded from the pursuer in the shortest time possible. In this two-player zero-sum game setting, we study the discontinuities of the value of the game near the boundary of the target set (the non-visibility region). In particular, we describe the transition between the usable part of the boundary of the target (where the value vanishes) and the non-usable part (where the value is positive). We show that the value enjoys a different behaviour depending on the regularity of the obstacles involved in the game. Namely, we prove that the boundary profile is continuous for the case of smooth obstacles, and that it exhibits a jump discontinuity when the obstacle contains corners. Moreover, we prove that, in the latter case, there is a semi-permeable barrier emanating from the interface between the usable and the non-usable part of the boundary of the target set.

math.AP↗

arXiv2022

Spectral decomposition of atomic structures in heterogeneous cryo-EM

We consider the problem of recovering the three-dimensional atomic structure of a flexible macromolecule from a heterogeneous cryo-EM dataset. The dataset contains noisy tomographic projections of the electrostatic potential of the macromolecule, taken from different viewing directions, and in the heterogeneous case, each image corresponds to a different conformation of the macromolecule. Under the assumption that the macromolecule can be modelled as a chain, or discrete curve (as it is for instance the case for a protein backbone with a single chain of amino-acids), we introduce a method to estimate the deformation of the atomic model with respect to a given conformation, which is assumed to be known a priori. Our method consists on estimating the torsion and bond angles of the atomic model in each conformation as a linear combination of the eigenfunctions of the Laplace operator in the manifold of conformations. These eigenfunctions can be approximated by means of a well-known technique in manifold learning, based on the construction of a graph Laplacian using the cryo-EM dataset. Finally, we test our approach with synthetic datasets, for which we recover the atomic model of two-dimensional and three-dimensional flexible structures from noisy tomographic projections.

eess.IV↗

arXiv2022

Sparsity in long-time control of neural ODEs

We consider the neural ODE and optimal control perspective of supervised learning, with $\ell^1$-control penalties, where rather than only minimizing a final cost (the \emph{empirical risk}) for the state, we integrate this cost over the entire time horizon. We prove that any optimal control (for this cost) vanishes beyond some positive stopping time. When seen in the discrete-time context, this result entails an \emph{ordered} sparsity pattern for the parameters of the associated residual neural network: ordered in the sense that these parameters are all $0$ beyond a certain layer. Furthermore, we provide a polynomial stability estimate for the empirical risk with respect to the time horizon. This can be seen as a \emph{turnpike property}, for nonsmooth dynamics and functionals with $\ell^1$-penalties, and without any smallness assumptions on the data, both of which are new in the literature.

cs.LG↗

arXiv2022

Reachable set for Hamilton-Jacobi equations with non-smooth Hamiltonian and scalar conservation laws

We 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.

math.AP↗

arXiv2022

Turnpike in Lipschitz-nonlinear optimal control

We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying system. Our strategy combines the construction of quasi-turnpike controls via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by applications in machine learning through deep residual neural networks, which may be fit within our setting. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.

math.OC↗

arXiv2021

Differentiability with respect to the initial condition for Hamilton-Jacobi equations

We prove that the viscosity solution to a Hamilton-Jacobi equation with a smooth convex Hamiltonian of the form $H(x,p)$ is differentiable with respect to the initial condition. Moreover, the directional Gâteaux derivatives can be explicitly computed almost everywhere in $\mathbb{R}^N$ by means of the optimality system of the associated optimal control problem. We also prove that, in the one-dimensional case in space and in the quadratic case in any space dimension, these directional Gâteaux derivatives actually correspond to the unique duality solution to the linear transport equation with discontinuous coefficient, resulting from the linearization of the Hamilton-Jacobi equation. The motivation behind these differentiability results arises from the following optimal inverse-design problem: given a time horizon $T>0$ and a target function $u_T$, construct an initial condition such that the corresponding viscosity solution at time $T$ minimizes the $L^2$-distance to $u_T$. Our differentiability results allow us to derive a necessary first-order optimality condition for this optimization problem, and the implementation of gradient-based methods to numerically approximate the optimal inverse design.

math.OC↗

arXiv2021

Multilevel Selective Harmonic Modulation via Optimal Control

We consider the Selective Harmonic Modulation (SHM) problem, consisting in the design of a staircase control signal with some prescribed frequency components. In this work, we propose a novel methodology to address SHM as an optimal control problem in which the admissible controls are piecewise constant functions, taking values only in a given finite set. In order to fulfill this constraint, we introduce a cost functional with piecewise affine penalization for the control, which, by means of Pontryagin's maximum principle, makes the optimal control have the desired staircase form. Moreover, the addition of the penalization term for the control provides uniqueness and continuity of the solution with respect to the target frequencies. Another advantage of our approach is that the number of switching angles and the waveform need not be determined a priori. Indeed, the solution to the optimal control problem is the entire control signal, and therefore, it determines the waveform and the location of the switches. We also provide several numerical examples in which the SHM problem is solved by means of our approach.

math.OC↗