arXiv subjects

Andrea Bonito

Publications and source records attributed to Andrea Bonito.

At least 19 recordsLinked to original sources

arXiv2026

Sampling and reconstruction of convex functions

We discuss optimal recovery for classes of multivariate convex functions from given point samples, as well as the sampling numbers of these classes, corresponding to optimal sample choices. Upper and lower bounds for either variant are established when the reconstruction error is measured in $L_p$ for $1\leq p\leq \infty$. These bounds match, sometimes up to logarithmic factors, and therefore characterize the respective optimal rate of decay. For classical smoothness classes such as Sobolev, H\"older or Besov spaces, it is well known that the optimal decay rate of sampling numbers can be achieved by sampling on uniform tensor product grids and using linear methods of reconstruction, such as piecewise polynomial interpolation. One of the main findings in this paper is that for classes of convex functions, these procedures generally produce suboptimal rates, except when $p=1$ and $p=\infty$, and are outperformed by nonlinear reconstruction methods that do not employ tensor product grids.

math.NA↗

arXiv2026

A near-optimal recovery algorithm for the Stokes equations with incomplete information on the boundary conditions

We address the problem of numerically approximating the velocity and pressure governed by the Stokes system when the boundary conditions are only partially known and thus do not uniquely determine the velocity-pressure couple. We propose an algorithm that takes advantage of available linear measurements of the velocity and pressure to construct a numerical approximation. This approximation is guaranteed to be near-optimal in the sense that it approximates the velocity-pressure couple that minimizes, in the energy norm, the distance to all other solutions satisfying the measurements and the Stokes system.

math.NA↗

arXiv2026

Error analysis for the approximation of a flow in deformable porous media with nonlinear strain-stress relation

We study a model describing the slow flow of a fluid through a deformable, porous, elastic solid undergoing small deformations. The stress-strain relationship of the solid incorporates nonlinear effects, formulated as a perturbation of the classical linear elasticity. To approximate the coupled system, we introduce a discrete scheme based on a first order semi-implicit time integration scheme combined with a standard finite element spatial discretization. We establish the existence and uniqueness of the discrete solution and derive a priori convergence estimates under the assumption that the nonlinear perturbations remain sufficiently small. Finally, we demonstrate the efficiency of the proposed scheme through numerical experiments that also highlight the nonlinear phenomena captured by the model.

math.NA↗

arXiv2025

Alleviating missing boundary conditions in elliptic partial differential equations using interior point measurements

We consider an optimal recovery problem for the Poisson problem when the boundary data is unknown. Compensating information is provided in the form of a finite number of measurements of the solution. A finite element algorithm for this problem was given in Binev et al. (2024), where measurements were assumed to be either bounded linear functionals of the solution or point measurements at locations lying anywhere in the closure of the computational domain. In contrast, we focus on the case of point measurements at locations lying in the interior of the domain. This lowers the regularity requirements placed on the solution. Also, a key ingredient in the recovery process is the finite element approximation of Riesz representers associated with the measurements. Our main result is a pointwise error estimate for the Riesz representers. We apply this to obtain improved estimates which measure the performance of the recovery algorithm in various norms.

math.NA↗

arXiv2025

Babu\v{s}ka's paradox in a nonlinear bending-folding model

The Babu\v{s}ka or plate paradox concerns the failure of convergence when a domain with curved boundary is approximated by polygonal domains in linear bending problems with simply supported boundary conditions. It can be explained via a boundary integral representation of the total Gaussian curvature that is part of the Kirchhoff--Love bending energy. It is shown that the paradox also occurs for a nonlinear bending-folding model which enforces vanishing Gaussian curvature. A simple remedy that is compatible with simplicial finite element methods to avoid incorrect convergence is devised.

math.NA↗

arXiv2024

Convergence and error control of consistent PINNs for elliptic PDEs

We provide an a priori analysis of collocation methods for solving elliptic boundary value problems. They begin with information in the form of point values of the data and utilize only this information to numerically approximate the solution u of the PDE. For such a method to provide an approximation with guaranteed error bounds, additional assumptions on the data, called model class assumptions, are needed. We determine the best error of approximating u in the energy norm, in terms of the total number of point samples, under all Besov class model assumptions for the right hand side and boundary data. We then turn to the study of numerical procedures and analyze whether a proposed numerical procedure achieves the optimal recovery error. We analyze numerical methods which generate the numerical approximation to $u$ by minimizing specified data driven loss functions over a set $\Sigma$ which is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a data driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis a new loss function is proposed. We also address the recent methods of Physics Informed Neural Networks. We prove that minimization of the new loss over restricted neural network spaces $\Sigma$ provides an optimal recovery of the solution $u$, provided that the optimization problem can be numerically executed and $\Sigma$ has sufficient approximation capabilities. We also analyze variants of the new loss function which are more practical for implementation. Finally, numerical examples illustrating the benefits of the proposed loss functions are given.

math.NA↗

arXiv2024

Approximating partial differential equations without boundary conditions

We consider the problem of numerically approximating the solutions to an elliptic partial differential equation (PDE) for which the boundary conditions are lacking. To alleviate this missing information, we assume to be given measurement functionals of the solution. In this context, a near optimal recovery algorithm based on the approximation of the Riesz representers of these functionals in some intermediate Hilbert spaces is proposed and analyzed in [Binev et al. 2024]. Inherent to this algorithm is the computation of $H^s$, $s>1/2$, inner products on the boundary of the computational domain. We take advantage of techniques borrowed from the analysis of fractional diffusion problems to design and analyze a fully practical near optimal algorithm not relying on the challenging computation of $H^s$ inner products.

math.NA↗

arXiv2024

Adaptive Finite Element Methods

This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second order elliptic PDEs and dimension d>1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs and beyond coercive problems to inf-sup stable AFEMs.

math.NA↗

arXiv2023

Solving PDEs with Incomplete Information

We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution. We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.

math.NA↗

arXiv2023

$Γ$-convergent LDG method for large bending deformations of bilayer plates

Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local discontinuous Galerkin (LDG) method which imposes a relaxed discrete isometry constraint and controls deformation gradients at barycenters of elements. We prove $Γ$-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect. We extend the $Γ$-convergence analysis to piecewise quadratic creases. We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.

math.NA↗

arXiv2023

Finite Element Methods for the Stretching and Bending of Thin Structures with Folding

In [Bonito et al., J. Comput. Phys. (2022)], a local discontinuous Galerkin method was proposed for approximating the large bending of prestrained plates, and in [Bonito et al., IMA J. Numer. Anal. (2023)] the numerical properties of this method were explored. These works considered deformations driven predominantly by bending. Thus, a bending energy with a metric constraint was considered. We extend these results to the case of an energy with both a bending component and a nonconvex stretching component, and we also consider folding across a crease. The proposed discretization of this energy features a continuous finite element space, as well as a discrete Hessian operator. We establish the $\Gamma$-convergence of the discrete to the continuous energy and also present an energy-decreasing gradient flow for finding critical points of the discrete energy. Finally, we provide numerical simulations illustrating the convergence of minimizers and the capabilities of the model.

math.NA↗

arXiv2023

Optimal Learning

This paper studies the problem of learning an unknown function $f$ from given data about $f$. The learning problem is to give an approximation $\hat f$ to $f$ that predicts the values of $f$ away from the data. There are numerous settings for this learning problem depending on (i) what additional information we have about $f$ (known as a model class assumption), (ii) how we measure the accuracy of how well $\hat f$ predicts $f$, (iii) what is known about the data and data sites, (iv) whether the data observations are polluted by noise. A mathematical description of the optimal performance possible (the smallest possible error of recovery) is known in the presence of a model class assumption. Under standard model class assumptions, it is shown in this paper that a near optimal $\hat f$ can be found by solving a certain discrete over-parameterized optimization problem with a penalty term. Here, near optimal means that the error is bounded by a fixed constant times the optimal error. This explains the advantage of over-parameterization which is commonly used in modern machine learning. The main results of this paper prove that over-parameterized learning with an appropriate loss function gives a near optimal approximation $\hat f$ of the function $f$ from which the data is collected. Quantitative bounds are given for how much over-parameterization needs to be employed and how the penalization needs to be scaled in order to guarantee a near optimal recovery of $f$. An extension of these results to the case where the data is polluted by additive deterministic noise is also given.

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arXiv2023

Numerical approximations of thin structure deformations

We review different (reduced) models for thin structures using bending as principal mechanism to undergo large deformations. Each model consists in the minimization of a fourth order energy, potentially subject to a nonconvex constraint. Equilibrium deformations are approximated using local discontinuous Galerkin (LDG) finite elements. The design of the discrete energies relies on a discrete Hessian operator defined on discontinuous functions with better approximation properties than the piecewise Hessian. Discrete gradient flows are put in place to drive the minimization process. They are chosen for their robustness and ability to preserve the nonconvex constraint. Several numerical experiments are presented to showcase the large variety of shapes that can be achieved with these models.

math.NA↗

arXiv2022

Numerical Approximation of Gaussian random fields on Closed Surfaces

We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances by several numerical experiments.

math.NA↗

arXiv2022

Approximation of the spectral fractional powers of the Laplace-Beltrami Operator

We consider numerical approximations of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consist of a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rates of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in $L^2$ and $H^1$. The performances of the algorithms are illustrated in different settings including the approximation of Gaussian fields on surfaces.

math.NA↗

arXiv2022

Error Estimates For A Linear Folding Model

An interior penalty discontinuous Galerkin method is devised to approximate minimizers of a linear folding model by discontinuous isoparametric finite element functions that account for an approximation of a folding arc. The numerical analysis of the discrete model includes an a priori error estimate in case of an accurate representation of the folding curve by the isoparametric mesh. Additional estimates show that geometric consistency errors may be controlled separately if the folding arc is approximated by piecewise polynomial curves. Various numerical experiments are carried out to validate the a priori error estimate for the folding model.

math.NA↗

arXiv2022

Modeling and simulation of thin sheet folding

The article addresses the mathematical modeling of the folding of a thin elastic sheet along a prescribed curved arc. A rigorous model reduction from a general hyperelastic material description is carried out under appropriate scaling conditions on the energy and the geometric properties of the folding arc in dependence on the small sheet thickness. The resulting two-dimensional model is a piecewise nonlinear Kirchhoff plate bending model with a continuity condition at the folding arc. A discontinuous Galerkin method and an iterative scheme are devised for the accurate numerical approximation of large deformations.

math.NA↗

arXiv2021

Numerical analysis of the LDG method for large deformations of prestrained plates

A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in our previous computational work. This paper presents a numerical analysis of this LDG method, focusing on the free boundary case. The problem consists of minimizing a fourth order bending energy subject to a nonlinear and nonconvex metric constraint. The energy is discretized using LDG and a discrete gradient flow is used for computing discrete minimizers. We first show $Γ$-convergence of the discrete energy to the continuous one. Then we prove that the discrete gradient flow decreases the energy at each step and computes discrete minimizers with control of the metric constraint defect. We also present a numerical scheme for initialization of the gradient flow, and discuss the conditional stability of it.

math.NA↗