About

Ronald DeVore is the Walter E. Koss Professor and Distinguished Professor of Mathematics at Texas A&M University. He is based in the Department of Mathematics, located in the Blocker Building 608F, College Station, Texas. His contact information includes an email at ronalddevore@tamu.edu and an office in Blocker Building 318D, with a phone number of (979) 862-4860. The webpage indicates his roles as a professor and his association with the university's mathematics department, but does not provide specific details about his research focus, background, or key contributions.

Research topics

• Artificial Intelligence
• Computer Science
• Mathematical optimization
• Applied mathematics
• Mathematical analysis
• Mathematics

Selected publications

• Convergence and error control of consistent PINNs for elliptic PDEs

  IMA Journal of Numerical Analysis · 2025-01-17 · 1 citations

  article

  Abstract We provide an a priori analysis of a certain class of numerical methods, commonly referred to as collocation methods, for solving elliptic boundary value problems. They begin with information in the form of point values of the right side $f$ of such equations and point values of the boundary function $g$ and utilize only this information to numerically approximate the solution $u$ of the partial differential equation (PDE). For such a method to provide an approximation to $u$ with guaranteed error bounds, additional assumptions on $f$ and $g$, called model class assumptions, are needed. We determine the best error (in the energy norm) of approximating $u$, 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 (nearly) achieves the optimal recovery (OR) error. In particular, we analyze numerical methods that generate the numerical approximation to $u$ by minimizing specified data-driven loss functions over a set $\varSigma $ that 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 loss function ${\cal L}^{*}$ is proposed. We also address the recent methods of physics informed neural networks. We prove that minimization of the new loss ${\cal L}^{*}$ over restricted neural network spaces $\varSigma $ provides an OR of the solution $u$, provided that the optimization problem can be numerically executed and $\varSigma $ has sufficient approximation capabilities. We also analyze variants of ${\cal L}^{*}$ that are more practical for implementation. Finally, numerical examples illustrating the benefits of the proposed loss functions are given.

  PublisherDOI

• Optimal Recovery Meets Minimax Estimation

  ArXiv.org · 2025-02-24

  preprintOpen access1st authorCorresponding

  A fundamental problem in statistics and machine learning is to estimate a function $f$ from possibly noisy observations of its point samples. The goal is to design a numerical algorithm to construct an approximation $\hat f$ to $f$ in a prescribed norm that asymptotically achieves the best possible error (as a function of the number $m$ of observations and the variance $σ^2$ of the noise). This problem has received considerable attention in both nonparametric statistics (noisy observations) and optimal recovery (noiseless observations). Quantitative bounds require assumptions on $f$, known as model class assumptions. Classical results assume that $f$ is in the unit ball of a Besov space. In nonparametric statistics, the best possible performance of an algorithm for finding $\hat f$ is known as the minimax rate and has been studied in this setting under the assumption that the noise is Gaussian. In optimal recovery, the best possible performance of an algorithm is known as the optimal recovery rate and has also been determined in this setting. While one would expect that the minimax rate recovers the optimal recovery rate when the noise level $σ$ tends to zero, it turns out that the current results on minimax rates do not carefully determine the dependence on $σ$ and the limit cannot be taken. This paper handles this issue and determines the noise-level-aware (NLA) minimax rates for Besov classes when error is measured in an $L_q$-norm with matching upper and lower bounds. The end result is a reconciliation between minimax rates and optimal recovery rates. The NLA minimax rate continuously depends on the noise level and recovers the optimal recovery rate when $σ$ tends to zero.

  PublisherOA PDFDOI

• Constructions of Bounded Solutions of $$ \textit{di}\upsilon $$ u = f in Critical Spaces

  2024-01-01

  book-chapter
  PublisherDOI

• A Note on Best n-term Approximation for Generalized Wiener Classes

  2024-01-01 · 1 citations

  book-chapter1st authorCorresponding
  PublisherDOI

• Prologue to Multiscale, Nonlinear and Adaptive Approximation II

  2024-01-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Introduction: Wolfgang Dahmen’s mathematical work (as of 2009)

  2024-01-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Constructions of bounded solutions of $div\, {\mathbf u}=f$ in critical spaces

  arXiv (Cornell University) · 2024-05-21

  preprintOpen access

  We construct uniformly bounded solutions of the equation $div\, {\mathbf u}=f$ for arbitrary data $f$ in the critical spaces $L^d(Ω)$, where $Ω$ is a domain of ${\mathbb R}^d$. This question was addressed by Bourgain & Brezis, [On the equation ${\rm div}\, Y=f$ and application to control of phases, JAMS 16(2) (2003) 393-426], who proved that although the problem has a uniformly bounded solution, it is critical in the sense that there exists no linear solution operator for general $L^d$-data. We first discuss the validity of this existence result under weaker conditions than $f\in L^d(Ω)$, and then focus our work on constructive processes for such uniformly bounded solutions. In the $d=2$ case, we present a direct one-step explicit construction, which generalizes for $d>2$ to a $(d-1)$-step construction based on induction. An explicit construction is proposed for compactly supported data in $L^{2,\infty}(Ω)$ in the $d=2$ case. We also present constructive approaches based on optimization of a certain loss functional adapted to the problem. This approach provides a two-step construction in the $d=2$ case. This optimization is used as the building block of a hierarchical multistep process introduced in [E. Tadmor, Hierarchical construction of bounded solutions in critical regularity spaces, CPAM 69(6) (2016) 1087-1109] that converges to a solution in more general situations.

  PublisherOA PDFDOI

• Weighted variation spaces and approximation by shallow ReLU networks

  Applied and Computational Harmonic Analysis · 2024-10-10 · 2 citations

  articleOpen access1st author

  We investigate the approximation of functions f on a bounded domain Ω ⊂ R d by the outputs of single-hidden-layer ReLU neural networks of width n . This form of nonlinear n -term dictionary approximation has been intensely studied since it is the simplest case of neural network approximation (NNA). There are several celebrated approximation results for this form of NNA that introduce novel model classes of functions on Ω whose approximation rates do not grow unbounded with the input dimension. These novel classes include Barron classes, and classes based on sparsity or variation such as the Radon-domain BV classes. The present paper is concerned with the definition of these novel model classes on domains Ω. The current definition of these model classes does not depend on the domain Ω. A new and more proper definition of model classes on domains is given by introducing the concept of weighted variation spaces. These new model classes are intrinsic to the domain itself. The importance of these new model classes is that they are strictly larger than the classical (domain-independent) classes. Yet, it is shown that they maintain the same NNA rates.

  PublisherDOI

• A note on best n-term approximation for generalized Wiener classes

  arXiv (Cornell University) · 2024-06-16

  preprintOpen access1st authorCorresponding

  We determine the best n-term approximation of generalized Wiener model classes in a Hilbert space $H $. This theory is then applied to several special cases.

  PublisherOA PDFDOI

• Convergence and error control of consistent PINNs for elliptic PDEs

  arXiv (Cornell University) · 2024-06-13

  preprintOpen access

  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 $Σ$ 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 $Σ$ provides an optimal recovery of the solution $u$, provided that the optimization problem can be numerically executed and $Σ$ 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.

  PublisherOA PDFDOI

Recent grants

• Numerical Methods for High Dimensional Partial Differential Equations

  NSF · $250k · 2015–2018

• FRG: Collaborative research: Algorithms for sparse data representations

  NSF · $189k · 2004–2008

• Collaborative Research: An ADT Proposal: Fast Point Cloud Surface Reconstruction Algorithms

  NSF · $708k · 2009–2013

• Numerical Methods for Parametric Partial Differential Equations

  NSF · $369k · 2018–2021

• ATD Collaborative Research: Theory and Algorithms for High Dimensional Learning

  NSF · $249k · 2012–2015

Frequent coauthors

• Albert Cohen

  95 shared

• Wolfgang Dahmen

  University of South Carolina

  61 shared

• Guergana Petrova

  Texas A&M University

  50 shared

• P. Wojtaszczyk

  30 shared

• Peter Binev

  22 shared

• В. А. Попов

  20 shared

• Carl de Boor

  University of Wisconsin–Madison

  19 shared

• Xiang Ming Yu

  15 shared

Labs

• Ronald DeVore's LabPI

  Research in mathematics, including algebraic geometry, arithmetic geometry, and number theory.