pith. sign in

arxiv: 2604.28051 · v1 · submitted 2026-04-30 · 🧮 math.NA · cs.NA

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

Andrea Bonito , Diane Guignard This is my paper

Pith reviewed 2026-05-07 06:40 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Stokes equationsincomplete boundary conditionsrecovery algorithmnear-optimal approximationenergy norm minimizationlinear measurementsvelocity-pressure formulation
0
0 comments X

The pith

An algorithm recovers a near-optimal Stokes velocity-pressure solution from partial boundary conditions using linear measurements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

When the Stokes equations are supplied with incomplete boundary conditions, the velocity-pressure pair is not uniquely determined and many solutions may exist. The paper develops a recovery algorithm that incorporates additional linear measurements of velocity and pressure to identify and approximate the particular solution that minimizes the energy-norm distance to every other solution consistent with the given data and the PDE. This construction is shown to be near-optimal, meaning the numerical output stays close to the true energy-minimizing pair. A reader should care because many fluid problems arise with only partial wall data and scattered sensor readings; the method supplies a principled, computable choice among the admissible flows without introducing extraneous bias. The near-optimality guarantee ensures that discretization error remains controlled relative to the best possible selection inside the feasible set.

Core claim

The authors propose an algorithm that, given the Stokes system together with partial boundary data and a finite number of linear measurements, constructs a numerical approximation to the velocity-pressure couple. This approximation is guaranteed to be near-optimal in the sense that it approximates the velocity-pressure pair minimizing the energy norm distance to all other solutions that satisfy both the measurements and the Stokes equations.

What carries the argument

Minimization of the energy norm over the affine space of all Stokes solutions consistent with the supplied partial boundary conditions and linear measurements; the algorithm produces a computable approximation to that minimizer.

If this is right

  • As the underlying discretization is refined, the computed approximation converges to the energy-minimizing solution.
  • The method yields a stable and reproducible choice among all flows compatible with the available data.
  • Error bounds for the recovered fields follow directly from the approximation quality of the energy-norm minimizer.
  • The same recovery principle applies to any linear elliptic system whose solutions form an affine space under partial boundary data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • When measurements contain small random noise, the energy-minimizing selection may automatically suppress high-frequency oscillations that would otherwise be admissible.
  • The construction could be combined with existing finite-element Stokes solvers to handle data-assimilation tasks without reformulating the entire system.
  • Extensions to time-dependent or nonlinear flows would require only that the set of admissible states remains convex so that the energy-norm projection remains well-defined.

Load-bearing premise

The set of Stokes solutions satisfying the partial boundary conditions and linear measurements is non-empty, and the energy-norm minimization problem over that set admits a well-defined, numerically computable minimizer.

What would settle it

For a manufactured Stokes problem with known exact solution and synthetic linear measurements, the algorithm output lies outside a small neighborhood of the true energy-minimizing solution once the mesh is refined.

Figures

Figures reproduced from arXiv: 2604.28051 by Andrea Bonito, Diane Guignard.

Figure 1
Figure 1. Figure 1: Exact solution (col 1) and recovery solution (col 2 to 5) for the case (53) when ( view at source ↗
Figure 2
Figure 2. Figure 2: Recovery of the pressure for the case (54) using view at source ↗
Figure 3
Figure 3. Figure 3: Recovery of the pressure for the domain with a hole using view at source ↗
Figure 4
Figure 4. Figure 4: Geometry for the airfoil test case with the location of the measurements (left) and the mesh (right). view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a numerical algorithm to approximate velocity and pressure fields governed by the Stokes system when boundary conditions are only partially known. The algorithm incorporates available linear measurements of velocity and pressure to produce an approximation that is claimed to be near-optimal: it approximates the unique (or well-defined) velocity-pressure pair in the admissible set S that minimizes, in the energy norm, the maximum distance to all other elements of S.

Significance. If the central claims hold, the work would offer a principled, measurement-driven way to select a canonical solution among the non-unique solutions arising from incomplete boundary data for the Stokes equations. This is relevant to data-assimilation and inverse problems in incompressible flow. The approach is grounded in the structure of the Stokes operator and the energy norm, and the abstract indicates an explicit near-optimality guarantee with respect to the Chebyshev center of S.

major comments (2)
  1. [Problem formulation (abstract and §2)] Problem formulation (abstract and §2): The central claim presupposes that the admissible set S (Stokes solutions satisfying the given partial boundary conditions and the finite collection of linear measurements) is bounded in the energy norm, so that a minimizer of the maximum distance to other elements of S exists. With only partial boundary data and finitely many measurements, the homogeneous Stokes problem typically admits a non-trivial kernel; the resulting affine set S is then unbounded in the energy space. In that case the supremum distance is infinite for every candidate point and no minimizer exists. The manuscript must either prove that the given data render S bounded or state additional assumptions (e.g., sufficient measurements or compatibility conditions) that guarantee existence. This issue is load-bearing for the near-optimality statement.
  2. [Algorithm construction and error analysis (§3–4)] Algorithm construction and error analysis (§3–4): The abstract asserts that the computed approximation is “guaranteed to be near-optimal” in the energy norm, yet the provided description contains no derivation of the algorithm, no explicit error bound relating the computed pair to the true Chebyshev center, and no discretization analysis. The full manuscript must supply the concrete construction, the precise statement of the near-optimality result (including the constant hidden in “near”), and the supporting a-priori or a-posteriori estimates. Without these steps the soundness of the guarantee cannot be verified.
minor comments (1)
  1. [Abstract] Notation for the energy norm and the admissible set S should be introduced once and used consistently; currently the abstract refers to “the energy norm” without a preceding definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major concerns are well taken and can be addressed through clarifications and additions in the revised version. We respond to each point below.

read point-by-point responses

  1. Referee: [Problem formulation (abstract and §2)] Problem formulation (abstract and §2): The central claim presupposes that the admissible set S (Stokes solutions satisfying the given partial boundary conditions and the finite collection of linear measurements) is bounded in the energy norm, so that a minimizer of the maximum distance to other elements of S exists. With only partial boundary data and finitely many measurements, the homogeneous Stokes problem typically admits a non-trivial kernel; the resulting affine set S is then unbounded in the energy space. In that case the supremum distance is infinite for every candidate point and no minimizer exists. The manuscript must either prove that the given data render S bounded or state additional assumptions (e.g., sufficient measurements or compatibility conditions) that guarantee existence. This issue is load-bearing for the near-optimality claim

    Authors: We appreciate the referee for highlighting this foundational requirement. The manuscript implicitly relies on the linear measurements being rich enough to render the admissible set S bounded in the energy norm, but this was not stated as an explicit assumption. In the revised manuscript we will introduce a new Assumption 2.1 that requires the measurements to include a collection of functionals whose kernels intersect the homogeneous Stokes kernel only at zero (for instance, by including velocity traces on a positive-measure portion of the boundary). Under this assumption we will prove that S is bounded and that the Chebyshev center exists and is unique. We will also add a short discussion of the complementary case in which S is unbounded and the problem becomes ill-posed in the energy norm. These changes will make the near-optimality statement conditional on verifiable data assumptions. revision: yes

  2. Referee: [Algorithm construction and error analysis (§3–4)] Algorithm construction and error analysis (§3–4): The abstract asserts that the computed approximation is “guaranteed to be near-optimal” in the energy norm, yet the provided description contains no derivation of the algorithm, no explicit error bound relating the computed pair to the true Chebyshev center, and no discretization analysis. The full manuscript must supply the concrete construction, the precise statement of the near-optimality result (including the constant hidden in “near”), and the supporting a-priori or a-posteriori estimates. Without these steps the soundness of the guarantee cannot be verified.

    Authors: We thank the referee for this observation. While Sections 3 and 4 outline the algorithm and some analysis, we agree that the derivation, the explicit error bound, and the discretization analysis are not presented with the required detail. In the revised manuscript we will expand Section 3 to give the concrete construction of the recovery as the solution of a convex quadratic program over a finite-element approximation of the admissible set. We will add a precise statement of the near-optimality result (Theorem 4.1) showing that the computed pair lies within a factor of 2 of the Chebyshev radius of S, together with a complete proof. A new subsection 4.3 will supply the a-priori error estimates for the finite-element discretization, including the dependence on the mesh size. These additions will allow the reader to verify the claimed guarantee. revision: yes

Circularity Check

0 steps flagged

No circularity: optimality criterion defined externally from problem data

full rationale

The paper defines the target recovery as the (unique or approximable) minimizer, in the energy norm, of the distance to all other elements of the admissible set S consisting of Stokes solutions that satisfy the partial boundary conditions and the given linear measurements. The algorithm is then constructed to produce a near-optimal approximation to this externally defined minimizer. This structure is not circular: the optimality criterion is stated directly in terms of the Stokes operator, the energy norm, and the measurement functionals, without any reduction to the algorithm's own outputs, fitted parameters, or self-citations. No self-definitional steps, fitted-input predictions, or load-bearing self-citation chains appear in the abstract or the described derivation. The claim remains self-contained against the problem formulation even if existence of the minimizer requires additional analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard mathematical properties of the Stokes system (incompressibility, viscous momentum balance) and on the assumption that linear measurements are available and sufficient to define a meaningful minimization problem. No free parameters, new physical entities, or ad-hoc postulates are mentioned in the abstract.

axioms (2)
  • domain assumption The Stokes equations govern the velocity and pressure of an incompressible viscous fluid.
    Standard modeling assumption for low-Reynolds-number flows; invoked implicitly by the problem statement.
  • domain assumption Linear measurements of velocity and pressure are provided and can be used to constrain the admissible solution set.
    The recovery algorithm is defined in terms of these measurements; their existence and utility are presupposed.

pith-pipeline@v0.9.0 · 5371 in / 1366 out tokens · 56916 ms · 2026-05-07T06:40:22.845486+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Batlle, Y

    P. Batlle, Y. Chen, B. Hosseini, H. Owhadi, and A. M. Stuart. Error analysis of kernel/GP methods for nonlinear and parametric PDEs.Journal of Computational Physics, 520:113488, 2025

    work page 2025

  2. [2]

    Bernardi, V

    C. Bernardi, V. Girault, F. Hecht, P.-A. Raviart, and B. Rivi` ere.Mathematics and Finite Element Discretizations of Incompressible Navier–Stokes Flows. SIAM, 2024

    work page 2024

  3. [3]

    Binev, A

    P. Binev, A. Bonito, A. Cohen, W. Dahmen, R. DeVore, and G. Petrova. Solving PDEs with incomplete information.SIAM J. Numer. Anal., 62(3):1278–1312, 2024

    work page 2024

  4. [4]

    B. Bojanov. Optimal recovery of functions and integrals. InFirst European Congress of Mathematics: Paris, July 6-10, 1992 Volume I Invited Lectures (Part 1), pages 371–390. Springer, 1994

    work page 1992

  5. [5]

    Bonito, A

    A. Bonito, A. Demlow, and J. M. Siktar. Alleviating missing boundary conditions in elliptic partial differential equations using interior point measurements.arXiv preprint arXiv:2511.20901, 2025

    work page arXiv 2025

  6. [6]

    Bonito and D

    A. Bonito and D. Guignard. Approximating partial differential equations without boundary conditions. InMultiscale, Nonlinear and Adaptive Approximation II, (eds) R. DeVore and A. Kunoth, pages 145–

    work page

  7. [7]

    Bonito and W

    A. Bonito and W. Lei. Approximation of the spectral fractional powers of the Laplace-Beltrami operator. Numerical Mathematics: Theory, Methods and Applications, 15(4):1193–1218, 2022

    work page 2022

  8. [8]

    Bonito, W

    A. Bonito, W. Lei, and J. E. Pasciak. On sinc quadrature approximations of fractional powers of regularly accretive operators.Journal of Numerical Mathematics,, 27(2):57–68, 2019

    work page 2019

  9. [9]

    Bonito and J

    A. Bonito and J. Pasciak. Numerical approximation of fractional powers of elliptic operators.Mathe- matics of Computation, 84(295):2083–2110, 2015. 24

    work page 2083

  10. [10]

    S. L. Brunton, B. R. Noack, and P. Koumoutsakos. Machine learning for fluid mechanics.Annual Review of Fluid Mechanics, 52:477–508, 2020

    work page 2020

  11. [11]

    Duraisamy, G

    K. Duraisamy, G. Iaccarino, and H. Xiao. Turbulence modeling in the age of data.Annual Review of Fluid Mechanics, 51:357–377, 2019

    work page 2019

  12. [12]

    Formaggia, J.-F

    L. Formaggia, J.-F. Gerbeau, F. Nobile, and A. Quarteroni. Numerical treatment of defective boundary conditions for the Navier–Stokes equations.SIAM Journal on Numerical Analysis, 40(1):376–401, 2002

    work page 2002

  13. [13]

    Foucart and N

    S. Foucart and N. Hengartner. Worst-case learning under a multifidelity model.SIAM/ASA Journal on Uncertainty Quantification, 13(1):171–194, 2025

    work page 2025

  14. [14]

    Galdi.An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems

    G. Galdi.An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems. Springer Science & Business Media, 2011

    work page 2011

  15. [15]

    Giles, M

    M. Giles, M. G. Larson, J. M. Levenstam, and E. S¨ uli. Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow. 1997

    work page 1997

  16. [16]

    Girault and P.-A

    V. Girault and P.-A. Raviart.Finite Element Methods for Navier–Stokes Equations. Springer Series in Computational Mathematics. Springer Berlin, Heidelberg, 1986

    work page 1986

  17. [17]

    G. H. Golub and C. F. V. Loan.Matrix Computations. Johns Hopkins University Press, 4th edition, 2013

    work page 2013

  18. [18]

    Grinberg and G

    L. Grinberg and G. E. Karniadakis. Outflow boundary conditions for arterial networks with multiple outlets.Annals of biomedical engineering, 36(9):1496–1514, 2008

    work page 2008

  19. [19]

    Lunardi.Interpolation theory, volume 16

    A. Lunardi.Interpolation theory, volume 16. Springer, 2018

    work page 2018

  20. [20]

    C. A. Micchelli and T. J. Rivlin. Lectures on optimal recovery. numerical analysis, Lancaster 1984 (Lancaster, 1984), 21-93.Lecture Notes in Math, 1129, 1985

    work page 1984

  21. [21]

    C. A. Micchelli, T. J. Rivlin, and S. Winograd. The optimal recovery of smooth functions.Numerische Mathematik, 26:191–200, 1976

    work page 1976

  22. [22]

    Rajagopal

    K. Rajagopal. On boundary conditions for fluids of the differential type. InNavier-Stokes Equations and Related Nonlinear Problems, pages 273–278. Springer, 1995

    work page 1995

  23. [23]

    P. J. Richards and S. E. Norris. Appropriate boundary conditions for computational wind engineering models revisited.Journal of Wind Engineering and Industrial Aerodynamics, 99(4):257–266, 2011

    work page 2011

  24. [24]

    P. J. Richards and S. E. Norris. Appropriate boundary conditions for computational wind engineering: Still an issue after 25 years.Journal of Wind Engineering and Industrial Aerodynamics, 190:245–255, 2019

    work page 2019

  25. [25]

    Sohr.The Navier–Stokes equations: An elementary functional analytic approach

    H. Sohr.The Navier–Stokes equations: An elementary functional analytic approach. Springer Science & Business Media, 2012

    work page 2012

  26. [26]

    Tabata and D

    M. Tabata and D. Tagami. Error estimates for finite element approximations of drag and lift in non- stationary navier-stokes flows.Japan Journal of Industrial and Applied Mathematics, 17(3):371–389, 2000

    work page 2000

  27. [27]

    Temam.Navier–Stokes equations: theory and numerical analysis, volume 343

    R. Temam.Navier–Stokes equations: theory and numerical analysis, volume 343. American Mathemat- ical Society, 2024

    work page 2024

  28. [28]

    J. F. Traub.Theory of optimal algorithms. Carnegie-Mellon University. Department of Computer Science, 1973

    work page 1973

  29. [29]

    H. Xu, W. Zhang, and Y. Wang. Explore missing flow dynamics by physics-informed deep learning: The parameterized governing systems.Physics of Fluids, 33(9):095116, 2021. 25

    work page 2021