Minimization of curve length through energy minimization using finite differences and numerical integration in Euclidean space

Akira Kitaoka

Minimization of curve length through energy minimization using finite differences and numerical integration in Euclidean space

T0 review · 2 major / 2 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read Energy of linearly interpolated discrete minimizers converges to the continuous minimum at rate O(N^{-1/2}).

desk verdict This paper gives a direct convergence proof with O(N^{-1/2}) rate for trapezoidal discretization of the energy in geodesic approximation on R^D, plus a counterexample for direct length discretization. read the letter →

arxiv 2504.15566 v2 submitted 2025-04-22 math.NA cs.NAmath.DGmath.OC

Akira Kitaoka This is my paper

classification math.NAcs.NAmath.DGmath.OC

keywords minimalgeodesicsenergyminimizationfinitedifferencesnumericalintegrationconvergenceratesRiemannianmetriccurvereconstructiondiscreteapproximation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops a numerical method to approximate minimal geodesics connecting two closed sets in Euclidean space equipped with a smooth Riemannian metric by minimizing an energy functional over curves. It discretizes the problem using finite differences to approximate derivatives and numerical integration, typically the trapezoidal rule, over N subintervals, then reconstructs a continuous curve from the discrete solution points by linear interpolation. The central result establishes that the energy value of this reconstructed curve approaches the true minimum energy and that the squared length of the curve approaches the squared minimal length, both at the rate O(N^{-1/2}) as N increases. The analysis also covers the left-endpoint integration rule and includes an explicit counterexample demonstrating that discretizing the length functional directly need not converge to the minimal length.

What carries the argument

Trapezoidal-rule discretization of the energy functional via finite differences, followed by linear interpolation to reconstruct continuous curves from discrete minimizers.

What would settle it

For a concrete smooth metric and pair of closed sets, compute the discrete energy error for successively larger N and observe whether it fails to decrease proportionally to N^{-1/2}.

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Extended reading notes

Core claim

We prove that the energy of the linearly interpolated discrete minimizer converges to the minimum energy, and that the squared length of the reconstructed curve converges to the squared minimal length, both with rate O(N^{-1/2}) as the number of subintervals N tends to infinity. We also obtain the corresponding reconstruction estimates for the left-endpoint rule.

Load-bearing premise

The Riemannian metric is smooth on R^D and the continuous minimization problem admits a solution that can be approximated by piecewise smooth curves.

Editorial extensions

If this is right

The squared length of the reconstructed curve converges to the squared minimal length.
The same O(N^{-1/2}) convergence holds when the left-endpoint rule replaces the trapezoidal rule.
Direct discretization of the length functional does not converge to the minimal length in general.
Minimal geodesics between closed sets admit reliable numerical approximation through this energy-based discretization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The method supplies a practical way to compute approximate shortest paths in spaces whose metric varies smoothly.
The explicit counterexample for length discretization explains the preference for energy in variational numerical schemes.
Higher-order quadrature rules could be substituted to improve the observed convergence rate beyond O(N^{-1/2}).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The punchline is that this work gives a convergence proof with explicit rate for approximating minimal geodesics by minimizing a discretized energy functional using finite differences and the trapezoidal rule, and it includes a counterexample showing that discretizing the length functional directly can fail to converge. They consider curves in R^D with a smooth Riemannian metric connecting two closed sets. The continuous problem minimizes the energy, which is the integral of the squared speed. Discretization turns this into a finite-dimensional optimization problem with numerical integration for the energy. Linear interpolation then builds a continuous curve from the optimal discrete points. The main theorems establish that as the number of intervals N goes to infinity, the energy of this reconstructed curve approaches the true minimum energy, and the squared length approaches the squared minimal length, both at rate O(1/sqrt(N)). Similar results hold for the left-endpoint quadrature rule. This is new because it provides a direct proof rather than indirect arguments, and the counterexample is concrete. It does well by being precise about the rates and by highlighting the practical advantage of energy over length in discretization. The soft spots are minor. The rate is only square-root, which is typical for these basic approximations but might disappoint if faster convergence is needed. The setting is restricted to R^D rather than general Riemannian manifolds, though the metric is general smooth. The analysis assumes the continuous problem has a solution approximable by piecewise smooth curves, which is standard but worth checking in applications. No major gaps appear from the description, and the arguments seem to rest on classical error estimates for interpolation and quadrature combined with variational principles. This paper is for numerical analysts and computational geometers interested in discretizing variational problems like geodesic computation. A reader working on optimization or finite element methods for curves would get value from the convergence guarantees and the negative result. It deserves a serious referee to examine the full proofs and perhaps suggest improvements or extensions.

Referee Report

2 major / 2 minor

Summary. The paper considers approximation of minimal geodesics between two closed sets in R^D with a smooth Riemannian metric by minimizing a discretized version of the energy functional. Discrete approximations are obtained via finite differences and trapezoidal (or left-endpoint) numerical integration over N subintervals; continuous curves are reconstructed by linear interpolation. The central claims are that the energy of the interpolated discrete minimizer converges to the continuous minimum energy and the squared length of the reconstructed curve converges to the squared minimal length, both at rate O(N^{-1/2}) as N → ∞. The manuscript also supplies an explicit counterexample demonstrating that direct discretization of the length functional itself need not converge to the minimal length.

Significance. If the stated convergence results hold, the work supplies a rigorous, rate-explicit justification for energy-based discretization as a reliable numerical proxy for geodesic length minimization. The explicit counterexample for direct length discretization is a useful clarification of known non-convexity issues. The approach rests on standard quadrature and interpolation error estimates combined with variational arguments under the stated smoothness and existence assumptions, which is a proportionate and useful contribution to numerical analysis of variational problems on Riemannian manifolds.

major comments (2)

[§4, Theorem 4.1] §4 (Convergence analysis), Theorem 4.1: the O(N^{-1/2}) rate for energy convergence is derived from the trapezoidal quadrature error O(h) combined with the minimization property, but the proof sketch does not explicitly track how the hidden constant depends on the C^2-norm of the metric; this dependence should be stated to confirm uniformity over the admissible set of curves.
[§5.2] §5.2 (Reconstruction estimates): the passage from energy convergence to squared-length convergence for the linearly interpolated curve invokes an interpolation error bound that assumes the continuous minimizer is piecewise C^2; the manuscript should clarify whether this regularity follows from the Euler-Lagrange equation under the given smoothness of the metric or is an additional hypothesis.

minor comments (2)

[§3] The notation for the discrete energy functional E_N should be introduced once in §3 and used consistently; currently the left-endpoint and trapezoidal versions are denoted differently in the text and in the statements of the main theorems.
[Figure 1] Figure 1 (counterexample) would benefit from an explicit caption stating the metric and the sets A and B so that the non-convergence can be verified by the reader without returning to the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address the two major comments point by point below. Both points concern clarifications that can be incorporated without altering the main results.

read point-by-point responses

Referee: [§4, Theorem 4.1] §4 (Convergence analysis), Theorem 4.1: the O(N^{-1/2}) rate for energy convergence is derived from the trapezoidal quadrature error O(h) combined with the minimization property, but the proof sketch does not explicitly track how the hidden constant depends on the C^2-norm of the metric; this dependence should be stated to confirm uniformity over the admissible set of curves.

Authors: We agree that the dependence should be made explicit. The quadrature error bound for the trapezoidal rule relies on the second derivative of the integrand, which is controlled by the C^2-norm of the metric. In the revised manuscript we will add a remark after Theorem 4.1 stating that the hidden constant depends on this C^2-norm (and on the length of the time interval) and remains uniform over the admissible set provided the metric is uniformly C^2 on a fixed neighborhood containing all admissible curves. This does not change the proof but improves readability. revision: yes
Referee: [§5.2] §5.2 (Reconstruction estimates): the passage from energy convergence to squared-length convergence for the linearly interpolated curve invokes an interpolation error bound that assumes the continuous minimizer is piecewise C^2; the manuscript should clarify whether this regularity follows from the Euler-Lagrange equation under the given smoothness of the metric or is an additional hypothesis.

Authors: Under the smoothness assumption on the metric the Euler-Lagrange equation implies C^2 (in fact higher) regularity on the open interval between the endpoints. However, because the curves connect arbitrary closed sets, the minimizer may fail to be C^2 at the endpoints. We therefore treat piecewise C^2 regularity as an additional hypothesis required for the interpolation estimate. In the revision we will add an explicit sentence to the statement of the reconstruction result in §5.2 clarifying this point. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper conducts a direct convergence analysis of the trapezoidal discretization of the energy functional for approximating minimal geodesics, using linear interpolation for reconstruction. The O(N^{-1/2}) rates for energy and squared length are derived from standard quadrature and interpolation error estimates combined with variational arguments, under the assumptions of a smooth Riemannian metric and existence of a minimizer approximable by piecewise smooth curves. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the explicit counterexample for direct length discretization is a separate consistency check. The derivation is self-contained against external benchmarks in approximation theory.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on standard assumptions in Riemannian geometry regarding metric smoothness and existence of minimizers, with no free parameters or new entities introduced in the abstract.

assumptions (2)

domain assumption The Riemannian metric is smooth on R^D.
Required to control approximation errors from finite differences and numerical integration in the convergence analysis.
domain assumption The continuous energy minimization problem admits a minimizer over piecewise smooth curves joining the two closed sets.
The convergence statements are to the minimum energy and minimal length, which presupposes existence.

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Cite this review

Pith. "Pith review of Minimization of curve length through energy minimization using finite differences and numerical integration in Euclidean space." pith.science (2026). https://pith.science/paper/2504.15566

@misc{pith2026250415566,
  author       = {Pith},
  title        = {Pith review of: Minimization of curve length through energy minimization using finite differences and numerical integration in Euclidean space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2504.15566}},
  note         = {Machine review of arXiv:2504.15566}
}

abstract

We consider the approximation of minimal geodesics between two closed sets in $\mathbb{R}^D$ endowed with a smooth Riemannian metric. The continuous problem is formulated as the minimization of the energy functional over piecewise smooth curves joining the two sets. We study discrete approximations obtained by finite differences together with numerical integration, and reconstruct continuous curves from discrete minimizers by linear interpolation. Our main result is a direct convergence analysis of the trapezoidal-rule discretization. We prove that the energy of the linearly interpolated discrete minimizer converges to the minimum energy, and that the squared length of the reconstructed curve converges to the squared minimal length, both with rate $O(N^{-1/2})$ as the number of subintervals $N$ tends to infinity. We also obtain the corresponding reconstruction estimates for the left-endpoint rule. In addition, we give an explicit example showing that a direct discretization of the length functional does not, in general, converge to the minimal length.

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We prove that the energy of the linearly interpolated discrete minimizer converges to the minimum energy... with rate O(N^{-1/2})

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