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arxiv: 2604.19910 · v1 · submitted 2026-04-21 · 🧮 math.NA · cs.NA

A Proximal Primal-Dual Approach to Generalized JKO Schemes for Doubly Nonlinear Parabolic Equations

Luis M. Brice\~no-Arias , Jos\'e A. Carrillo , Dante Kalise , Francisco J. Silva , Li Wang This is my paper

Pith reviewed 2026-05-10 01:20 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords proximal primal-dualJKO schemesdoubly nonlinear parabolic equationsproximal operatorsgradient flowsnumerical approximationp-Laplace equationrelativistic heat equation
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The pith

A proximal primal-dual method yields explicit formulas for proximal operators in generalized JKO schemes for doubly nonlinear equations.

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

The paper proposes using optimization strategies based on proximal primal-dual algorithms to numerically solve a family of nonlinear partial differential equations that arise as gradient flows with general costs. These include the p-Laplace equation and flux-limited equations such as the relativistic heat equation. Explicit formulas for the proximal operators are derived to enable efficient numerical approximation of these steepest descent evolutions. A reader would care because this variational approach provides a way to simulate these complex dynamics while recovering the expected qualitative behaviors in known cases.

Core claim

By recasting the generalized JKO scheme as a proximal primal-dual optimization problem, explicit formulas for the proximal operators with general costs are obtained, which are then used to approximate the solutions of doubly nonlinear parabolic equations in a computationally tractable manner.

What carries the argument

The proximal primal-dual algorithm applied to the generalized JKO scheme, which derives and applies proximal operators for general costs to discretize the gradient-flow dynamics.

If this is right

  • The method produces numerical solutions for the p-Laplace equation as a special case.
  • Flux-limited equations including the relativistic heat equation can be handled within the same framework.
  • Validation occurs by recovering the qualitative behavior of known steepest descent evolutions.
  • The scheme extends variational optimization techniques to a broad family of doubly nonlinear parabolic PDEs.

Where Pith is reading between the lines

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

  • The explicit proximal formulas may simplify implementation for other gradient flows with similar cost structures not explicitly tested here.
  • This optimization lens could connect to related numerical methods for measure-valued or nonlinear diffusion problems.
  • Parameter studies on the general cost functions might become feasible in applications where direct discretization is costly.

Load-bearing premise

Explicit and efficiently computable formulas for the proximal operators exist for the general family of costs considered and the discrete scheme reproduces the continuous dynamics without uncontrolled artifacts.

What would settle it

A numerical test on a known case such as the heat equation or p-Laplace flow where the computed solutions deviate from the exact or expected qualitative behavior would falsify the approach.

Figures

Figures reproduced from arXiv: 2604.19910 by Dante Kalise, Francisco J. Silva, Jos\'e A. Carrillo, Li Wang, Luis M. Brice\~no-Arias.

Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: 6. Discussion and open problems. In this work we have studied the use of primal-dual proximal operator methods for the numerical approximation of doubly 22 [PITH_FULL_IMAGE:figures/full_fig_p022_5_9.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the $p$-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by computing explicit formulas for proximal operators with general costs amenable to efficient numerical approximation. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.

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 / 2 minor

Summary. The manuscript proposes a proximal primal-dual variational method to discretize generalized JKO gradient flows for doubly nonlinear parabolic equations, including the p-Laplace equation and flux-limited equations such as the relativistic heat equation. Explicit formulas for proximal operators associated with general costs are derived to enable efficient numerical approximation, with validation performed by recovering the expected qualitative features (e.g., finite propagation speed) on a small number of textbook cases.

Significance. If the explicit proximal formulas are correct and the resulting scheme is shown to be consistent and convergent, the approach would provide a useful optimization-based framework for treating a broad family of nonlinear PDEs as gradient flows with general costs, extending classical JKO schemes to doubly nonlinear settings.

major comments (2)
  1. [Numerical validation] Numerical validation section: the scheme is tested only by qualitative recovery of known behaviors (finite propagation speed, etc.) on a few instances; no a-priori error estimates, no convergence tables under mesh/time-step refinement, and no quantitative comparison to exact solutions or high-resolution references are provided. This directly affects the central claim that the proximal discretization faithfully reproduces the continuous dynamics without uncontrolled artifacts.
  2. [§3] Proximal-operator derivations (abstract and §3): while explicit formulas are asserted to exist and be amenable to efficient approximation for general costs, the manuscript supplies neither the full derivations nor an analysis of their computational cost or stability for arbitrary costs beyond the validated special cases. This is load-bearing for the claim of broad applicability.
minor comments (2)
  1. [Introduction] Notation for the general cost functional and the associated proximal operator could be introduced with a single consolidated definition early in the paper to improve readability.
  2. [Introduction] A few references to recent work on proximal methods for flux-limited equations appear to be missing; adding them would strengthen the literature context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments point by point below, indicating the revisions we intend to incorporate.

read point-by-point responses

  1. Referee: Numerical validation section: the scheme is tested only by qualitative recovery of known behaviors (finite propagation speed, etc.) on a few instances; no a-priori error estimates, no convergence tables under mesh/time-step refinement, and no quantitative comparison to exact solutions or high-resolution references are provided. This directly affects the central claim that the proximal discretization faithfully reproduces the continuous dynamics without uncontrolled artifacts.

    Authors: We acknowledge that the present numerical section emphasizes qualitative recovery of features such as finite propagation speed to illustrate the scheme's fidelity to the underlying doubly nonlinear dynamics. For many equations in the family, closed-form exact solutions are unavailable, which limits direct quantitative error measurement. Nevertheless, we agree that additional quantitative evidence would strengthen the central claim. In the revised manuscript we will add a new subsection containing mesh-refinement studies and L^1-error tables for the linear heat equation (where exact solutions exist) and for the p-Laplacian with p=2, together with comparisons against high-resolution finite-volume references for the relativistic heat equation. We will also explicitly note that a priori error estimates lie outside the scope of this work, whose focus is the construction of explicit proximal operators, and flag this as an important direction for future analysis. revision: partial

  2. Referee: Proximal-operator derivations (abstract and §3): while explicit formulas are asserted to exist and be amenable to efficient approximation for general costs, the manuscript supplies neither the full derivations nor an analysis of their computational cost or stability for arbitrary costs beyond the validated special cases. This is load-bearing for the claim of broad applicability.

    Authors: Section 3 does derive the proximal operators by reducing the variational problem to a scalar optimization whose solution yields an explicit formula for each cost. To address the referee's concern, we will expand §3 with a complete, self-contained derivation (including the first-order optimality conditions and the reduction to a one-dimensional problem) and move the most technical steps to a new appendix. We will also add a short subsection that quantifies the computational cost: for costs satisfying standard convexity and growth assumptions the proximal step is solved by a safeguarded Newton or bisection method whose complexity is linear in the number of spatial degrees of freedom. Stability of the resulting time-stepping scheme under a mild CFL-type restriction derived from the proximal mapping will likewise be stated. These additions will make the general applicability explicit while retaining the validated special cases as illustrative examples. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on standard proximal-operator theory applied to external PDE models

full rationale

The paper computes explicit proximal-operator formulas for general costs and applies them to discretize doubly nonlinear gradient flows. These formulas are derived directly from the given cost functions without re-using the target PDE solution or fitted parameters as inputs. Validation consists of qualitative reproduction of known behaviors on textbook cases, which constitutes external checking rather than self-referential fitting. No self-definitional equations, no prediction of fitted quantities, and no load-bearing self-citations appear in the derivation chain. The scheme is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard existence theory for proximal operators and JKO schemes without introducing new free parameters, ad-hoc axioms, or invented entities.

axioms (1)
  • domain assumption Existence and uniqueness of proximal operators for the chosen family of costs and energies.
    Implicit in the variational formulation of the gradient flows.

pith-pipeline@v0.9.0 · 5402 in / 1109 out tokens · 24722 ms · 2026-05-10T01:20:18.366485+00:00 · methodology

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