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arxiv: 2606.27232 · v1 · pith:RM42FTQ3 · submitted 2026-06-25 · math.NA · cs.NA· math.FA

Quasi-Feynman formulas that provide fast converging Chernoff approximations to solution of parabolic differential equation on the real line

Ivan D. Remizov , Alexandr V. Vedenin This is my paper

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classification math.NA cs.NAmath.FA
keywords Chernoff product formulaquasi-Feynman formulasparabolic PDEquadratic convergenceRiemann integralssemigroup approximationsvariable coefficientsuniform convergence
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The pith

Chernoff approximations using quasi-Feynman formulas converge uniformly at quadratic rate for parabolic PDEs on the real line.

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

The paper constructs explicit approximations to solutions of second-order parabolic PDEs with variable coefficients on the real line by applying Chernoff's product formula to a newly defined operator-valued function expressed through Riemann integrals. These constructions produce a class of quasi-Feynman formulas that avoid grids and Galerkin projections while relying on semigroup theory and bounded integrals. Under the assumption of sufficient smoothness on the initial data and coefficients, the approximations are proved to converge uniformly in space and time at a quadratic rate rather than the conventional first-order rate. The work supplies both refined error estimates and an analytical tool that can be implemented directly in numerical practice. Symbolic computations are used to confirm the theoretical rates.

Core claim

The central claim is that a new operator-valued function defined by proper Riemann integrals over a bounded interval, when inserted into Chernoff's product formula, generates quasi-Feynman formulas whose iterates converge uniformly to the PDE solution at quadratic order in the time step whenever the data and coefficients are sufficiently smooth.

What carries the argument

A new operator-valued function defined through proper Riemann integrals over a bounded interval, inserted into Chernoff's product formula to generate the sequence of approximations.

If this is right

  • The approximations can be evaluated by repeated application of bounded integrals without spatial discretization.
  • The error bound improves from order one to order two in the time step under the stated smoothness conditions.
  • The same construction supplies an explicit analytic representation that is neither grid-based nor projection-based.
  • The method extends the range of Chernoff formulas that remain practical for variable-coefficient parabolic problems on unbounded domains.

Where Pith is reading between the lines

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

  • The integral construction might be adapted to higher-dimensional domains by replacing the one-dimensional Riemann integral with a suitable multi-dimensional analogue.
  • Quadratic convergence would reduce the number of product-formula steps needed to reach a target accuracy compared with standard first-order Chernoff schemes.
  • The approach could be tested on related evolution equations whose generators admit similar bounded-integral representations.

Load-bearing premise

The initial data and coefficients are sufficiently smooth.

What would settle it

A concrete counter-example or numerical run in which the observed uniform error for a smooth initial datum and coefficient decays only linearly with the time step would falsify the quadratic-rate claim.

Figures

Figures reproduced from arXiv: 2606.27232 by Alexandr V. Vedenin, Ivan D. Remizov.

Figure 1
Figure 1. Figure 1: Verification of the Chernoff function from Theorem 5 using Wolfram Mathematica 13.3. [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Verification of the Chernoff function from Theorem 5 using Wolfram Mathematica 13.3. [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Verification of the Chernoff function from Theorem 5 using Wolfram Mathematica 13.3. [PITH_FULL_IMAGE:figures/full_fig_p035_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Verification of the Chernoff function from Theorem 5 using Wolfram Mathematica 13.3. [PITH_FULL_IMAGE:figures/full_fig_p036_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Verification of the Chernoff function from Theorem 5 using Wolfram Mathematica 13.3. [PITH_FULL_IMAGE:figures/full_fig_p037_5.png] view at source ↗
read the original abstract

We construct explicit approximations to the solution of a second-order parabolic partial differential equation on the real line with variable coefficients. The method is based on Chernoff's product formula and uses a new operator-valued function defined through proper Riemann integrals over a bounded interval, which makes the approach readily usable in numerical practice. For sufficiently smooth initial data and coefficients, we prove that the resulting Chernoff approximations converge uniformly in space and time with a quadratic rate, improving the standard first-order estimate. The construction yields a new class of quasi-Feynman formulas that are neither grid-based nor Galerkin-type, but instead rely on semigroup theory and multiple bounded integrals. The theoretical findings are validated by symbolic computation, and the paper contributes both refined error bounds and a practical analytical tool for parabolic problems.

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

0 major / 2 minor

Summary. The manuscript constructs explicit approximations to solutions of second-order parabolic PDEs on the real line with variable coefficients via Chernoff's product formula. It introduces a new operator-valued function defined through proper Riemann integrals over a bounded interval and proves that, for sufficiently smooth initial data and coefficients, the resulting approximations converge uniformly in space and time at a quadratic rate (improving the standard first-order Chernoff estimate). The construction yields a class of quasi-Feynman formulas relying on semigroup theory and multiple bounded integrals; results are validated by symbolic computation.

Significance. If the quadratic convergence holds, the work supplies refined error bounds and a practical, non-grid, non-Galerkin analytical tool for parabolic problems. The explicit use of bounded Riemann integrals and the validation by symbolic computation are concrete strengths that enhance usability and verifiability.

minor comments (2)
  1. [Abstract] The abstract states that the approximations 'converge uniformly in space and time with a quadratic rate' but does not indicate the precise function spaces or the dependence of the constant on the smoothness parameters; a brief clarification in the introduction or §2 would help readers locate the exact statement of the main theorem.
  2. [§3] The description of the new operator-valued function via 'proper Riemann integrals over a bounded interval' is central to the construction; ensure that the definition (likely in §3) explicitly states the integrability conditions on the coefficients so that the operator is well-defined for the claimed class of variable-coefficient problems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation rests on Chernoff's product formula (standard semigroup theory) to construct explicit operator approximations via Riemann integrals, then proves uniform quadratic convergence under an explicit smoothness hypothesis on initial data and coefficients. No equation reduces to its own input by definition, no parameter is fitted then relabeled as a prediction, and no load-bearing step depends on a self-citation chain. The result is a standard error analysis that remains independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Ledger constructed from abstract only; no explicit free parameters, axioms, or invented entities are quantified in the provided text.

axioms (1)
  • domain assumption Chernoff's product formula applies to the semigroup generated by the second-order differential operator with variable coefficients
    Stated as the foundation for constructing the approximations.
invented entities (1)
  • new operator-valued function defined through proper Riemann integrals over a bounded interval no independent evidence
    purpose: To generate the quasi-Feynman formulas that yield the quadratic convergence
    Introduced in the abstract as the key technical device

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discussion (0)

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Reference graph

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