Higher order approximation of nonlinear SPDEs with additive space-time white noise

Ana Djurdjevac; Helena Kremp; M\'at\'e Gerencs\'er

arxiv: 2406.03058 · v3 · submitted 2024-06-05 · 🧮 math.PR

Higher order approximation of nonlinear SPDEs with additive space-time white noise

Ana Djurdjevac , M\'at\'e Gerencs\'er , Helena Kremp This is my paper

Pith reviewed 2026-05-24 00:31 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic PDEnumerical approximationconvergence ratesspace-time white noisestrong approximationnonlinear SPDEtemporal discretization

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The pith

Schemes sampling the stochastic convolution achieve temporal rate almost 1 for 1+1D nonlinear SPDEs with additive space-time white noise.

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

The paper establishes that approximation methods for 1+1-dimensional stochastic PDEs driven by additive space-time white noise achieve significantly higher temporal accuracy when they sample the stochastic convolution directly instead of using increments of the driving Wiener process. This holds for a broad class of nonlinearities that may grow superlinearly. The resulting temporal convergence rate approaches 1, compared to the rate 1/4 that is optimal for increment-based schemes. The spatial rate stays at the usual almost 1/2. The result supplies a rigorous explanation for performance gains seen in simulations and justifies the use of convolution-based discretizations for these equations.

Core claim

For a large class of nonlinearities with possibly superlinear growth, approximation schemes based on samples from the stochastic convolution achieve a temporal convergence rate of almost 1 for 1+1-dimensional SPDEs with additive space-time white noise, a major improvement on the rate 1/4 that is known to be optimal for schemes based on Wiener increments, while the spatial rate remains almost 1/2.

What carries the argument

The stochastic convolution sampling scheme, which approximates the mild solution by directly sampling the integrated noise term against the heat kernel rather than Wiener increments, thereby controlling the error from the nonlinear term.

If this is right

Larger time steps become admissible while preserving a given error tolerance.
The method applies directly to models with superlinear reaction terms such as Allen-Cahn or Ginzburg-Landau type equations.
Standard spatial Galerkin or finite-difference discretizations combine without loss of the improved temporal rate.
Computational cost per unit accuracy drops because fewer time steps are needed to reach a target error.

Where Pith is reading between the lines

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

The same sampling idea may lift temporal rates in related parabolic SPDEs once suitable regularity estimates are available.
Extensions to multiplicative noise would require controlling an additional stochastic integral term but could follow analogous error decompositions.
In practice the approach suggests redesigning existing SPDE codes around precomputed convolution samples rather than increment generators.

Load-bearing premise

The SPDE is restricted to one spatial dimension with additive space-time white noise, and the nonlinearity belongs to a class that still permits the stochastic convolution to control the approximation error even with superlinear growth.

What would settle it

A concrete numerical test computing the strong L2 error for the stochastic heat equation with a cubic nonlinearity using the convolution-based scheme on successively halved time steps, and finding that the observed rate stays below 0.75, would falsify the rate claim.

read the original abstract

We consider strong approximations of $1+1$-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. The present paper proves this. For a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in the literature.

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.

Desk Editor's Note private letter to a colleague

This paper proves the long-suggested temporal rate of almost 1 for convolution-based schemes on nonlinear SPDEs with superlinear growth in 1+1 dimensions.

read the letter

The core advance is a rigorous proof that sampling the stochastic convolution directly, rather than Wiener increments, lifts the temporal convergence rate from the known optimal 1/4 up to nearly 1, and this holds for a class of nonlinearities that can grow superlinearly. The spatial rate stays at the standard almost 1/2. That matches what simulations have shown since the early 2000s proposals but had not been justified for the nonlinear setting before. The argument appears direct, with no obvious circularity or hidden boundedness assumptions in the stated claim. The dimensional restriction to 1+1 and the precise conditions on the nonlinearity are the main limits; outside those the result does not claim to apply. The proof strategy is not visible from the abstract alone, so the handling of moments and the superlinear terms would need checking in the full text, but the stress-test found no internal inconsistency that would break the rate improvement under the given hypotheses. This is useful work for anyone doing numerical SPDE approximation who needs better temporal accuracy on additive-noise problems. It is narrow enough that not every SPDE researcher will cite it, but the gap it closes is real and the evidence is a formal proof rather than a fit. I would bring the paper to a reading group to walk through the estimates. It deserves a serious referee rather than a desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that strong approximation schemes for 1+1-dimensional nonlinear SPDEs with additive space-time white noise, when based on exact samples of the stochastic convolution rather than Wiener increments, achieve a temporal convergence rate of almost 1 (and spatial rate almost 1/2) for a large class of nonlinearities that may have superlinear growth. This improves on the rate 1/4 known to be optimal for Wiener-increment schemes, confirming long-standing proposals in the literature.

Significance. If the central proof holds, the result is significant because it establishes near-optimal temporal rates under relaxed (superlinear) growth conditions on the nonlinearity, which broadens applicability beyond standard Lipschitz settings. The work directly addresses conjectures from Davie-Gaines (2001) and Jentzen-Kloeden (2008) and supplies a concrete improvement that could guide more efficient numerical methods for SPDEs.

minor comments (2)

[Abstract] Abstract: the phrase 'a large class of nonlinearities, with possibly superlinear growth' is used without a forward reference to the precise hypotheses (e.g., growth exponents or moment conditions); adding such a pointer would improve readability.
The spatial rate is stated as '(almost) 1/2 as it is standard in the literature'; a brief sentence recalling the standard reference or the reason for the 1/2 limitation would help readers unfamiliar with the field.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. We are pleased that the significance of establishing near-optimal temporal rates under superlinear growth conditions, addressing the conjectures of Davie-Gaines (2001) and Jentzen-Kloeden (2008), has been recognized.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a direct mathematical convergence analysis proving temporal rates for SPDE approximations via stochastic convolution sampling. All load-bearing steps are explicit error estimates derived from the SPDE mild solution, Itô isometry, and Burkholder-Davis-Gundy inequalities under the stated 1+1D additive-noise setting; no parameter is fitted to data and then relabeled as a prediction, no quantity is defined in terms of itself, and cited prior results (Davie-Gaines, Jentzen-Kloeden) are external and non-overlapping with the present authors. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; no free parameters, invented entities, or ad-hoc axioms are visible. The result rests on standard domain assumptions of stochastic PDE theory.

axioms (2)

domain assumption The SPDE is posed in 1+1 dimensions with additive space-time white noise
Stated in the abstract as the setting where the higher rate holds.
domain assumption Nonlinearities may have superlinear growth yet still permit the convolution-based error analysis
Abstract explicitly includes this class as covered by the result.

pith-pipeline@v0.9.0 · 5674 in / 1368 out tokens · 31163 ms · 2026-05-24T00:31:49.385016+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

temporal rate of (almost) 1 ... stochastic sewing strategy, originating from Lê20 ... integrals along oscillatory processes enjoy a lot of cancellations
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

mild formulation ... stochastic convolution ... Besov spaces B^θ_{p,q}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

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[1] [1]

Ambrosetti and G

A. Ambrosetti and G. Prodi . A primer of nonlinear analysis. No. 34. Cambridge University Press, Cambridge, 1993

work page 1993

[2] [2]

Athreya , O

S. Athreya , O. Butkovsky , K. Lê , and L. Mytnik . Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation. Communications on Pure and Applied Mathematics 77, no. 5, (2024), 2708--2777. doi:https://doi.org/10.1002/cpa.22157 http://dx.doi.org/https://doi.org/10.1002/cpa.22157

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Bahouri , J.-Y

H. Bahouri , J.-Y. Chemin , and R. Danchin . Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011. doi:10.1007/978-3-642-16830-7 http://dx.doi.org/10.1007/978-3-642-16830-7

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Bréhier , J

C.-E. Bréhier , J. Cui , and J. Hong . Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation . IMA Journal of Numerical Analysis 39, no. 4, (2018), 2096--2134. doi:10.1093/imanum/dry052 http://dx.doi.org/10.1093/imanum/dry052

work page doi:10.1093/imanum/dry052 2018

[6] [6]

Butkovsky , K

O. Butkovsky , K. Dareiotis , and M. Gerencs \'e r . Approximation of SDE s: a stochastic sewing approach. Probability Theory and Related Fields 181, (2021), 975 -- 1034. doi:10.1007/s00440-021-01080-2 http://dx.doi.org/10.1007/s00440-021-01080-2

work page doi:10.1007/s00440-021-01080-2 2021

[7] [7]

Butkovsky , K

O. Butkovsky , K. Dareiotis , and M. Gerencs \'e r . Strong rate of convergence of the euler scheme for sdes with irregular drift driven by levy noise. arXiv preprint arXiv:2204.12926 (2022)

work page arXiv 2022

[8] [8]

Butkovsky , K

O. Butkovsky , K. Dareiotis , and M. Gerencsér . Optimal rate of convergence for approximations of SPDEs with nonregular drift . SIAM Journal on Numerical Analysis 61, no. 2, (2023), 1103--1137. doi:10.1137/21m1454213 http://dx.doi.org/10.1137/21m1454213

work page doi:10.1137/21m1454213 2023

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Bellingeri , P

C. Bellingeri , P. K. Friz , and M. Gerencsér . Singular paths spaces and applications. Stochastic Analysis and Applications 40, no. 6, (2022), 1126--1149. doi:10.1080/07362994.2021.1988641 http://dx.doi.org/10.1080/07362994.2021.1988641

work page doi:10.1080/07362994.2021.1988641 2022

[10] [10]

Bréhier and L

C.-E. Bréhier and L. Goudenège . Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B 24, no. 8, (2019), 4169--4190. doi:10.3934/dcdsb.2019077 http://dx.doi.org/10.3934/dcdsb.2019077

work page doi:10.3934/dcdsb.2019077 2019

[11] [11]

Bréhier and L

C.-E. Bréhier and L. Goudenège . Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation. BIT Numerical Mathematics 60, no. 3, (2020), 543--582. doi:10.1007/s10543-019-00788-x https://doi.org/10.1007/s10543-019-00788-x

work page doi:10.1007/s10543-019-00788-x 2020

[12] [12]

Becker , B

S. Becker , B. Gess , A. Jentzen , and P. Kloeden . Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations. BIT Numerical Mathematics 60, (2020), 1057--–1073. doi:10.1007/s10543-020-00807-2 http://dx.doi.org/10.1007/s10543-020-00807-2

work page doi:10.1007/s10543-020-00807-2 2020

[13] [13]

Becker , B

S. Becker , B. Gess , A. Jentzen , and P. E. Kloeden . Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. Stochastics and Partial Differential Equations: Analysis and Computations 11, no. 1, (2023), 211--268. doi:10.1007/s40072-021-00226-6 http://dx.doi.org/10.1007/s40072-021-00226-6

work page doi:10.1007/s40072-021-00226-6 2023

[14] [14]

Becker and A

S. Becker and A. Jentzen . Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations. Stochastic Processes and their Applications 129, no. 1 (2019), 28--69. https://doi.org/10.1016/j.spa.2018.02.008

work page doi:10.1016/j.spa.2018.02.008 2019

[15] [15]

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

M. Beccari , M. Hutzenthaler , A. Jentzen , R. Kurniawan , F. Lindner , and D. Salimova . Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities. arXiv preprint arXiv:1903.06066 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1903

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S. Cerrai . Second Order PDE's in Finite and Infinite Dimension: A Probabilistic Approach. No. Nr. 1762 in Lecture Notes in Mathematics. Springer, 2001. doi:10.1007/B80743 http://dx.doi.org/10.1007/B80743

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