Pith. sign in

REVIEW 3 major objections 3 minor 34 references

Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For superlinear SPDEs with multiplicative noise, Galerkin tamed Euler achieves weak error τ^ρ+λ_N^{-(ρ+γ/2)} uniformly in time, and the same rate controls the invariant-measure gap.

desk verdict First uniform-in-time weak and ergodic error rates for superlinear SPDEs with multiplicative noise; the proof is detailed and plausible, but two load-bearing ingredients are deferred to unpublished or unproved sources. read the letter →

arxiv 2607.19250 v2 pith:EJMZPJQL submitted 2026-07-21 math.NA cs.NA

Jingjing Cai , Zhihui Liu , Xiaoming Wu This is my paper · ORCID
classification math.NAcs.NA MSC 60H3560H1565M60
keywords superlinearSPDEmultiplicativenoiseuniform-in-timeweakconvergencetamedEulermethodGalerkindiscretizationMalliavincalculusbackwardKolmogorovequationergodicerrorestimate
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

This paper proves that a simple fully discrete scheme for superlinear SPDEs driven by multiplicative noise—spectral Galerkin in space, linearly implicit Euler in time, with a tamed nonlinearity—approximates expectations of smooth observables with an error of order τ^ρ + λ_N^{-(ρ+γ/2)} uniformly over all time steps, where ρ can be any number below 1. The result is the first uniform-in-time weak convergence estimate for an equation class that previously forced a choice between superlinear drift (only additive noise) and multiplicative noise (only globally Lipschitz drift). A corollary transfers the same rate to the distance between the exact and the numerical invariant measures, so long-time statistical averages computed from the scheme inherit the bound. The rate is essentially sharp: when the spatial regularity parameter γ is zero, the weak order doubles the strong order of the same scheme.

What carries the argument

The central object is the GTEM scheme X_{j+1}^N = X_j^N + τ A_N X_{j+1}^N + τ P_N F_τ(X_j^N) + P_N G(X_j^N) δ_j W, which is linear-implicit only in the Laplacian and therefore cheap to implement. The proof rides on three components: (1) uniform-in-time moment, Hölder, and Malliavin estimates for the Galerkin solution, the scheme, and its continuous interpolation; (2) uniform-in-time regularity estimates for the first three derivatives of the backward Kolmogorov equation solution U^M(t,x)=Eφ(X^M(t,x)); (3) a weak-error decomposition via Itô's formula, in which the singular stochastic-history terms are controlled by a Malliavin integration-by-parts formula and exponentially weighted discrete c

What would settle it

Take the 2D stochastic Allen–Cahn equation with multiplicative noise as in the paper's experiments and measure the temporal weak error at T=1,10,50 for τ from 2^{-6} down to 2^{-10}, using coupled coarse and fine reference paths. If the slopes fall below 0.8 at any T (or degrade as T grows) when λ1 > K3+Λw, the claimed uniform-in-time rate fails; conversely, if the slopes stay near 1 when λ1 ≤ K3+Λw, the proposed condition is over-pessimistic and only the 'sufficient' status of the theorem is in question.

Watch

Extended reading notes

Core claim

Theorem 2.1 states that under monotonicity and polynomial-growth assumptions on the drift and Lipschitz assumptions on the diffusion, together with the explicit dissipativity condition λ1 > K3 + Λw, the Galerkin tamed Euler method (GTEM) satisfies |E[φ(X(t_m)) − φ(X_m^N)]| ≤ C(1 + ‖X0‖^{...})(1 + t_m^{-ρ})(τ^ρ + λ_N^{-(ρ+γ/2)}) for every ρ ∈ (0,1) and φ ∈ C_b^3(H), uniformly in m, N, and τ. The same rate bounds the gap between the exact and numerical invariant measures, |∫_H φ dπ − ∫_{V_N} φ dπ^N_τ|, in Corollary 2.1. The temporal weak order is essentially 1 and the spatial weak order essentially 1+γ/2, which doubles the strong orders known for the same scheme when γ=0.

Load-bearing premise

Everything hinges on the sufficient dissipativity condition λ1 > K3 + Λw, which demands the first Dirichlet eigenvalue of the domain exceed a constant built from the drift and diffusion coefficients; without it, the uniform moment estimates, exponential mixing, and the error decomposition are not established.

Editorial extensions

If this is right

  • First uniform-in-time weak convergence rate, together with an ergodic error estimate, for a numerical method applied to SPDEs that are simultaneously superlinear in drift and multiplicative in noise.
  • The weak order is essentially 1 in time and 1+γ/2 in space; when γ=0 this doubles the strong order, indicating the error is dominated by the diffusion approximation.
  • Ergodic averages computed with GTEM inherit the same rate for C_b^3 observables, so long-time statistical simulation of stochastic Allen–Cahn type equations is quantitatively reliable.
  • The proof produces a reusable template: Malliavin integration by parts plus BKE regularity yields uniform-in-time weak estimates, a template that may extend to nonlinearity-explicit tamed schemes beyond the specific GTEM.

Reading between the lines

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

  • The paper's own threshold Λw is a coarse unified number across dimensions; in a fixed dimension with smaller noise constants, a sharper dissipativity condition would hold, so the method should apply to domains with smaller λ1 than the theorem requires.
  • The doubling of the strong order suggests that the weak error is controlled by the state-dependent diffusion, not the superlinear drift; testing higher-order strong integrators for the diffusion term might push the temporal weak order beyond 1.
  • The Malliavin-IBP template likely carries to other nonlinearity-explicit schemes, for example tamed exponential Euler or accelerated exponential Euler, giving uniform-in-time weak rates for the same equation class without the linear-implicit solve.
  • The t_m^{-ρ} singularity at small times is an artifact of the BKE estimates; a separate short-time argument might remove it for m≥1, yielding a fully uniform bound.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The manuscript analyzes a nonlinearity-explicit Galerkin tamed Euler method (GTEM) for semilinear parabolic SPDEs with superlinear drift and multiplicative noise. Under a sufficient dissipativity condition λ1 > K3 + Λw, it claims a uniform-in-time weak error bound |E[φ(X(t_m)) − φ(X_m^N)]| ≤ C(1 + t_m^{-ρ})(τ^ρ + λ_N^{-(ρ+γ/2)}) for every ρ∈(0,1) and bounded three-times differentiable test functions, with constants independent of m, N, and τ. Corollary 2.1 transfers this rate to the error between the exact and numerical invariant measures. The proof combines UIT moment, Hölder, and Malliavin estimates for the Galerkin and fully discrete solutions, regularity estimates for the backward Kolmogorov equation, a weak-error decomposition, and Malliavin integration by parts. Numerical experiments for the stochastic Allen–Cahn equation illustrate the temporal rates and ergodicity.

Significance. If correct, Theorem 2.1 appears to be the first uniform-in-time weak convergence rate for a numerical scheme applied to an SPDE with both superlinear drift and multiplicative noise, and the ergodic error estimate is a natural and valuable byproduct. The manuscript contains a substantial technical apparatus: detailed multi-step proofs with explicit constants, a careful treatment of short-time singularities and exponential decay, and separate appendices for the auxiliary smoothing, convolution, and one-step estimates. These are genuine strengths. The main caveat is that two load-bearing ingredients are imported from same-group or unpublished sources — Lemma 3.1 and [29, Theorem 4.1] — which limits the verifiability of the central claim from the text alone.

major comments (3)
  1. [Section 3.1, Lemma 3.1] Lemma 3.1 supplies the UIT moment bounds for the spectral Galerkin solution and is used throughout Section 4, e.g., in (4.1) and in the estimate of the initial-condition term. It is stated without proof, with the text saying 'we omit the details... analogous to [26, Proposition 2.1] and [28, Proposition 3.1]'. Those results are for the exact equation, not the Galerkin approximation, and [26], [28] are same-group papers/preprints. The reader cannot check whether the eigenvalue conditions stated in Lemma 3.1 — λ1 > K3 + (p−1)/2 K6 for γ=0 and λ1 > K3 + (p(q+1)−1)/2 K6 for γ∈(0,1) — are compatible with the global condition λ1 > K3 + Λw in Theorem 2.1 for the high moments Ψ_t^(j) needed in Lemmas 4.2–4.4. Please include a complete proof or a precise statement of the external result together with its full assumptions.
  2. [Section 4, proof of Theorem 2.1] The reduction from the exact solution to the spectral Galerkin solution uses the bound |Eφ(X(t_m))−Eφ(X^M(t_m))| ≤ C(t_m,X0) λ_M^{-(1+γ)/2}, attributed to [29, Theorem 4.1]. This bound is load-bearing for the spatial rate in (2.32). However, [29] is an unpublished preprint (arXiv:2502.19117) and the theorem is not stated. The paper does not verify that the hypotheses of [29, Theorem 4.1] — e.g., additive versus multiplicative noise, monotonicity constants, and spatial regularity — coincide with Assumptions 2.1–2.2 and the Galerkin equation (2.27). If the constants or the rate exponent differ, Theorem 2.1's spatial order does not follow. Please include the full statement of [29, Theorem 4.1] or provide a self-contained proof in this setting.
  3. [Section 4, Lemmas 4.2–4.4 and Section 2, Remark 2.1] The weak-error lemmas rely on (2.12) from [11, Lemma 3.2], which is again a same-group preprint (arXiv:2606.09173), and on the detailed Malliavin/BKE estimates with several coupled small parameters. In particular, Lemma 3.3 and Corollary 3.1 impose constraints such as ε1+ε2+κ_{d,ϱ,χ}+δ < 1/2, and Section 4 requires a choice of ϱ∈(1/4,1/2), χ, ε, δ with ρ < 1/2+ϱ and ε < 1/2+ϱ−ρ. While these inequalities are consistent for every ρ<1, the margins shrink as ρ→1 and the proof of Theorem 2.1 only says 'Combining Lemmas 4.1–4.4'. Please provide an explicit global ordering of all parameters (ϱ, χ, ε, δ, α) to confirm that the final constants are finite and independent of m,N,τ.
minor comments (3)
  1. [Section 4] The phrase 'Let ρ∈(0,1) be arbitrary' is repeated in the opening of Section 4 and in the proof of Theorem 2.1. More importantly, the parameter choice that makes the local estimates (Lemmas 4.1–4.4) yield exactly the rate τ^ρ + λ_N^{-(ρ+γ/2)} is not written down explicitly. Add a short 'choice of parameters' paragraph.
  2. [Section 5] The text says 'fitted slopes are compared with the rate τ^ρ' but does not report the fitted slopes or any error bars. Reporting the numerical slopes, e.g., for each T and observable, would make the sharpness claim quantitative and easier to assess.
  3. [References] Several load-bearing references are same-group arXiv preprints ([11], [22], [29], [30]). Please update publication statuses where available, or mark them as 'to appear' / 'submitted' so the reader can assess the dependency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; Theorem 2.1 is derived from new BKE/Malliavin estimates, with prior same-author results used only as external lemmas.

full rationale

The derivation chain of Theorem 2.1 is not circular. The UIT weak error is obtained from a weak error decomposition (Section 4) using (i) UIT moment, Hölder and Malliavin estimates for the Galerkin and tamed schemes (Section 3), (ii) BKE derivative estimates (Lemma 3.3, Corollary 3.1) proven in the paper, and (iii) local error estimates (Lemmas 4.1–4.4). These estimates are independent of the claimed convergence rate; the rate follows by summing the local errors. The paper explicitly defers Lemma 3.1 to [26, Prop 2.1] and [28, Prop 3.1], and uses [29, Thm 4.1] for the exact-solution Galerkin projection; these are prior results (some same-author) with stated assumptions, not restatements of Theorem 2.1, so they are external evidence rather than circular inputs. The dissipativity condition λ1 > K3 + Λ_w is an assumption, not a fitted output. Numerical experiments are illustrative and are not used to define the rate. Therefore no step reduces to its input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The theorem is conditional on standard monotone-coefficient assumptions and an explicit dissipativity inequality. No hidden fitted parameters enter the main proof; the constants in the numerical reference lines are not part of the central claim. The tamed drift f_τ is a known construction inherited from [29].

assumptions (6)
  • domain assumption Assumption 2.1: f is C^3, monotone, polynomially growing with the specific coercivity and taming inequalities (2.2)-(2.4).
    This defines the class of superlinear drifts the theorem applies to; it is not derived.
  • domain assumption Assumption 2.2(1): G is C^3 with bounded derivatives up to third order and Lipschitz conditions (2.20)-(2.23).
    The Malliavin IBP terms and third BKE derivative estimates require these bounded derivatives.
  • domain assumption Assumption 2.2(2): G: H^{1+γ} → L_2^{1+γ} grows linearly for γ satisfying (2.6).
    Used in the spatial error splitting and in (4.15) to control the diffusion defect with the spatial regularity γ.
  • domain assumption Dissipativity condition λ1 > K3 + Λw before Theorem 2.1.
    Sufficient condition for UIT moment bounds, exponential mixing, and the weak-error estimates; not derived from the equation.
  • standard math Sobolev embeddings (2.5)-(2.6), including the restriction γ=0 in d=1, 0<γ in d=2, γ>1/2 in d=3.
    Standard Sobolev embedding; the dimension-dependent γ restriction is part of the problem data.
  • domain assumption Existence, uniqueness, and exponential ergodicity of exact and numerical invariant measures, imported from [26, Theorem 2.1] and [11, Lemma 5.1, Theorem 5.1].
    Corollary 2.1 needs invariant measures; the paper does not prove these ergodicity results independently.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise." pith.science (2026). https://pith.science/paper/EJMZPJQL

@misc{pith2026260719250,
  author       = {Pith},
  title        = {Pith review of: Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EJMZPJQL}},
  note         = {Machine review of arXiv:2607.19250}
}
abstract

For a class of superlinear SPDEs driven by multiplicative noise, we prove an (essentially) sharp uniform-in-time (UIT) weak convergence rate for the nonlinearity-explicit Galerkin tamed Euler method (GTEM). Under standard monotonicity assumptions, the proof combines Malliavin calculus with regularity theory for the associated backward Kolmogorov equation (BKE), leading to UIT moment, H\"older, and Malliavin estimates, along with regularity estimates for the BKE solution. These estimates, together with a weak error decomposition and Malliavin integration by parts (IBP) formula, then yield a UIT weak convergence rate $\tau^\rho+\lambda_N^{-(\rho+\gamma/2)}$ for any $\rho \in (0,1)$, where $\gamma\in[0,1)$ quantifies the assumed spatial Sobolev regularity. Consequently, we obtain a sharp ergodic error estimate between the exact and numerical invariant measures. Numerical experiments support the theory.

Figures

Figures reproduced from arXiv: 2607.19250 by the authors.

Figure 1
Figure 1. Temporal weak errors. For fixed signs ηk,ℓ ∈ {−1, 1}, M ∈ N+, and α, p > 0, set u M α,p = X M k,ℓ=1 ηk,ℓek,ℓ (k 2 + ℓ 2) α logp (e + √ k 2 + ℓ 2) , u¯ M α,p = uM α,p ∥uMα,p∥ , and take X0,time = 0.8e1,1 + ¯u 64 1.05,1.2 . We fix N = 32, NQ = 64, T = {1, 10, 50}, τref = 2−12. The tested time steps are τ ∈ {2 −6 , 2 −7 , 2 −8 , 2 −9}. The coarse and ref￾erence paths use the same truncated Wiener path, with coarse incr… view at source ↗
Figure 2
Figure 2. Ergodicity test. Appendix Appendix A. Proof of the estimates in Remark 2.2. The estimates (2.13)– (2.15) follow directly from [29, Lemma 4.2]. By Assumption 2.1 and the definition (2.10) of fτ , one has |fτ (ξ)| ≤ C(1+|ξ| q+1) and |f ′ τ (ξ)| ≤ C(1+|ξ| q ), ξ ∈ R. Hence, the chain rule yields (2.16) and (2.17). It remains to prove (2.18) and (2.19). By the duality between H˙ −2ϱ and H˙ 2ϱ , it is enough to estimate … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 3 linked inside Pith

  1. [26]

    Z. Liu. Numerical ergodicity and uniform estimate of monotone SPDEs driven by multiplica- tive noise.J. Comput. Math., 44(1):84–102, 2026

    2026

  2. [28]

    Liu and Z

    Z. Liu and Z. Qiao. Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise.Stoch. Partial Differ. Equ. Anal. Comput., 9(3):559–602, 2021

    2021

  3. [29]

    Liu and J

    Z. Liu and J. Shen. Geometric ergodicity and optimal error estimates for a class of novel tamed schemes to super-linear stochastic PDEs.arXiv: 2502.19117, 2025

    arXiv 2025

  4. [1]

    Andersson and S

    A. Andersson and S. Larsson. Weak convergence for a spatial approximation of the nonlinear stochastic heat equation.Math. Comp., 85(299):1335–1358, 2016

    2016

  5. [2]

    Anton, D

    R. Anton, D. Cohen, and L. Quer-Sardanyons. A fully discrete approximation of the one- dimensional stochastic heat equation.IMA J. Numer. Anal., 40(1):247–284, 2020

    2020

  6. [3]

    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.Stoch. Partial Differ. Equ. Anal. Comput., 11(1):211–268, 2023

    2023

  7. [4]

    Becker and A

    S. Becker and A. Jentzen. Strong convergence rates for nonlinearity-truncated Euler-type ap- proximations of stochastic Ginzburg-Landau equations.Stochastic Process. Appl., 129(1):28– 69, 2019

    2019

  8. [5]

    Br´ ehier

    C.-E. Br´ ehier. Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme.ESAIM Math. Model. Numer. Anal., 56(1):151– 175, 2022

    2022

Show all 34 references
  1. [6]

    Br´ ehier

    C.-E. Br´ ehier. Total variation error bounds for the accelerated exponential Euler scheme ap- proximation of parabolic semilinear SPDEs.SIAM J. Numer. Anal., 62(3):1171–1190, 2024

    2024

  2. [7]

    Br´ ehier

    C.-E. Br´ ehier. Analysis of a modified regularity-preserving Euler scheme for parabolic semi- linear SPDEs: total variation error bounds for the numerical approximation of the invariant distribution.Found. Comput. Math., 25(2):511–586, 2025

    2025

  3. [8]

    Br´ ehier, J

    C.-E. Br´ ehier, J. Cui, and J. Hong. Strong convergence rates of semidiscrete splitting approx- imations for the stochastic Allen-Cahn equation.IMA J. Numer. Anal., 39(4):2096–2134, 2019

  4. [9]

    Br´ ehier and A

    C.-E. Br´ ehier and A. Debussche. Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient.J. Math. Pures Appl. (9), 119:193–254, 2018

    2018

  5. [10]

    Breit and A

    D. Breit and A. Prohl. Weak error analysis for the stochastic Allen-Cahn equation.Stoch. Partial Differ. Equ. Anal. Comput., 12(4):2181–2245, 2024

    2024

  6. [11]

    Cai and Z

    J. Cai and Z. Liu. Uniform-in-time strong error estimates of tamed-FEM to superlinear SPDEs driven by multiplicative noise.arXiv:2606.09173, 2026

    2026 arXiv

  7. [12]

    Cerrai.Second order PDE’s in finite and infinite dimension, volume 1762 ofLecture Notes in Mathematics

    S. Cerrai.Second order PDE’s in finite and infinite dimension, volume 1762 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 2001. A probabilistic approach

    2001

  8. [13]

    Cui and J

    J. Cui and J. Hong. Strong and weak convergence rates of a spatial approximation for stochas- tic partial differential equation with one-sided Lipschitz coefficient.SIAM J. Numer. Anal., 57(4):1815–1841, 2019

    2019

  9. [14]

    J. Cui, J. Hong, and L. Sun. Weak convergence and invariant measure of a full discretiza- tion for parabolic SPDEs with non-globally Lipschitz coefficients.Stochastic Process. Appl., 134:55–93, 2021

    2021

  10. [15]

    Da Prato and J

    G. Da Prato and J. Zabczyk.Ergodicity for infinite-dimensional systems, volume 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1996

    1996

  11. [16]

    Da Prato and J

    G. Da Prato and J. Zabczyk.Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2014

    2014

  12. [17]

    Debussche

    A. Debussche. Weak approximation of stochastic partial differential equations: the nonlinear case.Math. Comp., 80(273):89–117, 2011

    2011

  13. [18]

    Debussche and J

    A. Debussche and J. Printems. Weak order for the discretization of the stochastic heat equa- tion.Math. Comp., 78(266):845–863, 2009

    2009

  14. [19]

    X. Feng, Y. Li, and Y. Zhang. Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise.SIAM J. Numer. Anal., 55(1):194–216, 2017

    2017

  15. [20]

    Huang and J

    C. Huang and J. Shen. Stability and convergence analysis of a fully discrete semi- implicit scheme for stochastic Allen-Cahn equations with multiplicative noise.Math. Comp., 92(344):2685–2713, 2023

    2023

  16. [21]

    Jentzen and P

    A. Jentzen and P. Puˇ snik. Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinear- ities.IMA J. Numer. Anal., 40(2):1005–1050, 2020

    2020

  17. [22]

    Jiang and X

    Y. Jiang and X. Wang. Uniform-in-time weak error estimates of explicit full-discretization schemes for SPDEs with non-globally Lipschitz coefficients.arXiv: 2504.21364, 2025. UNIFORM WEAK AND ERGODIC ERROR ESTIMATES OF SUPERLINEAR SPDES 35

    2025 arXiv

  18. [23]

    Liu and Z

    H. Liu and Z. Liu. Inversions of stochastic processes from their ergodic measures of nonlinear SDEs.J. Inverse Ill-Posed Probl., 34(2):321–332, 2026

    2026

  19. [24]

    Liu.L p-convergence rate of backward Euler schemes for monotone SDEs.BIT, 62(4):1573– 1590, 2022

    Z. Liu.L p-convergence rate of backward Euler schemes for monotone SDEs.BIT, 62(4):1573– 1590, 2022

    2022

  20. [25]

    Z. Liu. Numerical ergodicity of stochastic Allen-Cahn equation driven by multiplicative white noise.Commun. Math. Res., 41(1):30–44, 2025

    2025

  21. [27]

    Liu and Z

    Z. Liu and Z. Qiao. Strong approximation of monotone stochastic partial differential equations driven by white noise.IMA J. Numer. Anal., 40(2):1074–1093, 2020

    2020

  22. [30]

    Qi and X

    R. Qi and X. Wang. Long-time error analysis of finite element fully discrete schemes for SPDEs with non-globally Lipschitz coefficients.arXiv:2603.23167, 2026

    2026

  23. [31]

    Wang and X

    M. Wang and X. Wang. A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients.IMA J. Numer. Anal., 45(1):516–579, 2025

    2025

  24. [32]

    X. Wang. An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation.Stochastic Process. Appl., 130(10):6271–6299, 2020

    2020

  25. [33]

    Wang and W

    Y. Wang and W. Cao. Approximation of the invariant measure for the stochastic Allen-Cahn equation via an explicit fully discrete scheme.BIT, 66(2):Paper No. 29, 32, 2026

    2026

  26. [34]

    Zhang, Y

    M. Zhang, Y. Zou, R. Zhang, and Y. Cao. Weak convergence of drift-implicit Euler and spectral Galerkin approximation to stochastic Allen-Cahn equation driven by multiplicative trace-class noise.J. Sci. Comput., 107(1):Paper No. 33, 47, 2026. Department of Mathematics, Southern...

    2026

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.