Pith. sign in

REVIEW 5 minor 51 references

Milstein-type Schemes for Hyperbolic SPDEs

T0 review · 0 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read Milstein schemes reach first-order convergence for hyperbolic SPDEs.

desk verdict This paper delivers the first rigorous rate-1 (up to log) pathwise-uniform error analysis for Milstein schemes in a hyperbolic SPDE setting, resolving a real open problem; the proof skeleton checks out, and the main caveats are the ones the authors themselves flag. read the letter →

arxiv 2512.19647 v4 pith:KCXQMEXP submitted 2025-12-22 math.NA cs.NAmath.APmath.FAmath.PR

Felix Kastner , Katharina Klioba This is my paper · ORCID
classification math.NAcs.NAmath.APmath.FAmath.PR MSC 60H1565C3065M1265M1535R60
keywords MilsteinschemehyperbolicSPDEcontractivesemigrouppathwiseuniformconvergencestrongratestochasticSchrödingerequationiteratedintegralsrationaltimediscretization
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 Milstein-type time discretizations, not just Euler-type methods, converge at up to first order for hyperbolic stochastic partial differential equations, where the leading operator generates a contractive but non-analytic semigroup. Previous rigorous results in this setting were stuck at rate 1/2 for schemes using only Wiener increments; the paper shows the extra iterated-integral correction in the Milstein scheme overcomes that barrier, with a pathwise-uniform L^p error of order h^α, up to a logarithmic factor for rational variants and with no logarithmic factor for the exponential variant. The proof covers stochastic Schrödinger, Maxwell, and transport equations, and numerical experiments for Schrödinger equations report rates close to 1. If correct, this resolves a decade-old open question about higher-order schemes for hyperbolic SPDEs and extends the Milstein theory beyond the parabolic case.

What carries the argument

The central object is the Milstein correction (G′G)(u_i)Δ²W_{i+1} — the iterated stochastic integral of the noise coefficient's derivative along the noise — added to each Euler step. The argument is carried by a two-space framework: the scheme is stable on a smoother subspace Y, while differentiability is used in a mixed way (directional differentiability on X, full differentiability from Y to X, and (2α−1)-order continuity of derivatives), which allows genuinely nonlinear Nemytskii-type operators to be treated. The rational-scheme analysis hinges on a logarithmic square-function estimate for discrete stochastic convolutions, which produces the √log(T/h) factor, and a discrete comparison lem

What would settle it

Simulate the linear stochastic Schrödinger setting of Section 7.1 with reference step size 2^{-16} and step sizes down to 2^{-12}; if the measured exponential-Milstein slope is below 0.9, the claimed order-one rate fails to show up numerically; if the rational-Milstein slope matches h√log(T/h) rather than h, the logarithmic factor is real.

Watch

Extended reading notes

Core claim

Under a set of regularity assumptions — a smoother subspace Y embedded in the state space X, Lipschitz drift and noise on X, linear growth and full differentiability on Y, and a (2α−1)-order continuity condition on the derivatives — the rational Milstein scheme converges at rate min{α,1} in pathwise uniform L^p norm, with the error bounded by a constant times h^α√log(T/h), while the exponential Milstein scheme achieves the same rate without the logarithmic factor. This is the first rigorous error analysis showing that Milstein schemes can break the 1/2-order barrier for hyperbolic SPDEs, improving on results for exponential Euler and rational schemes that use only Wiener increments.

Load-bearing premise

The result depends on the derivatives of the drift and noise being smooth enough on a smaller subspace Y — continuity of order 2α−1 — and on the combined noise sensitivity G′G being Lipschitz on the state space; without that, the Milstein correction cannot be controlled and only the old 1/2 rate is known.

Editorial extensions

If this is right

  • For hyperbolic SPDEs with contractive non-analytic semigroups, Milstein schemes achieve convergence rate min{α,1} in pathwise uniform L^p sense, improving the previously best-known 1/2 rate.
  • The exponential Milstein scheme attains rate 1 without any logarithmic factor when Y = D(A), matching the order of the Itô–Taylor expansion in the SDE case.
  • Rational A-stable schemes such as implicit Euler and Crank–Nicolson Milstein variants inherit the same rates, with Crank–Nicolson reaching 2/3 for Maxwell's equations under natural regularity assumptions.
  • The error estimates are pathwise uniform (max over time inside the expectation), not just pointwise root-mean-square bounds, and hold in every L^p moment for p in [2,∞).
  • For linear stochastic Schrödinger equations with sufficiently smooth potential and noise, the exponential Milstein scheme converges at rate 1, and numerical experiments confirm rates close to 1.

Reading between the lines

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

  • Editorial inference: The same two-space template should extend to locally Lipschitz nonlinearities via a stopping-time argument; the paper's own Section 7.3 simulation with a Nemytskii nonlinearity outside the assumptions already shows numerical rates near 1, suggesting the regularity conditions are sufficient rather than necessary.
  • Editorial inference: The logarithmic factor for rational schemes appears tied to the logarithmic square-function estimate; a sharper analysis of the discrete stochastic convolution might remove it for A-stable rational schemes as well, yielding a unified no-log theory.
  • Editorial inference: For stochastic wave equations, the authors note their approach does not improve on rate 3/2 obtained by a specialized scheme; combining the Milstein correction with the wave equation's extra smoothing is a natural next step to recover such higher rates within a unified framework.
  • Editorial inference: For non-commutative noise, exact simulation of the iterated integral is not currently known; combining the present error analysis with a derivative-free Milstein approximation would give a fully implementable first-order scheme for hyperbolic SPDEs.
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

0 major / 5 minor

Summary. The paper develops a temporal Milstein theory for semilinear hyperbolic SPDEs of the form dU + AU dt = F(U)dt + G(U)dW on a Hilbert space X, where -A generates a contractive, not necessarily analytic C0-semigroup. It introduces rational and exponential Milstein schemes, proves pathwise-uniform strong error bounds in L^p(Ω) of order min{α,1} up to a logarithmic factor for rational schemes and without the logarithmic factor for the exponential scheme, under a two-space Kato framework Y↪X with α-Hölder type regularity assumptions (Assumptions A1-A4). The central result, Theorem 5.3 and its exponential counterpart Theorem 5.7, is applied to stochastic Schrödinger, Maxwell, and transport equations, and numerical experiments for the stochastic Schrödinger equation confirm the predicted rates.

Significance. If the result holds, it resolves an open problem posed in [26, p. 324] and [31, Rem. 6.7] by giving the first rigorous convergence rates above 1/2 for Milstein schemes in the hyperbolic, non-analytic semigroup setting. The proof is substantial: it introduces two auxiliary processes, a three-term error split, stochastic Fubini, maximal inequalities for stochastic convolutions, a logarithmic square-function estimate, and a discrete Grönwall closure. The manuscript is unusually careful about constants, Gâteaux vs. Fréchet differentiability, and the distinction between pointwise and pathwise-uniform errors. It also provides publicly available code (DOI 10.5281/ZENODO.18229440) and openly discusses limitations, including the exact-simulation problem for iterated integrals in the non-commutative case and the linear-growth-on-Y restriction that excludes some Nemytskii nonlinearities. These caveats narrow the practical scope but do not contradict the internal mathematical claims.

minor comments (5)
  1. [Lemma 5.2] The assumption states g ∈ L∞(0,T;L^p(Ω;L_2^{(2)}(H,X))), but g is the linear noise value taking values in L_2(H,X), and the proof uses |||g|||∞,p,Y, i.e. the L_2(H,Y)-norm. The bilinear space L_2^{(2)} appears to be a typo; it should be L_2(H,X) (or L_2(H,Y) under the Y-invariance already assumed in A2(c)).
  2. [Section 7, last paragraph] The paper openly states that for non-commutative noise the exact iterated integrals cannot currently be simulated, so the implemented scheme in that case is not the analyzed scheme. Since this is a significant practical caveat, it should be stated more prominently in the introduction or abstract, not only in the numerical section.
  3. [Proposition 4.4] The application of the discrete Grönwall lemma is compressed. After φ(j) ≤ c + C(h∑φ(i)^2)^{1/2}, the identification β = C√h gives β²j ≤ C²T, which is what yields the (1+C²T)^{1/2} exp((1+C²T)/2) factor. Stating this identification explicitly would improve readability.
  4. [Theorem 1.1] The introduction states Y↪D(A^α) continuously, while Assumption A1 uses the real interpolation space D_A(α,∞) for α<1. The relationship should be clarified, since D(A^α) embeds into D_A(α,∞) but the two conditions are not identical in the statement of the theorem.
  5. [Section 7.1, Figure 1 and Table 2] The numerical rates are obtained from 100 samples and a reference solution at h=2^{-14}. Reporting confidence intervals or standard errors would strengthen the comparison against the theoretical rates, especially for the small step sizes where the reference error may bias the observed rates.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence proof derives the error bound from stated regularity and approximation assumptions without fitted parameters or definitional reductions.

full rationale

The derivation chain is self-contained and non-circular. Theorem 5.3 splits the error E(m) into E1 (solution regularity/Taylor expansion), E2 (difference v1-v2), and E3 (semigroup approximation), then closes the estimate with the discrete Grönwall inequality (Lemma 2.11). All constants are expressed in terms of the assumed Lipschitz/Hölder constants, the semigroup approximation constant C_alpha, the well-posedness/stability constants C_WP and C_stab, and the fixed L_p constants B_p. The rate alpha is a hypothesis parameter in Assumptions A1-A4 (scheme approximation order on Y and (2alpha-1)-Hölder regularity of F' and G'), not a fitted or inferred quantity; the theorem derives h^alpha from those inputs rather than defining them in terms of the target error. Lemma 5.6 uses the logarithmic square function estimate from [31, Prop. 2.3] and [14, Thm. 3.1], while Lemma 2.11 comes from [31, Lem. 2.7]; these are published, independently usable tools, and their self-citation status does not make the present conclusion equivalent to its assumptions. The exponential Milstein improvement (Theorem 5.7) follows structurally because E3 vanishes when R=S. The paper explicitly acknowledges limitations - e.g., Nemytskii nonlinearities on Y=H^1 prohibit optimal rates, and exact simulation of iterated integrals is currently known only under a commutativity condition - but these are implementation/scope gaps, not circular steps. No equation is defined in terms of the error being estimated, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters: all constants in Theorem 5.3 are derived from the stated Lipschitz, Hölder, embedding, and stability bounds; the rate α is a hypothesis (scheme order on Y), not fitted to data. No invented entities: the auxiliary processes v¹, v² (Def. 5.4) are proof devices, and the iterated stochastic integrals Δ²W are standard objects. The cost of the theorem is paid in assumptions — heavy regularity on derivatives and G′G — not in fitted quantities; the convergence rate follows from the assumptions rather than being tuned to match numerics.

assumptions (8)
  • standard math Maximal inequality for stochastic convolutions with quasi-contractive semigroups (Thm. 2.3)
    Stated with constants B_2=2, B_p=4√p, cited to [23] and [31, p. 2068]; used in Lemmas 5.1, 5.2, 5.5, 5.6 and throughout Theorem 5.3.
  • standard math Logarithmic square function estimate (Prop. 2.4) with constant K=4 exp(1+1/(2e))
    Cited to [46, Prop. 2.7], [31, Prop. 2.3], [14, Thm. 3.1]; source of the √log(T/h) factor for rational schemes in Lemma 5.6. One source ([31]) is co-authored by the present second author, but independent versions exist.
  • standard math Stochastic Fubini theorem [47, Thm. 2.2]
    Invoked at estimate (5.17) for T_{F,3,2} to exchange temporal and stochastic integrals; essential for pathwise-uniform L^p errors for general p.
  • domain assumption Semigroup difference decay ∥S(t)−S(s)∥_{L(Y,X)} ≤ 2C_Y(t−s)^α from Y ↪ D_A(α,∞) or Y ↪ D(A) (Lemma 3.1)
    The substitute for parabolic smoothing; the property that makes the hyperbolic setting tractable. Used in Lemmas 5.1–5.2 and estimates (5.13), (5.19)–(5.22).
  • domain assumption Contractivity of the scheme R on both X and Y, and of S on Y (Assumptions A1, A3)
    Needed for stability (Prop. 4.4) and Lemmas 5.5–5.6; satisfied by EXE, IE, CN and A-stable rational schemes via Prop. 2.10 under |r|≤1 on C⁻.
  • domain assumption (2α−1)-Hölder continuity of Gâteaux derivatives on Y and Lipschitz continuity of G′G on X (Assumptions A4(a),(b); A2(e))
    Load-bearing, Milstein-specific regularity enabling the second-order Taylor expansions (Prop. 3.7) used in (5.15), (5.18), (5.24), (5.25). Restrictive for Nemytskii noises; the paper's own Sec. 7.3 shows an excluded example.
  • standard math Well-posedness of (SEE) in X and Y (Thm. 4.2)
    Folklore well-posedness for globally Lipschitz nonlinearities, cited from [17, Thm. 7.5] and [31, Thm. 4.3–4.4]; stated for completeness.
  • standard math Discrete Grönwall inequality (Lemma 2.11)
    Cited from [31, Lem. 2.7] based on [36, Lem. A.3]; closes the error recursion E(m) ≤ C̃₁h + (C̃₂+C̃₃√log)h^α + C₄(hΣE(i)²)^{1/2} in Theorem 5.3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Milstein-type Schemes for Hyperbolic SPDEs." pith.science (2026). https://pith.science/paper/KCXQMEXP

@misc{pith2026251219647,
  author       = {Pith},
  title        = {Pith review of: Milstein-type Schemes for Hyperbolic SPDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCXQMEXP}},
  note         = {Machine review of arXiv:2512.19647}
}
abstract

This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a contractive, not necessarily analytic $C_0$-semigroup. Optimal convergence rates are derived for the pathwise uniform strong error \[ E_h^\infty := \Big(\mathbb{E}\Big[\max_{1\le j \le M}\|U_{t_j}-u_j\|_X^p\Big]\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\lesssim h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \lesssim h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs and from exponential to rational Milstein schemes. Moreover, root-mean-square error estimates are strengthened to pathwise uniform estimates. Numerical experiments validate the convergence rates for the stochastic Schr\"odinger equation. Further applications to Maxwell's and transport equations are included.

Figures

Figures reproduced from arXiv: 2512.19647 by the authors.

Figure 1
Figure 1. Numerical errors for the stochastic Schr¨odinger equation with (a) a potential (b) a nonlocal nonlinearity and (c) a Nemytskii-type nonlinearity. predictions. Here and in the subsequent simulations, we also state the numerical convergence rate if only the step sizes 2−5 to 2−7 are taken into account. It can be observed that these are higher. Both observations may be attributed to the fact that, for small step sizes,… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 23 canonical work pages

  1. [31]

    Klioba and M

    K. Klioba and M. Veraar. Pathwise uniform convergence of time discretization schemes for SPDEs.IMA J. Numer. Anal., pages 2060–2131, 2024.doi:10.1093/imanum/drae055

    work page doi:10.1093/imanum/drae055

  2. [1]

    Ambrosetti and G

    A. Ambrosetti and G. Prodi.A Primer of Nonlinear Analysis. Number 34 in Cambridge studies in advanced mathematics. Cambridge Univ. Press, Cambridge, 1. paperback edition, 1995

    1995

  3. [2]

    Anton and D

    R. Anton and D. Cohen. Exponential integrators for stochastic Schr¨ odinger equations driven by Itˆ o noise.J. Comput. Math., 36(2):276–309, 2018.doi:10.4208/jcm.1701-m2016-0525

    work page doi:10.4208/jcm.1701-m2016-0525 2018

  4. [3]

    Banjai, G

    L. Banjai, G. Lord, and J. Molla. Strong convergence of a Verlet integrator for the semilinear stochastic wave equation.SIAM J. Numer. Anal., 59(4):1976–2003, 2021.doi:10.1137/20M1364746

    work page doi:10.1137/20m1364746 1976

  5. [4]

    Barth and A

    A. Barth and A. Lang. Milstein approximation for advection-diffusion equations driven by multiplicative noncon- tinuous martingale noises.Appl. Math. Optim., 66(3):387–413, 2012.doi:10.1007/s00245-012-9176-y

    work page doi:10.1007/s00245-012-9176-y 2012

  6. [5]

    Barth and A

    A. Barth and A. Lang.L p and almost sure convergence of a Milstein scheme for stochastic partial differential equations.Stoch. Proc. Appl., 123(5):1563–1587, 2013.doi:10.1016/j.spa.2013.01.003

    work page doi:10.1016/j.spa.2013.01.003 2013

  7. [6]

    Bezanson, A

    J. Bezanson, A. Edelman, S. Karpinski, and V.B. Shah. Julia: A fresh approach to numerical computing.SIAM Rev., 59(1):65–98, 2017.doi:10.1137/141000671

    doi:10.1137/141000671 2017

  8. [7]

    Br´ ehier and D

    C.-E. Br´ ehier and D. Cohen. Analysis of a splitting scheme for a class of nonlinear stochastic Schr¨ odinger equa- tions.Appl. Numer. Math., 186:57–83, 2023.doi:10.1016/j.apnum.2023.01.002

    doi:10.1016/j.apnum.2023.01.002 2023

Show all 51 references
  1. [8]

    Brenner and V

    P. Brenner and V. Thom´ ee. On rational approximations of semigroups.SIAM J. Numer. Anal., 16(4):683–694, 1979

    1979

  2. [9]

    Cohen, J

    D. Cohen, J. Cui, J. Hong, and L. Sun. Exponential integrators for stochastic Maxwell’s equations driven by Itˆ o noise.J. Comput. Phys., 410:109382, 21, 2020.doi:10.1016/j.jcp.2020.109382

    2020

  3. [10]

    Cohen and A

    D. Cohen and A. Lang. Numerical approximation and simulation of the stochastic wave equation on the sphere. Calcolo, 59(3):Paper No. 32, 32, 2022.doi:10.1007/s10092-022-00472-7

    2022 doi

  4. [11]

    Cohen, S

    D. Cohen, S. Larsson, and M. Sigg. A trigonometric method for the linear stochastic wave equation.SIAM J. Numer. Anal., 51(1):204–222, 2013.doi:10.1137/12087030X. MILSTEIN-TYPE SCHEMES FOR HYPERBOLIC SPDES 38

    2013 doi

  5. [12]

    Cohen and L

    D. Cohen and L. Quer-Sardanyons. A fully discrete approximation of the one-dimensional stochastic wave equa- tion.IMA J. Numer. Anal., 36(1):400–420, 2016.doi:10.1093/imanum/drv006

    2016 doi

  6. [13]

    S. Cox, M. Hutzenthaler, A. Jentzen, J. van Neerven, and T. Welti. Convergence in H¨ older norms with applications to Monte Carlo methods in infinite dimensions.IMA J. Numer. Anal., 41(1):493–548, 04 2020.doi:10.1093/ imanum/drz063

    2020

  7. [14]

    Cox and J

    S. Cox and J. van Winden. Sharp supremum and H¨ older bounds for stochastic integrals indexed by a parameter, 2024.arXiv:2409.13615

    2024 arXiv

  8. [15]

    J. Cui. Explicit approximation for stochastic nonlinear schr¨ odinger equation.J. Differential Equations, 419:1–39, 2025.doi:10.1016/j.jde.2024.11.022

    2025 doi

  9. [16]

    Da Prato, A

    G. Da Prato, A. Jentzen, and M. R¨ ockner. A mild Itˆ o formula for SPDEs.Trans. Amer. Math. Soc., 372(6):3755– 3807, 2019.doi:10.1090/tran/7165

    2019 doi

  10. [17]

    Da Prato and J

    G. Da Prato and J. Zabczyk.Stochastic equations in infinite dimensions, volume 152 ofEncyclopedia of Math- ematics and its Applications. Cambridge University Press, Cambridge, second edition, 2014.doi:10.1017/ CBO9781107295513

    2014

  11. [18]

    Dahlquist

    G.G. Dahlquist. A special stability problem for linear multistep methods.Nordisk Tidskr. Informationsbehandling (BIT), 3:27–43, 1963.doi:10.1007/bf01963532

    1963 doi

  12. [19]

    Djurdjevac, M

    A. Djurdjevac, M. Gerencs´ er, and H. Kremp. Higher order approximation of nonlinear spdes with additive space- time white noise, 2024.arXiv:2406.03058

    2024 arXiv

  13. [20]

    Engel and R

    K.-J. Engel and R. Nagel.One-parameter semigroups for linear evolution equations, volume 194 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2000.doi:10.1007/b97696

    doi

  14. [21]

    Feng, A.A

    X. Feng, A.A. Panda, and A. Prohl. Higher order time discretization for the stochastic semilinear wave equation with multiplicative noise.IMA J. Numer. Anal., 44(2):836–885, 2024.doi:10.1093/imanum/drad024

    2024 doi

  15. [22]

    Gy¨ ongy and A

    I. Gy¨ ongy and A. Millet. Rate of convergence of space time approximations for stochastic evolution equations. Potential Anal., 30(1):29–64, 2009.doi:10.1007/s11118-008-9105-5

    2009 doi

  16. [23]

    Hausenblas and J

    E. Hausenblas and J. Seidler. Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability.Stoch. Anal. Appl., 26(1):98–119, 2008.doi:10.1080/07362990701673047

    2008 doi

  17. [24]

    J. Hong, B. Hou, and L. Sun. Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise.J. Comput. Phys., 451:Paper No. 110829, 20, 2022.doi:10.1016/j.jcp.2021.110829

    2022

  18. [25]

    Hyt¨ onen, J.M.A.M

    T.P. Hyt¨ onen, J.M.A.M. van Neerven, M.C. Veraar, and L. Weis.Analysis in Banach Spaces. Volume II. Proba- bilistic Methods and Operator Theory, volume 67 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer, 2017.doi:10.1007/978-3-319-69808-3

    2017 doi

  19. [26]

    Jentzen and M

    A. Jentzen and M. R¨ ockner. A Milstein scheme for SPDEs.Found. Comput. Math., 15(2):313–362, 2015.doi: 10.1007/s10208-015-9247-y

    2015 doi

  20. [27]

    Kamrani and D

    M. Kamrani and D. Bl¨ omker. Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition.J. Comput. Appl. Math., 323:123–135, 2017.doi: 10.1016/j.cam.2017.04.012

    2017 doi

  21. [28]

    Kastner and K

    F. Kastner and K. Klioba. Simulation code: Milstein-type schemes for hyperbolic SPDEs, 2026.doi:10.5281/ ZENODO.18229440

    2026

  22. [29]

    Kastner and A

    F. Kastner and A. R¨ oßler. An analysis of approximation algorithms for iterated stochastic integrals and a Julia andmatlabsimulation toolbox.Numer. Algorithms, 93(1):27–66, 2023.doi:10.1007/s11075-022-01401-z

    2023 doi

  23. [30]

    T. Kato. Quasi-linear equations of evolution, with applications to partial differential equations. InSpectral theory and differential equations (Proc. Sympos.), Lecture Notes in Math., Vol. 448, pages 25–70. Springer, Berlin, 1975

    1975

  24. [32]

    Klioba and M

    K. Klioba and M. Veraar. Temporal approximation of stochastic evolution equations with irregular nonlinearities. J. Evol. Equ., 24(2):Paper No. 43, 2024.doi:10.1007/s00028-024-00975-6

    2024 doi

  25. [33]

    Kloeden and E

    P.E. Kloeden and E. Platen.Numerical solution of stochastic differential equations. Applications of mathematics

  26. [34]

    Springer, Berlin, 3rd edition, 1999

    1999

  27. [35]

    Kov´ acs, A

    M. Kov´ acs, A. Lang, and A. Petersson. Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise.ESAIM Math. Model. Numer. Anal., 54(6):2199–2227, 2020. doi:10.1051/m2an/2020012

    2020

  28. [36]

    R. Kruse. Consistency and stability of a Milstein–Galerkin finite element scheme for semilinear SPDE.Stoch. Partial Differ. Equ. Anal. Comput., 2(4):471–516, 2014.doi:10.1007/s40072-014-0037-3

    2014 doi

  29. [37]

    Kruse.Strong and weak approximation of semilinear stochastic evolution equations

    R. Kruse.Strong and weak approximation of semilinear stochastic evolution equations. Springer, 2014.doi: 10.1007/978-3-319-02231-4

    2014 doi

  30. [38]

    K¨ uhn and R.L

    F. K¨ uhn and R.L. Schilling. Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups.Studia Math., 262(1):93–119, 2022.doi:10.4064/sm210127-23-3

    2022 doi

  31. [39]

    Leonhard and A

    C. Leonhard and A. R¨ oßler. Iterated stochastic integrals in infinite dimensions: approximation and error estimates. Stoch. Partial Differ. Equ.: Anal. Comput., 7(2):209–239, 9 2018.doi:10.1007/s40072-018-0126-9

    2018 doi

  32. [40]

    Li and X

    J. Li and X. Li. Exponential integrators for stochastic Schr¨ odinger equations.Phys. Rev. E, 101(1):013312, 10, 2020.doi:10.1103/physreve.101.013312. MILSTEIN-TYPE SCHEMES FOR HYPERBOLIC SPDES 39

    2020 doi

  33. [41]

    Lunardi.Analytic semigroups and optimal regularity in parabolic problems

    A. Lunardi.Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkh¨ auser Verlag, Basel, 1995

    1995

  34. [42]

    Mil’shtejn

    G.N. Mil’shtejn. Approximate integration of stochastic differential equations.Theory of Probability & its Appli- cations, 19(3):557–562, 1975.doi:10.1137/1119062

    1975 doi

  35. [43]

    Monk.Finite element methods for Maxwell’s equations

    P. Monk.Finite element methods for Maxwell’s equations. Oxford University Press, 2003.doi:10.1093/acprof: oso/9780198508885.001.0001

    2003

  36. [44]

    M¨ uller-Gronbach

    T. M¨ uller-Gronbach. The optimal uniform approximation of systems of stochastic differential equations.Ann. Appl. Probab., 12(2):664–690, 2002.doi:10.1214/aoap/1026915620

    2002

  37. [45]

    Taylor.Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, vol- ume 81 ofMathematical Surveys and Monographs

    M.E. Taylor.Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, vol- ume 81 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. doi:10.1090/surv/081

    doi

  38. [46]

    Triebel.Interpolation theory, function spaces, differential operators

    H. Triebel.Interpolation theory, function spaces, differential operators. Johann Ambrosius Barth, Heidelberg, second edition, 1995

    1995

  39. [47]

    van Neerven and M.C

    J.M.A.M. van Neerven and M.C. Veraar. Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes.Stoch. Partial Differ. Equ. Anal. Comput., 10(2):516–581, 2022. doi:10.1007/s40072-021-00204-y

    2022 doi

  40. [48]

    M. Veraar. The stochastic Fubini theorem revisited.Stochastics, 84(4):543–551, 2012.doi:10.1080/17442508. 2011.618883

    2012

  41. [49]

    von Hallern and A

    C. von Hallern and A. R¨ ossler. An analysis of the Milstein scheme for SPDEs without a commutative noise condition. InMonte Carlo and quasi-Monte Carlo methods, volume 324 ofSpringer Proc. Math. Stat., pages 503–521. Springer, Cham, 2020.doi:10.1007/978-3-030-43465-6_25

    2020 doi

  42. [50]

    von Hallern and A

    C. von Hallern and A. R¨ oßler. A derivative-free Milstein type approximation method for SPDEs covering the non-commutative noise case.Stoch. Partial Differ. Equ. Anal. Comput., 11(4):1672–1731, 2023.doi:10.1007/ s40072-022-00274-6

    2023

  43. [51]

    Wang and S

    X. Wang and S. Gan. A Runge–Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise.Numer. Algorithms, 62(2):193–223, 2013.doi:10.1007/s11075-012-9568-8. Department of Mathematics, University of Hamburg, Bundesstraße 55, ...

    2013 doi

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.