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Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise
Lukas Anzeletti, Máté Gerencsér, Helena Kremp
TL;DR
Spatial convergence for SPDE discretisations has remained limited by solution regularity, with no scheme previously achieving a spatial strong rate beyond that regularity. The paper constructs a fully discrete scheme using an explicitly simulable Gaussian approximation of the noise and achieves temporal rate M^-1+ε and spatial rate N^-3/2+ε.
Problem
SPDE discretisations face convergence limitations in both time and space, and no scheme had achieved a spatial strong rate superior to the solution’s spatial regularity.
Method
The scheme approximates the irregular noise component with an intermediate Gaussian process whose grid covariance is explicit and whose remainder decays at sufficiently high temporal and spatial rates.
Results
M^-1+ε in time and N^-3/2+ε in space are achieved for any ε>0, improving on the spatial rate 1/2 while retaining the temporal improvement.
Takeaways & Limitations
The method overcomes the spatial order 1/2 barrier and provides matching rates for equations with bounded and superlinearly growing nonlinearities.
Takeaways & Limitations
Convergence comparisons depend on the chosen error criterion, since there is no canonical choice of distance d.
Abstract
from arXiv · showhide
We introduce a fully discrete numerical scheme for semilinear SPDEs with additive space-time white noise that overcomes the previous order barrier for the spatial convergence rate. The scheme achieves a strong convergence rate of $M^{-1+ε}$ in time and $N^{-3/2+ε}$ in space for any $ε>0$, where $M^{-1}$ and $N^{-1}$ are the temporal, respectively the spatial, meshsizes. This substantially improves the standard spatial error bounds of order $N^{-1/2}$ in the literature.
1 Introduction
The paper addresses spatial convergence barriers in additive space-time white-noise SPDEs by constructing a scheme that improves both spatial and temporal rates, including for nonlinearities beyond the globally Lipschitz setting.
- Motivation: SPDE approximation rates are limited by solution regularity in both time and space, with prior spatial schemes unable to exceed the solution’s spatial regularity.The paper explicitly frames surpassing this barrier as its central objective.
- Order-barrier framework: The analysis separates Gaussian-noise generation from deterministic computation, formalizing order barriers through the selected noise observables and an error pseudometric.A lower bound for the resulting approximation class expresses the barrier imposed by those observables.
- Caveat: Comparisons across convergence results remain dependent on the chosen error metric, since no canonical distance is available.Possible criteria include continuous, discrete, and pointwise distances.
- Main contribution: The proposed scheme achieves a full error estimate of order M^-1+ε + N^-3/2+ε for any ε>0, substantially exceeding the previous spatial rate 1/2.The earlier improved temporal schemes retained a spatial rate of 1/2.
- Scope: The paper treats both globally Lipschitz nonlinearities and the stochastic Allen-Cahn case with superlinear growth.Its second main theorem matches the rate established for the bounded-nonlinearity case.
- Numerical approach: The scheme is based on an intermediate Gaussian process with explicit grid covariance, making the approximation genuinely implementable while controlling the remainder.It uses the intermediate process to approximate the irregular noise component on the space-time grid.
2 Preliminaries
The preliminaries develop the function-space, stochastic, and discretization estimates needed for the main convergence results. Particular emphasis is placed on spatial discretization estimates and transferring regularity bounds to restricted Ornstein–Uhlenbeck processes.
- Analytic and probabilistic tools: The proofs collect Sobolev, Besov, semigroup, stochastic sewing, and regularity estimates for continuous and discrete settings.These ingredients support the analysis of the solution, the Ornstein–Uhlenbeck process, and the remainder v = u − O.
- Spatial discretization: The discretization has an error order matching standard Fourier truncation, though under more restrictive exponent conditions.This estimate concerns the discretization as an operator on functions on T.
- Restricted stochastic process: Bounds for the restricted Ornstein–Uhlenbeck process in negative-regularity spaces are crucial for retaining temporal rate 1.The estimates suggest a tradeoff between temporal increments and the spatial mesh N^-1 for Θ_N O_t.
- Discrete function spaces: The discrete function-space framework defines continuous and discrete Sobolev and Besov norms, together with restriction and extension operators.The spaces C_N(T), discrete Littlewood–Paley blocks, and the operators Θ_N and Ψ_N organize the spatial approximation.
- Well-posedness and regularity: The well-posedness estimates establish a unique mild solution and provide regularity bounds for u and v = u − O under the stated assumptions.The constants depend on parameters including T, p, λ, ε, m, and K.
3 Bounded nonlinearity
For bounded nonlinearities, the paper decomposes the total error into spatial discretisation, temporal discretisation, and input-noise approximation errors, then proves bounds for each component. The resulting analysis supports the main bounded-nonlinearity convergence theorem and explains why the temporal and spatial rates do not follow parabolic scaling.
- Auxiliary estimates: The analysis also establishes well-posedness and regularity bounds for the auxiliary finite-dimensional processes used in the bounded-nonlinearity scheme.The stated lemmas cover well-posedness and moment or regularity estimates for the discretised processes.
- Error decomposition: The error is decomposed into spatial discretisation, temporal discretisation, and noise-approximation components.The spatial and temporal terms are bounded separately, while the noise term is handled by a dedicated lemma.
- Spatial error: The bounded-nonlinearity spatial analysis applies stochastic sewing to obtain a bound for the spatial error.The argument uses regularity estimates, semigroup bounds, composition estimates, and a Girsanov-based reduction.
- Main result: Theorem 1.1 follows by combining the noise, spatial, and temporal error bounds under the stated assumptions.The proof explicitly combines Lemmas 3.2, 3.4, and 3.6 with Assumption 1.1 a).
- Rate mechanism: The temporal and spatial rates do not exhibit parabolic scaling because the spatial and temporal sewing arguments estimate their integrands in different spaces.The spatial integrand is estimated in L2(T), whereas the temporal integrand is estimated in H^-1/2+ε(ΠN), allowing a better-than-half spatial temporal rate.
5 A concrete choice of the input noise
The paper constructs an implementable approximation of the Ornstein–Uhlenbeck input noise using Fourier modes and independent Gaussian variables. It verifies the approximation's regularity and closeness to the exact noise, then incorporates it into schemes with computational effort O(MN log N).
- Noise construction: The input noise approximation is built by truncating and simulating Fourier components of the Ornstein–Uhlenbeck process.The construction separates projected and remainder components and uses the discrete Fourier representation on an even spatial grid.
- Gaussian structure: The simulated Fourier coefficients have explicit Gaussian distributions, variances, and independence relations across time intervals and modes.The covariance structure is explicit, with conjugate relations for paired Fourier modes.
- Regularity verification: Under M^-1 ≥ cN^-2, the constructed noise satisfies the required regularity bounds.The regularity verification uses Fourier-multiplier estimates and the relation between the temporal and spatial mesh sizes.
- Approximation error: Under M^-1 ≥ cN^-2+ε, the approximation error between the exact and constructed noise decays as e^-c′N^ε in the stated maximum Lp norm.The bound is established uniformly over the temporal grid points.
- Scheme implementation: The full bounded- and polynomial-growth schemes combine the simulated noise with recursively updated deterministic components.The full schemes are written as Uk = Vk + Ok and Uk = Xk + Ok, respectively.
- Computational cost: The total computational effort for both schemes is O(MN log N).The Fourier transforms account for the logarithmic factor.
Declaration on AI usage
The paper states that AI tools were used only to search for references on heat kernel estimates.
- AI usage: AI tools were used exclusively for searching for references on heat kernel estimates.The declaration limits AI use to literature-reference searches.