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Numerical Ergodicity and Optimal Strong Error Estimates for a Class of Novel Tamed Schemes to Superlinear SPDEs
Zhihui Liu, Jie Shen
TL;DR
Superlinear SPDEs with coefficients outside classical Lipschitz and linear-growth assumptions pose a challenging numerical-ergodicity problem. The paper constructs tamed schemes that preserve the original Lyapunov structure and proves stability, inherited unique ergodicity, and optimal strong convergence rates for both noise cases.
Problem
Numerical ergodicity remains challenging for superlinear SPDEs whose coefficients violate classical Lipschitz continuity and linear-growth assumptions.
Source: §1 · p. 1Method
The paper develops explicit-in-nonlinearity tamed schemes that tame nodal values and preserve coupled monotonicity and the original Lyapunov structure.
Source: §1 · p. 2Results
The Galerkin-based fully discrete schemes preserve the original Lyapunov structure and inherit unique ergodicity under a nondegeneracy condition, with optimal spatial and temporal strong convergence orders for both noise cases.
Source: §1 · p. 3Takeaways & Limitations
The schemes provide an approach to approximating ergodic limits when explicit invariant measures for infinite-dimensional stochastic systems are typically infeasible to obtain.
Source: §1 · pp. 1, 3Under standard monotone/Lipschitz conditions, strong convergence rates depend on T in the general setting; uniform bounds are cited under a strong dissipativity condition.
Source: §4.1 · p. 14
Abstract
from arXiv · showhide
We construct a class of novel tamed schemes for superlinear stochastic partial differential equations (SPDEs), including the stochastic Allen--Cahn equation driven by either multiplicative or additive noise. The schemes preserve the same Lyapunov structure as the original system, and we rigorously establish their longtime unconditional stability. Furthermore, we prove that the corresponding Galerkin-based fully discrete tamed schemes inherit the unique ergodicity of the underlying SPDEs and achieve optimal strong convergence rates in both the multiplicative and additive noise cases.
1. Introduction
The paper targets explicit numerical schemes for superlinear SPDEs that preserve longtime ergodicity while approximating solutions strongly. Its tamed schemes retain the equation’s Lyapunov structure, are unconditionally stable over infinite time horizons, and achieve optimal strong convergence rates for multiplicative and additive noise.
1. Introduction
- Classical Euler–Maruyama schemes can blow up in p-th moments for p ≥ 2 on these superlinear equations, and the square-function Lyapunov functional is unsuitable for that scheme.Source: §1 · p. 2
- The schemes tame nodal values rather than norms at each step, and their taming functions preserve coupled monotonicity that conventional taming disrupts.The authors present this design as more efficient to implement than existing tamed schemes.Source: §1 · p. 2
- The temporal-tamed scheme and its Galerkin full discretizations are proved unconditionally stable over any time interval, including infinite horizons.Source: §1 · p. 2
- The Galerkin-based fully discrete schemes inherit unique ergodicity by preserving the original equation’s Lyapunov structure, together with equivalence of transition probabilities under a nondegeneracy condition.Source: §1 · p. 3
- Strong error analysis for additive noise requires Ls-regularity of the exact solution with s > 2 and ˙H1-regularity of an auxiliary process; the stated 3D regularity cannot be controlled by the ˙H1-norm alone.The cited regularity requirement is not needed for the multiplicative-noise result.Source: §1 · p. 3
- Explicit tamed schemes preserve the Lyapunov structure and unique ergodicity of superlinear SPDEs, with optimal spatial and temporal strong convergence orders for multiplicative and additive noise.The results cover one, two, and three space dimensions.Source: §1 · p. 3
2. Preliminaries
This section sets the functional framework and coefficient assumptions, then defines an explicit-in-nonlinearity tamed EM scheme for the stochastic evolution equation.
2.1. Notations.
- The notation section introduces Lebesgue spaces and the Dirichlet Laplacian framework used for the SPDE.It also defines a ∧ b as min{a, b}.Source: §2.1 · p. 4
2. Preliminaries
- The preliminaries define the Dirichlet Laplacian and fractional Sobolev spaces, and specify a trace-class Wiener process setting.The analysis uses the space H and its Sobolev domains, with Q self-adjoint and positive-definite.Source: §2.1 · p. 4
2.2. Assumptions.
- The coefficient assumptions impose coupled coercivity and polynomial-growth conditions, with dimension-dependent restrictions on q for the Sobolev embeddings.They include the stochastic Allen–Cahn double-well drift with a quadratic-growth diffusion of sufficiently small leading coefficient.Source: §2.2 · pp. 4–5
2.3. Tamed Schemes.
- The tamed EM scheme applies tamed drift and diffusion functions to the previous state, and is a linearly implicit Markov chain.The step uses temporal increment τ and Wiener increment δ_{n−1}W; the taming functions are defined in (2.11).Source: §2.3 · pp. 5–6
3. Unique Ergodicity of Tamed Schemes
The section establishes Lyapunov stability for the tamed schemes and their Galerkin finite-element discretizations, then proves unique ergodicity under a nondegeneracy condition.
3.1. Lyapunov Structure.
- The temporal tamed EM scheme preserves the original equation’s Lyapunov structure and is unconditionally stable over an infinite time horizon.Its one-step estimate controls both the state norm and its gradient norm.Source: §3.1 · pp. 6–8
3.2. Application to Galerkin-based tamed full-discretizations.
- Galerkin FEM full discretizations retain Lyapunov estimates without additional conditions and extend the scheme to spatially discretized states.The paper treats both tamed-FEM and drift-tamed-FEM; the latter applies when the diffusion has linear growth.Source: §3.2 · pp. 8–9, §3.1 · p. 8
3.3. Unique Ergodicity of Tamed-FEM and Drift-Tamed-FEM.
- The unique-ergodicity argument combines the schemes’ Lyapunov structure with regular transition kernels whose measures are equivalent under nondegenerate noise.The paper notes that geometric ergodicity could be proved with additional differentiability assumptions by verifying a minorization condition.Source: §3.3 · p. 10
- The tamed-FEM and drift-tamed-FEM are uniquely ergodic for their stated step-size ranges under the additional nondegeneracy assumption.The transition kernels are regular, while the analysis does not cover the degenerate standard linear multiplicative-noise case.Source: §3.3 · pp. 9–10
4. Strong Error Estimates in Multiplicative Noise Case
The section establishes finite-time strong error estimates for the drift-tamed finite-element scheme under coefficient and regularity conditions. It develops auxiliary-process moment bounds and treats multiplicative noise.
4.1. A Priori Estimates.
- The error analysis imposes a uniform upper bound on the derivative of the tamed drift; the condition includes odd polynomials with negative leading coefficients.The derivative bound is identified as crucial for optimal strong error estimates.Source: §4 · p. 11
- Under standard monotone/Lipschitz conditions, the strong convergence estimates depend on T; uniform-in-T bounds are cited under a stronger dissipativity condition.The authors describe this T-dependence as seemingly unavoidable in the general setting.Source: §4.1 · p. 14
4.2. Auxiliary Process and Moments’ Estimates.
- An H1 moment estimate for the auxiliary process supplies higher-order moment bounds needed in the subsequent error analysis.The proof directly establishes the auxiliary process's H1 norm, eliminating the dimension restriction noted for an earlier approach.Source: §4.2 · pp. 15–16
4.3. Strong Error Estimate.
- Theorem 4.1 establishes strong convergence of the drift-tamed-FEM to the exact solution for initial data in H^{1+γ} under the stated coefficient assumptions.The result covers γ ∈ [0, 1], with an additional assumption when γ = 1.Source: §4.3 · p. 19
5. High-order Convergence Rate in Additive Noise Case
The section analyzes strong convergence for constant additive noise and seeks a higher-order result under an additional growth condition on the drift’s second derivative.
5. High-order Convergence Rate in Additive Noise Case
- The additive-noise analysis considers a constant diffusion operator G.The equation is written with diffusion term GdW.Source: §5 · p. 20
- The paper establishes a higher-order convergence result for the drift-tamed-FEM scheme applied to Eq. (5.1), under an additional polynomial growth condition on the second derivative f′′ of f.Source: §5 · p. 21
- For 3D stochastic Allen–Cahn with q = 2, a term in the estimate cannot be controlled using only the solution’s H1 norm.The section also notes an additional higher-integrability requirement for the additive-noise estimates in this case.Source: §5 · p. 23