Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise
For researchers working on numerical methods for SPDEs, this provides the first sharp uniform-in-time weak and ergodic error estimates for superlinear problems with multiplicative noise.
The paper proves a uniform-in-time weak convergence rate for a nonlinearity-explicit full discretization of superlinear SPDEs with multiplicative noise, achieving rate τ^ρ+λ_N^{-ρ} for any ρ∈(0,1), and obtains a sharp ergodic error estimate between exact and numerical invariant measures.
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, Hölder, 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 $τ^ρ+λ_N^{-ρ}$ for any $ρ\in (0,1)$. Consequently, we obtain a sharp ergodic error estimate between the exact and numerical invariant measures. Numerical experiments support the theory.