Quantifying the effect of noise perturbation for the stochastic Burgers equation with additive trace-class noise
This work addresses error analysis for stochastic PDEs, specifically for researchers in applied mathematics and numerical analysis, but it is incremental as it extends known error bounds to a specific equation.
The paper tackles the problem of quantifying errors from noise perturbations in the stochastic Burgers equation with additive trace-class noise, establishing upper bounds for weak and strong errors, such as weak error of O(||(-A)^{-1^{-}}(Q1-Q2)||) and strong error of O(||(-A)^{-1/2^{-}}|Q1^{1/2}-Q2^{1/2}|||).
We establish upper bounds for the weak and strong error resulting from a perturbation of the noise driving the stochastic Burgers equation, where we assume the noise to be additive and of trace class and the initial value to be sufficiently regular. More specifically, replacing the covariance operator of the driving noise $Q_1 \in \mathcal{L}_1(L^2)$ in the Burgers equation by a covariance operator $Q_2 \in \mathcal{L}_1(L^2)$ results in a weak error of $\mathcal{O}\big(\| (-A)^{-1^{-} } (Q_1-Q_2) \|_{\mathcal{L}_1(L^2)}\big)$ and a strong error of $\mathcal{O}\big(\big\| (-A)^{-1/2^{-}}\big|Q_1^{1/2} -Q_2^{1/2}\big| \big\|_{\mathcal{L}_2(L^2)}\big)$. Here $\|\cdot \|_{\mathcal{L}_1}$ is the trace class norm, $\|\cdot \|_{\mathcal{L}_2}$ is the Hilbert-Schmidt norm, and $A$ is the one-dimensional Dirichlet Laplacian that represents the leading term in the Burgers equation. In particular, our results provide upper bounds for the weak and strong error arising when approximating the trace class noise by finite-dimensional noise; the rates we obtain reflect the general philosophy that the weak convergence rate should be twice the strong rate.