On posterior consistency of data assimilation with Gaussian process priors: the 2D-Navier-Stokes equations (Q6621548)

The authors consider the 2D flat torus \(\Omega =[0,2\pi ]^{2}\) and the incompressible Navier-Stokes equations written as:\(\frac{\partial }{\partial t}u-\nu \Delta u+(u\cdot \nabla )u=f-\nabla p\), \(\nabla \cdot u=0\), on \( (0,T]\times \Omega \), \(\int_{\Omega }u(t,\cdot )=0\) for all \(t\in (0,T]\), with the initial condition \(u(0,\cdot )=\theta \in V=\{u\in H^{1}(\Omega )^{2}:\nabla \cdot u=0\), \(\int_{\Omega }u=0\}\) on \(\Omega \), \(p\) a scalar pressure, \(\nu >0\) a fixed viscosity constant, and \(f\in V\) a forcing term. The authors rewrite this problem as: find a solution \(u\in V\) of the nonlinear evolution equation in \(V\) written as: \(\frac{du}{dt}+\nu Au+B(u,u)=f\), \(u(0)=\theta \), where \(A=-P\Delta \), and \(B=P[(u\cdot \nabla )u]\), \(P\) being the projection onto the Hilbert subspace \(H=\{u\in L^{2}(\Omega )^{2}:\nabla \cdot u=0\), \(\int_{\Omega }u=0\}\), and \(\Delta \) the Laplacian operator. The authors recall that this last problem has a unique strong solution \(u\in C([0,T],V)\cap L^{2}((0,T],\mathcal{D}(A))\), with \(\mathcal{D}(A)=H^{2}(\Omega )^{2}\cap V\), such that \(\frac{du}{dt}\in L^{2}((0,T],H)\). This solution satisfies some estimates. The authors also recall a Lipschitz stability estimate for this problem in terms of the initial conditions. The authors assume that the initial condition \(\theta \) is unknown and that it is replaced by a Gaussian random field over \(\Omega \) . They consider a Borel probability measure \(\Pi ^{\prime }\) on \(V\cap H^{2}(\Omega )^{2}\) as the law of the centred Gaussian random vector field \( (\theta ^{\prime }(x)=(\theta _{1}^{\prime }(x),\theta _{2}^{\prime }(x)):x\in \Omega )\) with reproducing kernel Hilbert space \(\mathcal{H}\) continuously imbedded into \(V\cap H^{\alpha }(\Omega )^{2}\) for some \(\alpha \geq 2\). They then take as prior \(\Pi =\Pi _{N}\) for \(\theta \) the law of the rescaled random vector field \(\theta =\theta ^{\prime }/N^{1/(2\alpha +2)}\). The authors give examples of priors with the associated posterior distributions, defined as: \(d\Pi (\theta \mid Z^{(N)})\propto e^{\ell _{N}(\theta )}d\Pi (\theta )\), where \(Z^{(N)}=(Y_{i},t_{i},X_{i})\) are statistical observations with \(Y_{i}=u_{\theta }(t_{i},X_{i})+\epsilon _{i}\) , and \(\ell _{N}(\theta )=-\frac{1}{2}\sum_{i=1}^{N}\left\vert Y_{i}-u_{\theta }(X_{i},t_{i})\right\vert ^{2}\), \(\theta \in V\). They prove logarithmic posterior rates, assuming hypotheses on the ground truth initial condition \(\theta _{0}\). This allows finally proving that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field.