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A scalable approach for variational data assimilation. (English) Zbl 1311.65056

The article deals with the following problem (the DA inverse problem): \[ v(t,y) = {\mathcal H}(u(t,y)), \quad 0 \leq t \leq T, \;y \in \Omega, \] where \({\mathcal H}\) is a given nonlinear operator, \(u(t,x)\) the state evolution of a preductive system, \(v(t,x)\) the observation data. Really, the following digitization \[ {\mathbf v}_k = {\mathbf H}[{\mathbf u}_k^{DA}] \] is considered; here \({\mathbf H} \in {\mathfrak R}^{NP \times nobs}\) is a matrix obtained by the first-order approximation of the Jacobian of \({\mathcal H}\), and \(nobs \ll NP\); \({\mathbf u}^{DA} = \{u(t_k,x_j)\}_{j=1,\dots,NP}^{DA}\), \(u(t,x):\;[0,T] \times \Omega \to {\mathfrak R}\) is a solution. This problem is reduced to the following 3D variational DA problem \[ {\mathbf u}^{DA} = \mathrm{argmin}_{{\mathbf u} \in {\mathfrak R}^{NP}}\, J({\mathbf u}) = \mathrm{argmin}_{{\mathbf u} \in {\mathfrak R}^{NP}}\, \big\{\|{\mathbf H}{\mathbf u} - {\mathbf v}\|_{\mathbf R}^2 + \lambda\|{\mathbf u} - {\mathbf u}^b\|_{\mathbf B}^2\big\} \] (\(\lambda\) is the regularization parameter, \(\|\cdot\|_{\mathbf B}\) and \(\|\cdot\|_{\mathbf R}\) are the weighted Euclidean norms on \({\mathfrak R}^{NP}\)). The authors, using the decomposition \[ \Omega = \bigcup_{i=1}^p \Omega_i, \qquad \Omega_i \cap \Omega_j = \Omega_{ij} \neq \emptyset,\tag{*} \] reduce the original (global) problem to the analogous (local) ones for \(\Omega_i\), \(i = 1,\dots,p\). In this connection, the authors state that \[ f(t,x) = \sum_{i=1}^p EQ_i[f_i^{RO}(t,x)], \qquad f:\;[0,T] \times \Omega \to {\mathfrak R}, \] where \(f \to f_i^{RO}\) is the restriction operator onto \(\Omega_i\) and \(g_i \to EQ_ig\) the extension operator from \(\Omega_i\) onto \(\Omega\): \(EO_i g_i(x) = g_i(x)\) for \(x \in \Omega_i\) and \(EO_i g_i(x) = 0\) elsewhere; the latter is evidently false due to (*).

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65J22 Numerical solution to inverse problems in abstract spaces
65Y05 Parallel numerical computation
47J30 Variational methods involving nonlinear operators

Software:

OceanVAR; L-BFGS
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Full Text: DOI

References:

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