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Next: 5.1.1 Forward or direct Up: 5. Automatic Differentiation Previous: 5. Automatic Differentiation   Contents


5.1 Some basic algebra

Let $ \cal{M} $ be a general nonlinear, model, i.e. a mapping from the $ m$ -dimensional space $ U \subset I\!\!R^m$ of input variables $ \vec{u}=(u_1,\ldots,u_m)$ (model parameters, initial conditions, boundary conditions such as forcing functions) to the $ n$ -dimensional space $ V \subset I\!\!R^n$ of model output variable $ \vec{v}=(v_1,\ldots,v_n)$ (model state, model diagnostics, objective function, ...) under consideration,

\begin{equation*}\begin{aligned}{\cal M} \, : & \, U \,\, \longrightarrow \, V \... ...\, \longmapsto \, \vec{v} \, = \, {\cal M}(\vec{u}) \end{aligned}\end{equation*}

The vectors $ \vec{u} \in U $ and $ v \in V $ may be represented w.r.t. some given basis vectors $ {\rm span} (U) = \{ {\vec{e}_i} \}_{i = 1, \ldots , m} $ and $ {\rm span} (V) = \{ {\vec{f}_j} \}_{j = 1, \ldots , n} $ as

$\displaystyle \vec{u} \, = \, \sum_{i=1}^{m} u_i \, {\vec{e}_i}, \qquad \vec{v} \, = \, \sum_{j=1}^{n} v_j \, {\vec{f}_j} $

Two routes may be followed to determine the sensitivity of the output variable $ \vec{v}$ to its input $ \vec{u}$ .


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