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1.1 Introduction

1.3 Continuous equations in `r' coordinates

2. Discretization and Algorithm

4. Software Architecture

5. Automatic Differentiation

6. Physical Parameterization and Packages

7. Diagnostics and tools

8. Interface with ECCO

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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 -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 -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,

$\begin{displaymath}\begin{split}{\cal M} \, : & \, U \,\, \longrightarrow \, V \... ...\, \longmapsto \, \vec{v} \, = \, {\cal M}(\vec{u}) \end{split}\end{displaymath}$

(5.1)

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 $\vec{u}$ .

Subsections

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