Field functions

A field function f_{dyn}:\cD \times \Rset^{\inputDim} \rightarrow \cD' \times \Rset^q where \cD \in \Rset^n and \cD' \in \Rset^p is defined by:

(1)\begin{aligned} f_{dyn}(\vect{t}, \vect{x}) = (t'(\vect{t}), v'(\vect{t}, \vect{x})) \end{aligned}

with t': \cD \rightarrow \cD' and v': \cD \times \Rset^{\inputDim} \rightarrow \Rset^q. A field function f_{dyn} transforms a multivariate stochastic process:

\begin{aligned} X: \Omega \times \cD \rightarrow \Rset^{\inputDim}\end{aligned}

where \cD \in \Rset^n is discretized according to the \cM into the multivariate stochastic process:

\begin{aligned} Y=f_{dyn}(X)\end{aligned}

such that:

\begin{aligned} Y: \Omega \times \cD' \rightarrow \Rset^q\end{aligned}

where the mesh \cD' \in \Rset^p is discretized according to the \cM'.

A field function f_{dyn} also acts on fields and produces fields of possibly different dimension (q\neq \inputDim) and mesh (\cD \neq \cD' or \cM \neq \cM').

Value function

A value function f_{spat}: \cD \times \Rset^{\inputDim} \rightarrow \cD \times \Rset^q is a particular field function that leaves the mesh of a field invariant and can be defined using a function g : \Rset^{\inputDim} \rightarrow \Rset^q such that:

(2)\begin{aligned} f_{spat}(\vect{t}, \vect{x})=(\vect{t}, g(\vect{x}))\end{aligned}

Let us note that the input dimension of f_{spat} is still d, the dimension of \vect{x}. Its output dimension is equal to q. The creation of a value function requires the function g and the integer n: the dimension of the vertices of the mesh \cM. These data are required to test the compatibility of the dimensions when a composite process is created using the value function.

Vertex value function

A vertex-value function f_{temp}: \cD \times \Rset^{\inputDim} \rightarrow \cD \times \Rset^q is a particular field function that leaves the mesh of a field invariant and is defined by a function h : \Rset^n \times \Rset^{\inputDim} \rightarrow \Rset^q such that:

(3)\begin{aligned} f_{temp}(\vect{t}, \vect{x})=(\vect{t}, h(\vect{t},\vect{x}))\end{aligned}

Let us note that the input dimension of f_{temp} is still d, the dimension of \vect{x}. Its output dimension is equal to q.