A stable integration scheme is essential for performing mesh-based integration in INLA. Here we describe the notation and implementation of the integration scheme used in the fmesher package, which is also employed in INLA for spatial models based on the finite element method (FEM).

This is extracted from the Appendix Coherent Disaggregation and Uncertainty Quantification for Spatially Misaligned Data.

Notation

We denote a function on a triangulation of space \(f(\cdot)\) with \(t = 1,\dots,n_t\) and \(v=1,2,3\), as the indices of a triangular mesh (fm_mesh_2d object) and the vertex respectively and \(\mathcal{T}_t\) referring to the \(t\)-th element. We further define \(f_{(t,v)}:= f(s_{(t,v)})\) and \(\sum_{t}^{n_t}\sum_{v \in \mathcal{T}_t } w_{(t,v)} = 1\), for some weights \(\{w_{(t,v)}\}\), and some vertices \(\{s_{(t,v)}\} \subset \Omega\) inside the sample space. The domain contribution can be approximated via integration points as, \(\int_{\Omega} f(s) \,ds \approx \sum_{t}^{n_t} \sum_{v \in \mathcal{T}_t } w_{(t,v)}f_{(t,v)}; \label{eq:intsch}\) A 3-column integer matrix with 1-based vertex indices for each triangle \((t,v)\) (with reference to mesh$loc) is stored under mesh$graph$tv in the mesh object under the fmesher package.

WIP…