---
title: "Custom mesh classes"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Custom mesh classes}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
set.seed(12345L)
```

```{r setup}
library(fmesher)
```

## Minimal interface

Users and package developers can add `fmesher` support to their own classes.
A minimal interface needs to define `fm_dof()` and `fm_basis()` methods.
Assuming the class is called `custom`, the methods should be named `fm_dof.custom()` and `fm_basis.custom()`.

- The `fm_dof.custom(x)` method *must* take an object `x` of class `custom` and
  return the number of degrees of freedom of the function space.
- The `fm_basis.custom(x, loc, ..., full = FALSE)` method *must* take an object `x` of
  class `custom` and return a `sparseMatrix` or `Matrix` matrix with each column
  containing the basis function evaluated at the locations determined by `loc`.

  The type of `loc` may be any type (or types) that is supported by the custom
  class.
  
  The `...` part can include further named arguments specific to the custom
  class. These must be optional arguments so that `fm_basis(x, loc)` works.
  
  When `full = TRUE`, a full `fm_basis` object *must* be returned, which is a
  list containing *at least* the basis matrix as `A`, and a logical vector,
  `ok`, indicating which `loc` values were valid evaluation points. The `A`
  matrix *must* be all-zero for invalid `loc`.
- With the above requirements fulfilled, the default `fm_evaluator()` and
  `fm_evaluate()` methods can be used to evaluate functions at any location,
  with the need fo the user to define any further methods.
  
  Special `fm_evaluator.custom()` and `fm_evaluate.custom()` methods may be
  defined if needed, e.g. to support semi-automated output reformatting.
  
### Example: Harmonic function space of order n

```{r}
# Custom class for harmonic functions up to order `n`
create_custom <- function(n) {
  stopifnot(n >= 0)
  structure(
    list(n = n),
    class = "custom"
  )
}
fm_dof.custom <- function(x) {
  # Return the number of degrees of freedom
  1L + 2L * x[["n"]]
}
fm_basis.custom <- function(x, loc, ..., full = FALSE) {
  # Return the evaluated basis functions
  A <- Matrix::Matrix(0.0, NROW(loc), fm_dof(x))
  ok <- !is.na(loc)
  A[ok, 1L] <- 1.0
  for (k in seq_len(x[["n"]])) {
    A[ok, 2 * k] <- cos(2 * pi * k * loc[ok])
    A[ok, 2 * k + 1L] <- sin(2 * pi * k * loc[ok])
  }
  result <- structure(
    list(
      A = A,
      ok = ok, # Required prior to version 0.2.0.9003
      loc = loc
    ),
    class = "fm_basis"
  )

  # Use the fm_basis method to extract the A matrix if full is FALSE:
  fm_basis(result, full = full)
}
```

Note: From version `0.2.0.9004`, the `fm_basis.matrix`, `fm_basis.Matrix`, and
`fm_basis.list` methods provide an easier way to construct the `fm_basis`
object, by creating the object and optionally extracting `A` in a single call:
```{r, eval=FALSE}
# 'matrix' and 'Matrix' methods:
fm_basis(
  A = A,
  ok = ok, # If missing or NULL, inferred to be all TRUE
  loc = loc, # Optional additional content
  full = full
)
# 'list' method:
fm_basis(
  list(
    A = A,
    ok = ok, # If missing or NULL, inferred to be all TRUE
    loc = loc
  ),
  full = full
)
```

#### Registering the methods

These S3 methods must be registered with the `S3method()` function in
scripts, and with special NAMESPACE tags in packages.
In a script, one should use
```{r}
.S3method("fm_dof", "custom", "fm_dof.custom")
.S3method("fm_basis", "custom", "fm_basis.custom")
```
In a package, if R is version 3.6 or newer, one can use roxygen2 tags
```{r}
#' @rawNamespace S3method(fmesher::fm_dof, custom)
#' @rawNamespace S3method(fmesher::fm_basis, custom)
```
or before each method, use `@exportS3Method`, like this:
```{r, eval = FALSE}
#' @title Degrees of freedom for custom mesh
#' @description the number of degrees of freedom
#' # The rest of the documentation goes here
#' @exportS3method fmesher::fm_dof
fm_dof.custom <- function(x) {
  1L + 2L * x[["n"]]
}
```
which semi-automates it.

We can the use the new methods with
```{r}
m <- create_custom(2)

# How many latent variables are needed?
fm_dof(m)

# Evaluate the basis functions at some locations:
fm_basis(m, seq(0, 1, length.out = 6))
fm_basis(m, seq(0, 1, length.out = 6), full = TRUE)

# Check if missing values are handled correctly:
fm_basis(m, c(0.1, NA, 0.2))
fm_basis(m, c(0.1, NA, 0.2), full = TRUE)
```

## Expanded implementations

The main additional method that can be defined is the `fm_int()` integration scheme method.
This must have the call structure `fm_int.custom(domain, samplers = NULL, name = "x", ...)`.

- The `domain` argument is the `custom` class object over which to integrate.
- The `samplers` argument is any object, typically an `sf` or `tibble` specifying
  one or more subsets of the domain, e.g. polygons. When `NULL`, the entire domain
  should be integrated.
- The `name` argument is a character string specifying the name of the integration
  point variable.
- The `...` arguments can be augmented with further optional arguments, e.g. options
  controlling the integration scheme construction and/or the output format.

`out <- fm_int(domain, samplers, name)` should return a
`data.frame`, `tibble`, or `sf` object with integration points
in a column with the name indicated by *<name>*, and additional columns `weight` with
corresponding integration weights, and a `.block` column.

- The *<name>* column format should be compatible with `fm_basis(domain, out[[name]])`.
- The `.block` column should be an integer vector indicating which subdomain
  each integration point belongs to, usable by `fm_block_eval()`:
```{r,echo=FALSE}
out <- data.frame(
  x = c(
    seq(0, 0.3, by = 0.1),
    seq(0.3, 0.5, by = 0.1),
    seq(0.5, 1, by = 0.1)
  ),
  weight = rep(
    c(0.05, 0.1, 0.05, 0.05, 0.1, 0.05, 0.05, 0.1, 0.05),
    c(1, 2, 1, 1, 1, 1, 1, 4, 1)
  ),
  .block = rep(c(1, 2, 3), c(4, 3, 6))
)
out
```
```{r,eval=TRUE}
values <- fm_evaluate(
  m,
  field = c(1, 1, 0, 0, 0),
  loc = out[["x"]]
)

# Blockwise aggregation:
fm_block_eval(
  block = out$.block,
  weights = out$weight,
  values = values
)

# Exact integrals:
c(0.3, 0.2, 0.5) +
  c(
    sin(2 * pi * 0.3),
    sin(2 * pi * 0.5) - sin(2 * pi * 0.3),
    sin(2 * pi) - sin(2 * pi * 0.5)
  ) / (2 * pi)
```
