Interval, Circle, and Tadpole graphs

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# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
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}

1 Import libraries

# to update
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(MetricGraph)
library(Matrix)
library(rSPDE)

library(dplyr)
library(tidyverse)
library(plotly)

library(grateful) # Cite all loaded packages

library(slackr)
source("keys.R")
slackr_setup(token = token) # token comes from keys.R

## [1] "Successfully connected to Slack"

capture.output(
  knitr::purl(here::here("functionality1.Rmd"), output = here::here("functionality1.R")),
  file = here::here("old/purl_log.txt")
)
source(here::here("functionality1.R"))

asppp <- 0.7

2 Interval graph

# Function to build an interval graph and create a mesh
gets_graph_interval <- function(n){
  edge <- rbind(c(0,0),c(1,0))
  edges = list(edge)
  graph <- metric_graph$new(edges = edges)
  graph$build_mesh(n = n)
  return(graph)
}

# Matern covariance function ff
matern.covariance <- function(h, kappa, nu, sigma) {
  if (nu == 1 / 2) {
    C <- sigma^2 * exp(-kappa * abs(h))
  } else {
    C <- (sigma^2 / (2^(nu - 1) * gamma(nu))) *
      ((kappa * abs(h))^nu) * besselK(kappa * abs(h), nu)
  }
  C[h == 0] <- sigma^2
  return(as.matrix(C))
}

# Folded.matern.covariance.1d
folded.matern.covariance.1d.local <- function(x, kappa, nu, sigma,
                                              L = 1, N = 10,
                                              boundary = c("neumann",
                                                           "dirichlet", "periodic")) {
  boundary <- tolower(boundary[1])
  if (!(boundary %in% c("neumann", "dirichlet", "periodic"))) {
    stop("The possible boundary conditions are 'neumann',
    'dirichlet' or 'periodic'!")
  }
  addi = t(outer(x, x, "+"))
  diff = t(outer(x, x, "-"))
  s1 <- sapply(-N:N, function(j) { 
    diff + 2 * j * L
  })
  s2 <- sapply(-N:N, function(j) {
    addi + 2 * j * L
  })
  if (boundary == "neumann") {
    C <- rowSums(matern.covariance(h = s1, kappa = kappa,
                                   nu = nu, sigma = sigma) +
                   matern.covariance(h = s2, kappa = kappa,
                                     nu = nu, sigma = sigma))
  } else if (boundary == "dirichlet") {
    C <- rowSums(matern.covariance(h = s1, kappa = kappa,
                                   nu = nu, sigma = sigma) -
                   matern.covariance(h = s2, kappa = kappa,
                                     nu = nu, sigma = sigma))
  } else {
    C <- rowSums(matern.covariance(h = s1,
                                   kappa = kappa, nu = nu, sigma = sigma))
  }
  return(matrix(C, nrow = length(x)))
}

# Function to get the true covariance matrix
gets_true_cov_mat = function(graph, kappa, nu, sigma, N, boundary){
  #h = c(graph$mesh$V[-2,1], graph$mesh$V[2,1])
  h = graph$mesh$V[,1]
  true_cov_mat = folded.matern.covariance.1d.local(x = h, kappa = kappa, nu = nu, sigma = sigma, N = N, boundary = boundary)
  return(true_cov_mat)
}


# Parameters of the approximation
n = 333
type = "covariance"
type_rational_approximation = "chebfun"
rho = 1
m = 4
nu = 0.6
sigma = 1
N.folded = 10
boundary = "neumann"

# Get the graph and build the mesh
graph = gets_graph_interval(n = n)

kappa = sqrt(8*nu)/rho
tau = sqrt(gamma(nu) / (sigma^2 * kappa^(2*nu) * (4*pi)^(1/2) * gamma(nu + 1/2)))  #sigma = 1, d = 1
alpha = nu + 1/2

# Get the true covariance
true_cov_mat = gets_true_cov_mat(graph = graph,
                                 kappa = kappa,
                                 nu = nu,
                                 sigma = sigma,
                                 N = N.folded,
                                 boundary = boundary)

# Get the approximate covariance matrix
op = matern.operators(alpha = alpha, 
                      kappa = kappa, 
                      tau = tau,
                      m = m, 
                      graph = graph,
                      type = type,
                      type_rational_approximation = type_rational_approximation)

appr_cov_mat = covariance_mesh(op)

# Plot the true covariance and the approximation
point  <- c(1,0.5)
c_cov <- cov_function_mesh(op, matrix(point,1,2))
loc <- graph$coordinates(PtE = point)
m1 <- which.min((graph$mesh$V[,1]-loc[1])^2 + (graph$mesh$V[,2]-loc[2])^2)
p = graph$plot_function(true_cov_mat[m1,], type = "plotly", line_width = 7, edge_width = 7, vertex_size =7) # blue is the true one
p_int <- graph$plot_function(c_cov, p = p, line_color = "red", type = "plotly", line_width = 7, edge_width = 7, vertex_size =7)  %>% 
  config(mathjax = 'cdn') %>% 
  layout(font = list(family = "Palatino"),
         scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              camera = list(eye = list(x = 2.3*asppp, 
                                       y = 2.3*asppp, 
                                       z = 2.3*asppp),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
           annotations = list(
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 1, z = 0,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 0.5, z = 0,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = 0, y = 0.6, z = 1.25,
               text = TeX("\\bullet\\mbox{ exact}"),
               textangle = 0, ax = 60, ay = 0,
               font = list(color = "#0000C8", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = 0, y = 0.745, z = 1.20,
               text = TeX("\\bullet\\mbox{ approximated}"),
               textangle = 0, ax = 60, ay = 0,
               font = list(color = "red", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))))

save(p_int, file = here::here("data_files/p_int.Rdata"))

# Compute the errors
L_inf_error = max(abs(true_cov_mat - appr_cov_mat))
L_2_error = sqrt(as.double(t(graph$mesh$weights)%*%(true_cov_mat - appr_cov_mat)^2%*%graph$mesh$weights))
print(L_inf_error)
print(L_2_error)

3 Circle graph

# Function to build a circle graph and create a mesh
gets_graph_circle <- function(n){
  r = 1/(pi)
  theta <- seq(from=-pi,to=pi,length.out = 100)
  edge <- cbind(1+r+r*cos(theta),r*sin(theta))
  edges = list(edge)
  graph <- metric_graph$new(edges = edges)
  graph$build_mesh(n = n)
  return(graph)
}

# Matern covariance function
matern.covariance <- function(h, kappa, nu, sigma) {
  if (nu == 1 / 2) {
    C <- sigma^2 * exp(-kappa * abs(h))
  } else {
    C <- (sigma^2 / (2^(nu - 1) * gamma(nu))) *
      ((kappa * abs(h))^nu) * besselK(kappa * abs(h), nu)
  }
  C[h == 0] <- sigma^2
  return(as.matrix(C))
}

# Folded.matern.covariance.1d 
folded.matern.covariance.1d.local <- function(x, kappa, nu, sigma, L = 1, N = 10, boundary = c("neumann",
                                                           "dirichlet", "periodic")) {
  boundary <- tolower(boundary[1])
  if (!(boundary %in% c("neumann", "dirichlet", "periodic"))) {
    stop("The possible boundary conditions are 'neumann',
    'dirichlet' or 'periodic'!")
  }
  addi = t(outer(x, x, "+"))
  diff = t(outer(x, x, "-"))
  s1 <- sapply(-N:N, function(j) { # s1 is a matrix of size length(h)x(2N+1)
    diff + 2 * j * L
  })
  s2 <- sapply(-N:N, function(j) {
    addi + 2 * j * L
  })
  if (boundary == "neumann") {
    C <- rowSums(matern.covariance(h = s1, kappa = kappa,
                                   nu = nu, sigma = sigma) +
                   matern.covariance(h = s2, kappa = kappa,
                                     nu = nu, sigma = sigma))
  } else if (boundary == "dirichlet") {
    C <- rowSums(matern.covariance(h = s1, kappa = kappa,
                                   nu = nu, sigma = sigma) -
                   matern.covariance(h = s2, kappa = kappa,
                                     nu = nu, sigma = sigma))
  } else {
    C <- rowSums(matern.covariance(h = s1,
                                   kappa = kappa, nu = nu, sigma = sigma))
  }
  return(matrix(C, nrow = length(x)))
}

# Function to get the true covariance matrix
gets_true_cov_mat = function(graph, kappa, nu, sigma, N, boundary){
  h = c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[,2])
  true_cov_mat = folded.matern.covariance.1d.local(x = h, kappa = kappa, nu = nu, sigma = sigma, N = N, boundary = boundary)
  return(true_cov_mat)
}


# Parameters of the approximation
n = 666
type = "covariance"
type_rational_approximation = "chebfun"
rho = 1
m = 4
nu = 0.6
sigma = 1
N.folded = 10
boundary = "periodic"

# Get the graph and build the mesh
graph = gets_graph_circle(n = n)


kappa = sqrt(8*nu)/rho
tau = sqrt(gamma(nu) / (sigma^2 * kappa^(2*nu) * (4*pi)^(1/2) * gamma(nu + 1/2)))  #sigma = 1, d = 1
alpha = nu + 1/2

# Get the "true" covariance matrix
true_cov_mat = gets_true_cov_mat(graph = graph,
                                 kappa = kappa,
                                 nu = nu,
                                 sigma = sigma,
                                 N = N.folded,
                                 boundary = boundary)

# Get the approximate covariance matrix
op = matern.operators(alpha = alpha, 
                      kappa = kappa, 
                      tau = tau,
                      m = m, 
                      graph = graph,
                      type = type,
                      type_rational_approximation = type_rational_approximation)
appr_cov_mat = covariance_mesh(op)

# Plot the true covariance and the approximation
point  <- c(1,0)
c_cov <- cov_function_mesh(op, matrix(point,1,2))
loc <- graph$coordinates(PtE = point)
m1 <- which.min((graph$mesh$V[,1]-loc[1])^2 + (graph$mesh$V[,2]-loc[2])^2)
p = graph$plot_function(true_cov_mat[m1,], type = "plotly", line_width = 7, edge_width = 7, vertex_size =7)
p_cir <- graph$plot_function(c_cov,p = p, line_color = "red", type = "plotly",line_width = 7, edge_width = 7, vertex_size =7) %>% 
  config(mathjax = 'cdn') %>% 
  layout(font = list(family = "Palatino"),
         scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              camera = list(eye = list(x = 2.3*asppp, 
                                       y = 2.3*asppp, 
                                       z = 2.3*asppp),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
           annotations = list(
             list(
               x = 0, y = 1, z = 0,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 1 + 2/pi, z = 0,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = -0.05, y = 1.085, z = 0.95,
               text = TeX("\\bullet\\mbox{ exact}"),
               textangle = 0, ax = 60, ay = 0,
               font = list(color = "#0000C8", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = -0.05, y = 1.2, z = 0.9,
               text = TeX("\\bullet\\mbox{ approximated}"),
               textangle = 0, ax = 60, ay = 0,
               font = list(color = "red", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))))

save(p_cir, file = here::here("data_files/p_cir.Rdata"))

# Compute the errors
L_inf_error = max(abs(true_cov_mat - appr_cov_mat))
L_2_error = sqrt(as.double(t(graph$mesh$weights)%*%(true_cov_mat - appr_cov_mat)^2%*%graph$mesh$weights))
print(L_inf_error)
print(L_2_error)

4 Tadpole graph

# Function to build a tadpole graph and create a mesh
gets_graph_tadpole <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 100)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$build_mesh(h = h)
  return(graph)
}

# Function to compute the eigenfunctions 
tadpole.eig <- function(k,graph){
x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 

if(k==0){ 
  f.e1 <- rep(1,length(x1)) 
  f.e2 <- rep(1,length(x2)) 
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  f = list(phi=f1/sqrt(3)) 
  
} else {
  f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
  f.e2 <- sin(pi*k*x2/2)                  
  
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  
  if((k %% 2)==1){ 
    f = list(phi=f1/sqrt(3)) 
  } else { 
    f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
    f.e2 <- cos(pi*k*x2/2)
    f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
    f <- list(phi=f1,psi=f2/sqrt(3/2))
  }
}

return(f)
}

# Function to get the true covariance matrix
gets_true_cov_mat <- function(graph, kappa, tau, alpha, n.overkill){
  Sigma.kl <- matrix(0,nrow = dim(graph$mesh$V)[1],ncol = dim(graph$mesh$V)[1])
  for(i in 0:n.overkill){
    phi <- tadpole.eig(i,graph)$phi
    Sigma.kl <- Sigma.kl + (1/(kappa^2 + (i*pi/2)^2)^(alpha))*phi%*%t(phi)
    if(i>0 && (i %% 2)==0){ 
      psi <- tadpole.eig(i,graph)$psi
      Sigma.kl <- Sigma.kl + (1/(kappa^2 + (i*pi/2)^2)^(alpha))*psi%*%t(psi)
    }
    
  }
  Sigma.kl <- Sigma.kl/tau^2
  return(Sigma.kl)
}

# Parameters of the approximation
h = 0.003
type = "covariance"
type_rational_approximation = "chebfun"
rho = 1
m = 4
nu = 0.6

# Get the graph and build the mesh
graph = gets_graph_tadpole(h = h)


n.overkill = 10000
kappa = sqrt(8*nu)/rho
tau = sqrt(gamma(nu) / (1^2 * kappa^(2*nu) * (4*pi)^(1/2) * gamma(nu + 1/2)))  #sigma = 1, d = 1
alpha = nu + 1/2


# Get the true covariance
true_cov_mat = gets_true_cov_mat(graph = graph,
                                 kappa = kappa,
                                 tau = tau,
                                 alpha = alpha,
                                 n.overkill = n.overkill)

# Get the approximate covariance matrix
op = matern.operators(alpha = alpha, 
                      kappa = kappa, 
                      tau = tau,
                      m = m, 
                      graph = graph,
                      type = type,
                      type_rational_approximation = type_rational_approximation)

appr_cov_mat = covariance_mesh(op)


# Plot the true covariance and the approximation
point  <- c(1,1)
c_cov <- cov_function_mesh(op, matrix(point,1,2))
loc <- graph$coordinates(PtE = point)
m1 <- which.min((graph$mesh$V[,1]-loc[1])^2 + (graph$mesh$V[,2]-loc[2])^2)
p <- graph$plot_function(true_cov_mat[m1,], type = "plotly", line_width = 7, edge_width = 7, vertex_size =7) # blue is the true one
p_tad <- graph$plot_function(c_cov, p = p, line_color = "red", type = "plotly",line_width = 7, edge_width = 7, vertex_size =7) %>% 
  config(mathjax = 'cdn') %>% 
  layout(font = list(family = "Palatino"),
         scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              camera = list(eye = list(x = 2.3*asppp, 
                                       y = 2.3*asppp, 
                                       z = 2.3*asppp),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
           annotations = list(
                  list(
                     y = 0, x = 0, z = 0,
                     text = TeX("v_1"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     y = 1, x = 0, z = 0,
                     text = TeX("v_2"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     y = 0.5, x = 0, z = 0,
                     text = TeX("e_1"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
                   list(
                     y = 1+2/pi, x = 0, z = 0,
                     text = TeX("e_2"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = -0.08, y = 1.135, z = 0.65,
               text = TeX("\\bullet\\mbox{ exact}"),
               textangle = 0, ax = 60, ay = 0,
               font = list(color = "#0000C8", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = -0.08, y = 1.25, z = 0.6,
               text = TeX("\\bullet\\mbox{ approximated}"),
               textangle = 0, ax = 60, ay = 0,
               font = list(color = "red", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))))

save(p_tad, file = here::here("data_files/p_tad.Rdata"))

# Compute the errors
L_inf_error = max(abs(true_cov_mat - appr_cov_mat))
L_2_error = sqrt(as.double(t(graph$mesh$weights)%*%(true_cov_mat - appr_cov_mat)^2%*%graph$mesh$weights))
print(L_inf_error)
print(L_2_error)

5 Exact vs Approximated

load(here::here("data_files/p_int.Rdata"))
p_int

Figure 1: Exact and approximated covariance function on the interval graph.

load(here::here("data_files/p_cir.Rdata"))
p_cir

Figure 2: Exact and approximated covariance function on the circle graph.

load(here::here("data_files/p_tad.Rdata"))
p_tad

Figure 3: Exact and approximated covariance function on the tadpole graph.

6 References

grateful::cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.5.2 (R Core Team 2025a) and the following R packages: cowplot v. 1.2.0 (Wilke 2025), ggmap v. 4.0.2 (Kahle and Wickham 2013), ggpubr v. 0.6.3 (Kassambara 2026), ggtext v. 0.1.2 (Wilke and Wiernik 2022), glue v. 1.8.0 (Hester and Bryan 2024), grid v. 4.5.2 (R Core Team 2025b), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 25.11.22 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018; Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.13.0 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.50 (Xie 2014, 2015, 2025), latex2exp v. 0.9.8 (Meschiari 2026), Matrix v. 1.7.3 (Bates, Maechler, and Jagan 2025), MetricGraph v. 1.5.0.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), OpenStreetMap v. 0.4.1 (Fellows and Stotz 2025), patchwork v. 1.3.1 (Pedersen 2025), plotly v. 4.11.0 (Sievert 2020), plotrix v. 3.8.14 (J 2006), renv v. 1.1.7 (Ushey and Wickham 2026), reshape2 v. 1.4.4 (Wickham 2007), reticulate v. 1.44.1 (Ushey, Allaire, and Tang 2025), rmarkdown v. 2.30 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2025), rSPDE v. 2.5.2.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.4.0 (Wickham, Pedersen, and Seidel 2025), sf v. 1.1.0 (E. Pebesma 2018; E. Pebesma and Bivand 2023), slackr v. 3.4.0 (Kaye et al. 2025), sp v. 2.2.1 (E. J. Pebesma and Bivand 2005; Bivand, Pebesma, and Gomez-Rubio 2013), tidyverse v. 2.0.0 (Wickham et al. 2019), tikzDevice v. 0.12.6 (Sharpsteen and Bracken 2023), viridis v. 0.6.5 (Garnier et al. 2024), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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