Numerical error

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1 Import libraries

library(MetricGraph)
library(Matrix)
library(rSPDE)

library(dplyr)
library(tidyverse)
library(plotly)
library(ggplot2)
library(latex2exp)
library(ggtext)

library(grateful) # Cite all loaded packages
library(here) # here::here() starts from the home directory

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")
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source(here::here("functionality1.R"))

2 Interval graph

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# library calls
library(MetricGraph)
library(Matrix)
library(rSPDE)

# function 1
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. Same as in the package
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 I edited
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 2
gets_true_cov_mat = function(graph, kappa, nu, sigma, N, boundary){
  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
n_vector = c(998, 198, 98, 8)
type_vector = c("covariance", "operator")
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
rho_vector = c(0.1, 0.5, 1, 2)
m_vector = c(0,1,2,3,4,5)
nu_vector = c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01))
boundary = "neumann"
sigma = 1
N.folded = 10

Error = list()
for (s in c(1,2,3,4)) { # loop over n_vector
  Error[[as.character(n_vector[s])]] = list()
  n = n_vector[s]
  graph = gets_graph_interval(n = n)
  for (i in c(1)) { # loop over type_vector
    Error[[as.character(n_vector[s])]][[type_vector[i]]] = list()
    type = type_vector[i]
    for (j in c(1,2,3)) { # loop over type_rational_approximation_vector
      Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]] = list()
      type_rational_approximation = type_rational_approximation_vector[j]
      for (k in c(1,2,3,4)) { # loop over rho_vector
        Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]] = list()
        rho = rho_vector[k]
        for (l in 1:length(m_vector)) { # loop over m_vector
          Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]] = matrix(NA, nrow = length(nu_vector), ncol = 2)
          colnames(Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]]) = c("L_inf_error", "L_2_error")
          m = m_vector[l]
          for (r in 1:length(nu_vector)) { # loop over nu_vector
            nu = nu_vector[r]
            
            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
            
            tryCatch({
              # getting true covariance
              true_cov_mat = gets_true_cov_mat(graph = graph,
                                               kappa = kappa,
                                               nu = nu,
                                               sigma = sigma,
                                               N = N.folded,
                                               boundary = boundary)
              
              # getting 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 = op$covariance_mesh()
            
            # computing 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))
            
            Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,1] = L_inf_error
            Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,2] = L_2_error
            }, error = function(err){
              warning(paste("Error occurred at iteration n=", n, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              print(paste("Error occurred at iteration n=", n, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,1] = NA
              Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,2] = NA
            })
            print(paste("n=", n, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu))
          }
        }
      }
    }
  }
}

Error_interval_DEF = Error
save(Error_interval_DEF, file = here::here("data_files/Error_interval_DEF.RData"))

2.1 Plotting the errors

load(here::here("data_files/Error_interval_DEF.RData"))

n_vector = c("998", "198", "98", "8") 
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
nu_max = 53
nu_vector = 0.5 + c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01))[1:nu_max]

n = "998"
t = "chebfun"


dat = rbind(
data.frame(nu = rep(nu_vector,5), 
           rho = rep(0.1, times = 5*nu_max),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["1"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["2"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["3"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["4"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(0.5, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["1"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["2"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["3"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["4"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(1, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["1"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["2"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["3"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["4"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(2, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["1"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["2"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["3"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["4"]][1:nu_max,1],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

## L2

data.frame(nu = rep(nu_vector,5), 
           rho = rep(0.1, times = 5*nu_max),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["1"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["2"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["3"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["4"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.1"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(0.5, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["1"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["2"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["3"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["4"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["0.5"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(1, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["1"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["2"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["3"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["4"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["1"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(2, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["1"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["2"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["3"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["4"]][1:nu_max,2],
                     Error_interval_DEF[[n]][["covariance"]][[t]][["2"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
)
)


dat$rho = factor(dat$rho, levels = c("0.1", "0.5", "1", "2"))
dat$L = factor(dat$L, levels = c("L2", "Linf"))
levels(dat$rho) = c("0.1" = "$\\rho = 0.1$", "0.5" = "$\\rho = 0.5$", "1" = "$\\rho = 1$", "2" = "$\\rho = 2$")
levels(dat$L) = c("L2" = "$L_2(\\Gamma\\times\\Gamma)$", "Linf" = "$L_\\infty(\\Gamma\\times\\Gamma)$")


p_err_int <- ggplot(dat, aes(nu, error, colour = as.factor(m))) + 
  facet_grid(L ~ rho) +
  geom_line(size = 1.3) +
  scale_y_log10(n.breaks = 5) +
  scale_x_continuous(n.breaks = 10) +
  theme_bw() +
  theme(panel.spacing = unit(0.3, "cm"), 
        text = element_text(family = "Palatino"),
        plot.title = element_text(hjust = 0.5, size = 16),
        strip.text = element_text(size = 12),        # Panel titles
        axis.title = element_text(size = 14),        # Axis titles
        axis.text = element_text(size = 12),         # Axis text
        legend.title = element_text(size = 14),      # Legend title
        legend.text = element_text(size = 12)) +
  labs(x = "$\\alpha\\mbox{ }(\\mbox{Smoothness parameter})$", y = "$\\mbox{Covariance Error}$", color = "$m$") +
  ggtitle("$\\mbox{Interval graph}$") 

myggsave(p_err_int, width = 12, height = 6)

knitr::include_graphics(here::here("data_files/tikzpic/p_err_int.pdf"))

Figure 1: Covariance error for the interval graph.

3 Circle graph

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# library calls
library(MetricGraph)
library(Matrix)
library(rSPDE)

# function 1
gets_graph_circle <- function(n){
  r = 1/(pi)
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge <- cbind(1+r+r*cos(theta),r*sin(theta))
  edges = list(edge)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = 2)
  graph$build_mesh(n = n)
  return(graph)
}

# matern covariance function. Same as in the package
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 I edited
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 2
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
n_vector = 2*c(998, 198, 98, 8)
type_vector = c("covariance", "operator")
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
rho_vector = c(0.1, 0.5, 1, 2)
m_vector = c(0,1,2,3,4,5)
nu_vector = c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01))
boundary = "periodic"
sigma = 1
N.folded = 10

Error = list()
for (s in c(1,2,3,4)) { # loop over n_vector
  Error[[as.character(n_vector[s])]] = list()
  n = n_vector[s]
  graph = gets_graph_circle(n = n)
  for (i in c(1)) { # loop over type_vector
    Error[[as.character(n_vector[s])]][[type_vector[i]]] = list()
    type = type_vector[i]
    for (j in c(1,2,3)) { # loop over type_rational_approximation_vector
      Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]] = list()
      type_rational_approximation = type_rational_approximation_vector[j]
      for (k in c(1,2,3,4)) { # loop over rho_vector
        Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]] = list()
        rho = rho_vector[k]
        for (l in 1:length(m_vector)) { # loop over m_vector
          Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]] = matrix(NA, nrow = length(nu_vector), ncol = 2)
          colnames(Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]]) = c("L_inf_error", "L_2_error")
          m = m_vector[l]
          for (r in 1:length(nu_vector)) { # loop over nu_vector
            nu = nu_vector[r]
            
            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
            
            tryCatch({
              # getting true covariance
              true_cov_mat = gets_true_cov_mat(graph = graph,
                                               kappa = kappa,
                                               nu = nu,
                                               sigma = sigma,
                                               N = N.folded,
                                               boundary = boundary)
              
              # getting 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 = op$covariance_mesh()
            
            # computing 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))
            
            Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,1] = L_inf_error
            Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,2] = L_2_error
            }, error = function(err){
              warning(paste("Error occurred at iteration n=", n, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              print(paste("Error occurred at iteration n=", n, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,1] = NA
              Error[[as.character(n_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][r,2] = NA
            })
            print(paste("n=", n, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu))
          }
        }
      }
    }
  }
}
Error_circle_DEF = Error
save(Error_circle_DEF, file = here::here("data_files/Error_circle_DEF.RData"))

3.1 Plotting the errors

load(here::here("data_files/Error_circle_DEF.RData"))

n_vector = c("1996", "396", "196", "16") 
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
nu_max = 53
nu_vector = 0.5 + c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01))[1:nu_max]

n = "1996"
t = "chebfun"
  

dat = rbind(
data.frame(nu = rep(nu_vector,5), 
           rho = rep(0.1, times = 5*nu_max),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["1"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["2"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["3"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["4"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(0.5, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["1"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["2"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["3"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["4"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(1, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["1"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["2"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["3"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["4"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(2, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["1"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["2"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["3"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["4"]][1:nu_max,1],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

## L2

data.frame(nu = rep(nu_vector,5), 
           rho = rep(0.1, times = 5*nu_max),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["1"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["2"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["3"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["4"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.1"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(0.5, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["1"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["2"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["3"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["4"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["0.5"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(1, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["1"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["2"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["3"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["4"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["1"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(2, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["1"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["2"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["3"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["4"]][1:nu_max,2],
                     Error_circle_DEF[[n]][["covariance"]][[t]][["2"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
)
)

dat$rho = factor(dat$rho, levels = c("0.1", "0.5", "1", "2"))
dat$L = factor(dat$L, levels = c("L2", "Linf"))
levels(dat$rho) = c("0.1" = "$\\rho = 0.1$", "0.5" = "$\\rho = 0.5$", "1" = "$\\rho = 1$", "2" = "$\\rho = 2$")
levels(dat$L) = c("L2" = "$L_2(\\Gamma\\times\\Gamma)$", "Linf" = "$L_\\infty(\\Gamma\\times\\Gamma)$")


p_err_cir <- ggplot(dat, aes(nu, error, colour = as.factor(m))) + 
  facet_grid(L ~ rho) +
  geom_line(size = 1.3) +
  scale_y_log10(n.breaks = 5) +
  scale_x_continuous(n.breaks = 10) +
  theme_bw() +
  theme(panel.spacing = unit(0.3, "cm"), 
        text = element_text(family = "Palatino"),
        plot.title = element_text(hjust = 0.5, size = 16),
        strip.text = element_text(size = 12),        # Panel titles
        axis.title = element_text(size = 14),        # Axis titles
        axis.text = element_text(size = 12),         # Axis text
        legend.title = element_text(size = 14),      # Legend title
        legend.text = element_text(size = 12)) +
  labs(x = "$\\alpha\\mbox{ }(\\mbox{Smoothness parameter})$", y = "$\\mbox{Covariance Error}$", color = "$m$") +
  ggtitle("$\\mbox{Circle graph}$") 

myggsave(p_err_cir, width = 12, height = 6)

knitr::include_graphics(here::here("data_files/tikzpic/p_err_cir.pdf"))

Figure 2: Covariance error for the circle graph.

4 Tadpole graph

4.1 rho = 0.1

---

Press Show to reveal the code chunks.

---

# library calls
library(MetricGraph)
library(Matrix)
library(rSPDE)

# function 1
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 2
gets_graph <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = c(1,2))
  graph$build_mesh(h=h)
  return(graph)
}

#function 3
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
h_vector = c(0.001, 0.005, 0.01, 0.1)
type_vector = c("covariance", "operator")
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
rho_vector = c(0.1, 0.5, 1, 2)
m_vector = c(0,1,2,3,4,5)
nu_vector = c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01))
n.overkill = 1000
sigma = 1

Error = list()
for (s in c(1,2,3,4)) { # loop over h_vector
  Error[[as.character(h_vector[s])]] = list()
  h = h_vector[s]
  graph = gets_graph(h = h)
  for (i in c(1)) { # loop over type_vector
    Error[[as.character(h_vector[s])]][[type_vector[i]]] = list()
    type = type_vector[i]
    for (j in c(1,2,3)) { # loop over type_rational_approximation_vector
      Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]] = list()
      type_rational_approximation = type_rational_approximation_vector[j]
      for (k in c(1)) { # loop over rho_vector
        Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]] = list()
        rho = rho_vector[k]
        for (l in 1:length(m_vector)) { # loop over m_vector
          Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]] = matrix(NA, nrow = length(nu_vector), ncol = 2)
          colnames(Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]]) = c("L_inf_error", "L_2_error")
          m = m_vector[l]
          for (n in 1:length(nu_vector)) { # loop over nu_vector
            nu = nu_vector[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
            
            tryCatch({
            # getting true covariance
            true_cov_mat = gets_true_cov_mat(graph = graph,
                                             kappa = kappa,
                                             tau = tau,
                                             alpha = alpha,
                                             n.overkill = n.overkill)
            
            # getting 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 = op$covariance_mesh()
            
            # computing 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))
            
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = L_inf_error
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = L_2_error
            }, error = function(err){
              warning(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              print(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = NA
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = NA
            })
            print(paste("h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu))
          }
        }
      }
    }
  }
}

Error_tadpole_DEF_rho0.1 = Error
save(Error_tadpole_DEF_rho0.1, file = here::here("data_files/Error_tadpole_DEF_rho0.1.RData"))

4.2 rho = 0.5

---

Press Show to reveal the code chunks.

---

# library calls
library(MetricGraph)
library(Matrix)
library(rSPDE)

# function 1
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 2
gets_graph <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = c(1,2))
  graph$build_mesh(h=h)
  return(graph)
}

#function 3
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
h_vector = c(0.001, 0.005, 0.01, 0.1)
type_vector = c("covariance", "operator")
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
rho_vector = c(0.1, 0.5, 1, 2)
m_vector = c(0,1,2,3,4,5)
nu_vector = c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01))
n.overkill = 1000
sigma = 1

Error = list()
for (s in c(1,2,3,4)) { # loop over h_vector
  Error[[as.character(h_vector[s])]] = list()
  h = h_vector[s]
  graph = gets_graph(h = h)
  for (i in c(1)) { # loop over type_vector
    Error[[as.character(h_vector[s])]][[type_vector[i]]] = list()
    type = type_vector[i]
    for (j in c(1,2,3)) { # loop over type_rational_approximation_vector
      Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]] = list()
      type_rational_approximation = type_rational_approximation_vector[j]
      for (k in c(2)) { # loop over rho_vector
        Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]] = list()
        rho = rho_vector[k]
        for (l in 1:length(m_vector)) { # loop over m_vector
          Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]] = matrix(NA, nrow = length(nu_vector), ncol = 2)
          colnames(Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]]) = c("L_inf_error", "L_2_error")
          m = m_vector[l]
          for (n in 1:length(nu_vector)) { # loop over nu_vector
            nu = nu_vector[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
            
            tryCatch({
            # getting true covariance
            true_cov_mat = gets_true_cov_mat(graph = graph,
                                             kappa = kappa,
                                             tau = tau,
                                             alpha = alpha,
                                             n.overkill = n.overkill)
            
            # getting 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 = op$covariance_mesh()
            
            # computing 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))
            
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = L_inf_error
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = L_2_error
            }, error = function(err){
              warning(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              print(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = NA
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = NA
            })
            print(paste("h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu))
          }
        }
      }
    }
  }
}

Error_tadpole_DEF_rho0.5 = Error
save(Error_tadpole_DEF_rho0.5, file = here::here("data_files/Error_tadpole_DEF_rho0.5.RData"))

4.3 rho = 1

---

Press Show to reveal the code chunks.

---

# library calls
library(MetricGraph)
library(Matrix)
library(rSPDE)

# function 1
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 2
gets_graph <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = c(1,2))
  graph$build_mesh(h=h)
  return(graph)
}

#function 3
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
h_vector = c(0.001, 0.005, 0.01, 0.1)
type_vector = c("covariance", "operator")
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
rho_vector = c(0.1, 0.5, 1, 2)
m_vector = c(0,1,2,3,4,5)
nu_vector = c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01)) 
n.overkill = 1000
sigma = 1

Error = list()
for (s in c(1,2,3,4)) { # loop over h_vector
  Error[[as.character(h_vector[s])]] = list()
  h = h_vector[s]
  graph = gets_graph(h = h)
  for (i in c(1)) { # loop over type_vector
    Error[[as.character(h_vector[s])]][[type_vector[i]]] = list()
    type = type_vector[i]
    for (j in c(1,2,3)) { # loop over type_rational_approximation_vector
      Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]] = list()
      type_rational_approximation = type_rational_approximation_vector[j]
      for (k in c(3)) { # loop over rho_vector
        Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]] = list()
        rho = rho_vector[k]
        for (l in 1:length(m_vector)) { # loop over m_vector
          Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]] = matrix(NA, nrow = length(nu_vector), ncol = 2)
          colnames(Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]]) = c("L_inf_error", "L_2_error")
          m = m_vector[l]
          for (n in 1:length(nu_vector)) { # loop over nu_vector
            nu = nu_vector[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
            
            tryCatch({
            # getting true covariance
            true_cov_mat = gets_true_cov_mat(graph = graph,
                                             kappa = kappa,
                                             tau = tau,
                                             alpha = alpha,
                                             n.overkill = n.overkill)
            
            # getting 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 = op$covariance_mesh()
            
            # computing 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))
            
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = L_inf_error
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = L_2_error
            }, error = function(err){
              warning(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              print(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = NA
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = NA
            })
            print(paste("h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu))
          }
        }
      }
    }
  }
}

Error_tadpole_DEF_rho1 = Error
save(Error_tadpole_DEF_rho1, file = here::here("data_files/Error_tadpole_DEF_rho1.RData"))

4.4 rho = 2

---

Press Show to reveal the code chunks.

---

# library calls
library(MetricGraph)
library(Matrix)
library(rSPDE)

# function 1
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 2
gets_graph <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = c(1,2))
  graph$build_mesh(h=h)
  return(graph)
}

#function 3
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
h_vector = c(0.001, 0.005, 0.01, 0.1)
type_vector = c("covariance", "operator")
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
rho_vector = c(0.1, 0.5, 1, 2)
m_vector = c(0,1,2,3,4,5)
nu_vector = c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01)) 
n.overkill = 1000
sigma = 1

Error = list()
for (s in c(1,2,3,4)) { # loop over h_vector
  Error[[as.character(h_vector[s])]] = list()
  h = h_vector[s]
  graph = gets_graph(h = h)
  for (i in c(1)) { # loop over type_vector
    Error[[as.character(h_vector[s])]][[type_vector[i]]] = list()
    type = type_vector[i]
    for (j in c(1,2,3)) { # loop over type_rational_approximation_vector
      Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]] = list()
      type_rational_approximation = type_rational_approximation_vector[j]
      for (k in c(4)) { # loop over rho_vector
        Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]] = list()
        rho = rho_vector[k]
        for (l in 1:length(m_vector)) { # loop over m_vector
          Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]] = matrix(NA, nrow = length(nu_vector), ncol = 2)
          colnames(Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]]) = c("L_inf_error", "L_2_error")
          m = m_vector[l]
          for (n in 1:length(nu_vector)) { # loop over nu_vector
            nu = nu_vector[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
            
            tryCatch({
            # getting true covariance
            true_cov_mat = gets_true_cov_mat(graph = graph,
                                             kappa = kappa,
                                             tau = tau,
                                             alpha = alpha,
                                             n.overkill = n.overkill)
            
            # getting 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 = op$covariance_mesh()
            
            # computing 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))
            
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = L_inf_error
            Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = L_2_error
            }, error = function(err){
              warning(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              print(paste("Error occurred at iteration h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu, "Error:", conditionMessage(err)))
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,1] = NA
              Error[[as.character(h_vector[s])]][[type_vector[i]]][[type_rational_approximation_vector[j]]][[as.character(rho_vector[k])]][[as.character(m_vector[l])]][n,2] = NA
            })
            print(paste("h=", h, ",type=", type, ",type_rational_approximation=", type_rational_approximation, ",rho=", rho, ",m=", m, ",nu=", nu))
          }
        }
      }
    }
  }
}

Error_tadpole_DEF_rho2 = Error
save(Error_tadpole_DEF_rho2, file = here::here("data_files/Error_tadpole_DEF_rho2.RData"))

4.5 Plotting the errors

load(here::here("data_files/Error_tadpole_DEF_rho0.1.RData"))
load(here::here("data_files/Error_tadpole_DEF_rho0.5.RData"))
load(here::here("data_files/Error_tadpole_DEF_rho1.RData"))
load(here::here("data_files/Error_tadpole_DEF_rho2.RData"))

h_vector = c("0.001", "0.005", "0.01", "0.1")
type_rational_approximation_vector = c("chebfun", "brasil", "chebfunLB")
nu_max = 53
nu_vector = 0.5 + c(seq(0.1,0.4,by=0.05), seq(0.41,0.59,by=0.01), seq(0.6,1.4,by=0.05), seq(1.41,1.59,by=0.01), seq(1.6,2.4,by=0.05), seq(2.41,2.49,by=0.01))[1:nu_max]

n = "0.001"
t = "chebfun"


dat = rbind(
data.frame(nu = rep(nu_vector,5), 
           rho = rep(0.1, times = 5*nu_max),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["1"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["2"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["3"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["4"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(0.5, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["1"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["2"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["3"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["4"]][1:nu_max,1],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(1, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["1"]][1:nu_max,1],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["2"]][1:nu_max,1],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["3"]][1:nu_max,1],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["4"]][1:nu_max,1],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(2, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["1"]][1:nu_max,1],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["2"]][1:nu_max,1],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["3"]][1:nu_max,1],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["4"]][1:nu_max,1],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["5"]][1:nu_max,1]),
           L = rep("Linf", times = 5*nu_max)
),

## L2

data.frame(nu = rep(nu_vector,5), 
           rho = rep(0.1, times = 5*nu_max),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["1"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["2"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["3"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["4"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.1[[n]][["covariance"]][[t]][["0.1"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(0.5, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["1"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["2"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["3"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["4"]][1:nu_max,2],
                     Error_tadpole_DEF_rho0.5[[n]][["covariance"]][[t]][["0.5"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(1, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["1"]][1:nu_max,2],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["2"]][1:nu_max,2],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["3"]][1:nu_max,2],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["4"]][1:nu_max,2],
                     Error_tadpole_DEF_rho1[[n]][["covariance"]][[t]][["1"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
),

data.frame(nu = rep(nu_vector,5), 
           rho = c(rep(2, times = 5*nu_max)),
           m = rep(1:5, each = nu_max),
           error = c(Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["1"]][1:nu_max,2],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["2"]][1:nu_max,2],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["3"]][1:nu_max,2],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["4"]][1:nu_max,2],
                     Error_tadpole_DEF_rho2[[n]][["covariance"]][[t]][["2"]][["5"]][1:nu_max,2]),
           L = rep("L2", times = 5*nu_max)
)
)


dat$rho = factor(dat$rho, levels = c("0.1", "0.5", "1", "2"))
dat$L = factor(dat$L, levels = c("L2", "Linf"))
levels(dat$rho) = c("0.1" = "$\\rho = 0.1$", "0.5" = "$\\rho = 0.5$", "1" = "$\\rho = 1$", "2" = "$\\rho = 2$")
levels(dat$L) = c("L2" = "$L_2(\\Gamma\\times\\Gamma)$", "Linf" = "$L_\\infty(\\Gamma\\times\\Gamma)$")


p_err_tad <- ggplot(dat, aes(nu, error, colour = as.factor(m))) + 
  facet_grid(L ~ rho) +
  geom_line(size = 1.3) +
  scale_y_log10(n.breaks = 5) +
  scale_x_continuous(n.breaks = 10) +
  theme_bw() +
  theme(panel.spacing = unit(0.3, "cm"), 
        text = element_text(family = "Palatino"),
        plot.title = element_text(hjust = 0.5, size = 16),
        strip.text = element_text(size = 12),        # Panel titles
        axis.title = element_text(size = 14),        # Axis titles
        axis.text = element_text(size = 12),         # Axis text
        legend.title = element_text(size = 14),      # Legend title
        legend.text = element_text(size = 12)) +
  labs(x = "$\\alpha\\mbox{ }(\\mbox{Smoothness parameter})$", y = "$\\mbox{Covariance Error}$", color = "$m$") +
  ggtitle("$\\mbox{Tadpole graph}$") 

myggsave(p_err_tad, width = 12, height = 6)

knitr::include_graphics(here::here("data_files/tikzpic/p_err_tad.pdf"))

Figure 3: Covariance error for the tadpole graph.

5 References

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