Convergence in 𝑚 for the optimal control variable

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# Define the captioner function
captioner <- function(caption) {
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library(MetricGraph)
library(ggplot2)
library(reshape2)
library(plotly)
library(patchwork)
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("control_functionality.Rmd"), output = here::here("control_functionality.R")),
  file = here::here("purl_log.txt")
)
source(here::here("control_functionality.R"))

# Parameters
T_final <- 2
kappa <- 4#16 # readRDS("old/kappa.RDS") #4
divider <- 4# readRDS("old/divider.RDS") # 1
mu <- 1
a <- - 1
b <- 1
N_finite = 4 # choose even
adjusted_N_finite <- N_finite + N_finite/2 + 1
# Coefficients for f and g
coeff_elliptic_g <- 20*(1:adjusted_N_finite)^-1
coeff_elliptic_g[-2] <- 0
coeff_elliptic_f <- rep(0, adjusted_N_finite)
coeff_elliptic_f[2] <- 10

# Time step and mesh size
m_vector <- c(2, 3, 4, 5, 6, 7, 8)

h_star <- 0.001

# Overkill parameters
overkill_time_step <- 0.1 * 2^-14
overkill_h <- h_star#(0.1 * 2^-14)^(1/2)


# Finest time and space mesh
overkill_time_seq <- seq(0, T_final, length.out = ((T_final - 0) / overkill_time_step + 1))
overkill_graph <- gets.graph.tadpole(h = overkill_h)

# Compute the weights on the finest mesh
overkill_graph$compute_fem() # This is needed to compute the weights
overkill_C <- overkill_graph$mesh$C

overkill_psi <- cos(overkill_time_seq)
overkill_phi <- sin(T_final - overkill_time_seq)
overkill_psi_prime <- - sin(overkill_time_seq)
overkill_phi_prime <- - cos(T_final - overkill_time_seq)

alpha_vector <- seq(1, 1.8, by = 0.2)

# Create a matrix to store the errors
errors_u_bar <- matrix(NA, nrow = length(m_vector), ncol = length(alpha_vector))
errors_p_bar <- matrix(NA, nrow = length(m_vector), ncol = length(alpha_vector))
errors_z_bar <- matrix(NA, nrow = length(m_vector), ncol = length(alpha_vector))
for (j in 1:length(alpha_vector)) {
  alpha <- alpha_vector[j] 
  beta <- alpha / 2

  # Compute the eigenvalues and eigenfunctions on the finest mesh
  overkill_eigen_params <- gets.eigen.params(N_finite = N_finite, 
                                             kappa = kappa, 
                                             alpha = alpha, 
                                             graph = overkill_graph)
  EIGENVAL_MINUS_ALPHA <- overkill_eigen_params$EIGENVAL_MINUS_ALPHA # Eigenvalues (they are independent of the meshes)
  overkill_EIGENFUN <- overkill_eigen_params$EIGENFUN # Eigenfunctions on the finest mesh
  
  # Compute the true solution on the finest mesh
  overkill_elliptic_f <- as.vector(overkill_EIGENFUN %*% coeff_elliptic_f)
  overkill_elliptic_g <- as.vector(overkill_EIGENFUN %*% coeff_elliptic_g)
  # Construct the corresponding elliptic solution u and v on the integration mesh
  overkill_elliptic_u <- as.vector(overkill_EIGENFUN %*% (coeff_elliptic_f * EIGENVAL_MINUS_ALPHA))
  overkill_elliptic_v <- as.vector(overkill_EIGENFUN %*% (coeff_elliptic_g * EIGENVAL_MINUS_ALPHA))
  overkill_u_bar <- outer(overkill_elliptic_u, overkill_psi)
  overkill_p_bar <- - mu * outer(overkill_elliptic_v, overkill_phi)
  overkill_z_bar <- matrix(pmax(a, pmin(b, - overkill_p_bar / mu)), dim(overkill_p_bar))
  
  maxabs <- max(abs(overkill_p_bar / mu))
  
  for (i in 1:length(m_vector)) {
    m <- m_vector[i]
    h <- exp(- pi * sqrt((1 - alpha / 2) * m))/divider 
    time_step <- (h^alpha)
    h <- largest_nested_h(h_star, h) # makes them nested
    time_seq <- seq(0, T_final, length.out = ((T_final - 0) / time_step + 1))
    graph <- gets.graph.tadpole(h = h)
    graph$compute_fem()
    G <- graph$mesh$G
    C <- graph$mesh$C
    L <- kappa^2*C + G
    # Construct the fractional operator, which is shared for the forward and adjoint problems
    my_op_frac <- my.fractional.operators.frac(L, 
                                               beta, 
                                               C, 
                                               scale.factor = kappa^2, 
                                               m = m, 
                                               time_step)
    eigen_params <- gets.eigen.params(N_finite = N_finite, 
                                      kappa = kappa, 
                                      alpha = alpha, 
                                      graph = graph)
    EIGENFUN <- eigen_params$EIGENFUN
    # Construct the right hand side functions f and g for the elliptic problem
    elliptic_f <- as.vector(EIGENFUN %*% coeff_elliptic_f)
    elliptic_g <- as.vector(EIGENFUN %*% coeff_elliptic_g)
    # Construct the corresponding elliptic solution u and v
    elliptic_u <- as.vector(EIGENFUN %*% (coeff_elliptic_f * EIGENVAL_MINUS_ALPHA))
    elliptic_v <- as.vector(EIGENFUN %*% (coeff_elliptic_g * EIGENVAL_MINUS_ALPHA))
    
    psi <- cos(time_seq)
    phi <- sin(T_final - time_seq)
    psi_prime <- - sin(time_seq)
    phi_prime <- - cos(T_final - time_seq)
    
    u_bar <- elliptic_u %*% t(psi)
    p_bar <- - mu * (elliptic_v %*% t(phi))
    z_bar <- matrix(pmax(a, pmin(b, - p_bar / mu)), dim(p_bar))

    # Construct the projection matrix
    Psi <- graph$fem_basis(overkill_graph$get_mesh_locations())
    R <- t(Psi) %*% overkill_C
    minus_p_bar_over_mu <- overkill_elliptic_v %*% t(phi)
    z_aux <- matrix(pmax(a, pmin(b, minus_p_bar_over_mu)), dim(minus_p_bar_over_mu))
    f <- overkill_elliptic_u %*% t(psi_prime) + overkill_elliptic_f %*% t(psi) - z_aux
    
    
    U_d <- overkill_elliptic_u %*% t(psi) -
      mu * (overkill_elliptic_v %*% t(phi_prime)) +
      mu * (overkill_elliptic_g %*% t(phi))
    
    V_d <- R %*% U_d
    u_0 <- elliptic_u
    F_proj <- R %*% f
    
    u_d <- outer(elliptic_u, psi) -
      mu * outer(elliptic_v, phi_prime) +
      mu * outer(elliptic_g, phi)
    
    
    tol <- 1e-12
    maxit <- 25
    verbose <- TRUE
    nested_spatial_mesh <- TRUE
    res <- solve_coupled_system_multi_tol(
      my_op_frac = my_op_frac,
      time_step = time_step, 
      time_seq = time_seq, 
      u_0 = u_0, 
      F_proj = F_proj, 
      Z_ini = F_proj*0,
      V_d = V_d, 
      u_d = u_d,
      Psi = Psi, 
      R = R,
      a = a, 
      b = b, 
      C = C,
      mu = mu,
      tol = tol, 
      maxit = maxit, 
      verbose = verbose,
      nested_spatial_mesh = nested_spatial_mesh,
      true_sol = list(
        u_bar = u_bar, 
        p_bar = p_bar, 
        z_bar = z_bar
      ))
    
    U_bar <- res$U
    P_bar <- res$P
    Z_bar <- res$Z
    
    projected_u_bar_piecewise <- construct_piecewise_projection(Psi %*% U_bar, time_seq, overkill_time_seq)
    projected_p_bar_piecewise <- construct_piecewise_projection(Psi %*% P_bar, time_seq, overkill_time_seq)
    projected_z_bar_piecewise <- construct_piecewise_projection(Psi %*% Z_bar, time_seq, overkill_time_seq)
    
    UU <- overkill_u_bar - projected_u_bar_piecewise
    PP <- overkill_p_bar - projected_p_bar_piecewise
    ZZ <- overkill_z_bar - projected_z_bar_piecewise
    
    rm(projected_u_bar_piecewise, projected_p_bar_piecewise, projected_z_bar_piecewise)
    
    DDD <- sqrt(as.double(overkill_time_step * sum(UU * (overkill_C %*% UU))))
    HHH <- sqrt(as.double(overkill_time_step * sum(PP * (overkill_C %*% PP))))
    KKK <- sqrt(as.double(overkill_time_step * sum(ZZ * (overkill_C %*% ZZ))))
    
    rm(UU, PP, ZZ)

    errors_u_bar[i,j] <- DDD
    errors_p_bar[i,j] <- HHH
    errors_z_bar[i,j] <- KKK
    print(paste0("m =", m, ", alpha =", alpha, ", h =", h, ", time_step =", time_step))
    slackr_msg(text = paste0("m =", m, ", alpha =", alpha, ", h =", h, ", time_step =", time_step, ", converged = ", res$converged, ", iterations = ", res$iterations, ", error u = ", DDD, ", error p = ", HHH, ", error z = ", KKK, ", maxabs = ", maxabs), channel = "#research")
  }
}
save(errors_u_bar, errors_p_bar, errors_z_bar, file = here::here("data_files/control_error_m_with_iteration.RData"))

1 Convergence results

# Load the errors data
load(here::here("data_files/control_error_m_with_iteration.RData"))

observed_rates_u_bar <- numeric(length(alpha_vector))
observed_rates_p_bar <- numeric(length(alpha_vector))
observed_rates_z_bar <- numeric(length(alpha_vector))

for (i in 1:length(alpha_vector)) {observed_rates_u_bar[i] <- coef(lm(log(errors_u_bar[, i]) ~ sqrt(m_vector)))[2]}
for (i in 1:length(alpha_vector)) {observed_rates_p_bar[i] <- coef(lm(log(errors_p_bar[, i]) ~ sqrt(m_vector)))[2]}
for (i in 1:length(alpha_vector)) {observed_rates_z_bar[i] <- coef(lm(log(errors_z_bar[, i]) ~ sqrt(m_vector)))[2]}

theoretical_rates <- - pi * alpha_vector * sqrt((1 - alpha_vector / 2))

p_u_bar <- error.convergence.plotter(x_axis_vector = m_vector, 
                               alpha_vector, 
                               errors_u_bar, 
                               theoretical_rates, 
                               observed_rates_u_bar,
                               line_equation_fun = exp_line_equation,
                               fig_title = expression(italic(bar(u))),
                               x_axis_label = expression(italic(m)),
                               apply_sqrt = TRUE)

p_p_bar <- error.convergence.plotter(x_axis_vector = m_vector, 
                               alpha_vector, 
                               errors_p_bar, 
                               theoretical_rates, 
                               observed_rates_p_bar,
                               line_equation_fun = exp_line_equation,
                               fig_title = expression(italic(bar(p))),
                               x_axis_label = expression(italic(m)),
                               apply_sqrt = TRUE)

p_z_bar <- error.convergence.plotter(x_axis_vector = m_vector, 
                               alpha_vector, 
                               errors_z_bar, 
                               theoretical_rates, 
                               observed_rates_z_bar,
                               line_equation_fun = exp_line_equation,
                               fig_title = expression(italic(bar(z))),
                               x_axis_label = expression(italic(m)),
                               apply_sqrt = TRUE)

p_m_u <- error.convergence.plotter(x_axis_vector = m_vector, 
                               alpha_vector, 
                               errors_u_bar, 
                               theoretical_rates, 
                               observed_rates_u_bar,
                               line_equation_fun = exp_line_equation,
                               fig_title = expression("Convergence in " * italic(m)),
                               x_axis_label = expression(italic(m)),
                               apply_sqrt = TRUE)
p_m_z <- error.convergence.plotter(x_axis_vector = m_vector, 
                               alpha_vector, 
                               errors_z_bar, 
                               theoretical_rates, 
                               observed_rates_z_bar,
                               line_equation_fun = exp_line_equation,
                               fig_title = expression("Convergence in " * italic(m)),
                               x_axis_label = expression(italic(m)),
                               apply_sqrt = TRUE)
save(p_m_u, file = here::here("data_files/p_m_u.RData"))
save(p_m_z, file = here::here("data_files/p_m_z.RData"))

p_all_m <- (p_u_bar | p_p_bar | p_z_bar) + 
  plot_annotation(
    title = expression("         Convergence in " * italic(m)),
    theme = theme(plot.title = element_text(size = 18, face = "bold", hjust = 0.5, 
        family = "Palatino"))
  )
p_all_m

Figure 1: Comparison of theoretical and observed convergence behavior for the \(L_2((0,T);L_2(\Gamma))\)-error with respect to \(m\) on a semi-\(\text{log}_{e}\) scale, with \(m\) plotted on a square-root scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of \(\alpha\) along with the corresponding theoretical (‘theo’), and observed (‘obs’) rates for each case.

#ggsave(here::here("data_files/control_conv_rates_m_u_bar_with_iteration.png"), width = 4, height = 5, plot = p_u_bar, dpi = 300)
#ggsave(here::here("data_files/control_conv_rates_m_p_bar_with_iteration.png"), width = 4, height = 5, plot = p_p_bar, dpi = 300)
#ggsave(here::here("data_files/control_conv_rates_m_z_bar_with_iteration.png"), width = 4, height = 5, plot = p_z_bar, dpi = 300)
ggsave(here::here("data_files/control_conv_rates_m_all_with_iteration.png"), width = 12, height = 6, plot = p_all_m, dpi = 300)

2 References

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