diff --git a/R/point-estimation.R b/R/point-estimation.R
index cffdcab..30e753f 100644
--- a/R/point-estimation.R
+++ b/R/point-estimation.R
@@ -6,12 +6,25 @@ compute_point_estimate <- function(pf,
   nparams <- pf$nparams
 
   if (!is.null(guess)) {
-    # uses user default cluster
-    opt <- optimParallel::optimParallel(
-      par = guess,
-      fn = pf$get_value,
-      control = list(fnscale = -1)
-    )
+    # if user did not set a default cluster we compute in sequential
+    if (is.null(parallel::getDefaultCluster())) {
+      cli::cli_alert_warning("Default cluster has not been set. You may consider setting one to run computation in parallel.")
+      opt <- stats::optim(
+        par = guess,
+        fn = pf$get_value,
+        method = "L-BFGS-B",
+        control = list(fnscale = -1)
+      )
+    }
+    else {
+      # uses user default cluster to compute in parallel
+      opt <- optimParallel::optimParallel(
+        par = guess,
+        fn = pf$get_value,
+        control = list(fnscale = -1),
+        parallel = list(cl = NULL, forward = FALSE, loginfo = FALSE)
+      )
+    }
     x0 <- opt$par
     fval <- opt$value
   } else {
@@ -21,6 +34,7 @@ compute_point_estimate <- function(pf,
     if (length(upper_bound) != nparams)
       abort("The number of provided upper bounds does not match the number of parameters.")
 
+    # if user default cluser is not set, will run in sequential, else run in parallel
     opt <- rgenoud::genoud(
       fn = pf$get_value,
       nvars = nparams,
@@ -31,7 +45,7 @@ compute_point_estimate <- function(pf,
       wait.generations = 2 * nparams + 1,
       BFGSburnin = 2 * nparams + 1,
       print.level = 0,
-      cluster = parallel::getDefaultCluster(),
+      cluster = if (!is.null(parallel::getDefaultCluster())) parallel::getDefaultCluster() else FALSE,
       balance = nparams > 2
     )
     opt <- compute_point_estimate(
diff --git a/R/sysdata.rda b/R/sysdata.rda
index 2270f22..c442eee 100644
Binary files a/R/sysdata.rda and b/R/sysdata.rda differ
diff --git a/data-raw/df_parallelization.rds b/data-raw/df_parallelization.rds
index 0c46cc2..9d25136 100644
Binary files a/data-raw/df_parallelization.rds and b/data-raw/df_parallelization.rds differ
diff --git a/data-raw/parallelization-vignette.R b/data-raw/parallelization-vignette.R
index 79c04ce..7e09d2d 100644
--- a/data-raw/parallelization-vignette.R
+++ b/data-raw/parallelization-vignette.R
@@ -8,8 +8,8 @@ library(progressr)
 library(tictoc)
 library(flipr)
 
-ngrid_in <- 50L
-nperms <- 5000
+ngrid_in <- 100L
+nperms <- 2000
 n1 <- 10
 set.seed(1234)
 x <- rnorm(n1, mean = 1, sd = 1)
@@ -25,16 +25,22 @@ stat_assignments <- list(delta = 1)
 
 # Inference on the mean without parallelization --------------------------------
 
+plan(sequential)
+setDefaultCluster(NULL)
+progressr::handlers(global = FALSE)
+
 pf <- PlausibilityFunction$new(
   null_spec = null_spec,
   stat_functions = stat_functions,
   stat_assignments = stat_assignments,
   x, y
 )
+pf$set_nperms(nperms)
 
-pf$set_point_estimate(mean(y) - mean(x))
+tic()
+pf$set_point_estimate()
+time_without_parallel <- toc()$callback_msg
 
-pf$set_nperms(nperms)
 pf$set_parameter_bounds(
   point_estimate = pf$point_estimate,
   conf_level = pf$max_conf_level
@@ -47,11 +53,11 @@ pf$set_grid(
 
 tic()
 pf$evaluate_grid(grid = pf$grid)
-time_without_parallelization <- toc()$callback_msg
+time_without_future <- toc()$callback_msg
 
 # Inference on the mean with parallelization -----------------------------------
 
-ncores <- 4
+ncores <- 3
 plan(multisession, workers = ncores)
 cl <- makeCluster(ncores)
 setDefaultCluster(cl)
@@ -63,10 +69,12 @@ pf <- PlausibilityFunction$new(
   stat_assignments = stat_assignments,
   x, y
 )
+pf$set_nperms(nperms)
 
-pf$set_point_estimate(mean(y) - mean(x))
+tic()
+pf$set_point_estimate()
+time_with_parallel <- toc()$callback_msg
 
-pf$set_nperms(nperms)
 pf$set_parameter_bounds(
   point_estimate = pf$point_estimate,
   conf_level = pf$max_conf_level
@@ -79,12 +87,16 @@ pf$set_grid(
 
 tic()
 pf$evaluate_grid(grid = pf$grid)
-time_with_parallelization <- toc()$callback_msg
+time_with_future <- toc()$callback_msg
+
+stopCluster(cl)
 
 df_parallelization <- list(
   delta = pf$grid$delta,
-  time_par = time_with_parallelization,
-  time_without_par = time_without_parallelization
+  time_without_parallel = time_without_parallel,
+  time_without_future = time_without_future,
+  time_with_parallel = time_with_parallel,
+  time_with_future = time_with_future
 )
 
 saveRDS(df_parallelization, "data-raw/df_parallelization.rds")
diff --git a/vignettes/parallelization.Rmd b/vignettes/parallelization.Rmd
index 4b52fc3..189d482 100644
--- a/vignettes/parallelization.Rmd
+++ b/vignettes/parallelization.Rmd
@@ -14,8 +14,10 @@ knitr::opts_chunk$set(
 )
 library(flipr)
 load("../R/sysdata.rda")
-time_without_parallelization <- df_parallelization$time_without_par
-time_with_parallelization <- df_parallelization$time_par
+time_without_parallel <- df_parallelization$time_without_parallel
+time_without_future <- df_parallelization$time_without_future
+time_with_parallel <- df_parallelization$time_with_parallel
+time_with_future <- df_parallelization$time_with_future
 ```
 
 The [**flipr**](https://permaverse.github.io/flipr/) package uses functions
@@ -25,21 +27,21 @@ user side. We illustrate here how to achieve asynchronous evaluation. We use the
 [**future**](https://future.futureverse.org/index.html) package to set the plan,
 the **parallel** package to define a default cluster, and the
 [**progressr**](https://progressr.futureverse.org/index.html) package to report
-progress updates.
+progress updates. 
 
-By setting the desired number of cores, we define the number of background R
-sessions that will be used to evaluate expressions in parallel. This number is
-used to set the multisession plan with the function `future::plan()` and to
-define a default cluster with `parallel::setDefaultCluster()`. Then, to enable
-the visualization of evaluation progress, we can put the code in the
-`progressr::with_progress()` function, or more simply set it for all the
-following code with the `progressr::handlers()` function. After these settings,
-[**flipr**](https://permaverse.github.io/flipr/) functions can be used, as shown
-in this example.
+More precisely, setting a default cluster with **parallel** is useful to allow 
+parallel computation of the point estimate if an estimation is needed. On the 
+other side, setting the plan with 
+[**future**](https://future.futureverse.org/index.html) allows parallel 
+computation when evaluating the plausibility function. A comparison of
+computation time with sequential and parallel computation for those two cases is
+done in the following.
+
+## Computation without parallel processing
 
 To show the benefit of parallel processing, we compare here the processing times
-necessary to evaluate a grid with a plausibility function. First, here is the
-computation without parallelization.
+necessary to compute a point estimation and to evaluate the grid for a 
+plausibility function. First, here is the computation without parallelization.
 
 ```{r, eval=FALSE}
 set.seed(1234)
@@ -58,24 +60,34 @@ pf <- PlausibilityFunction$new(
   stat_assignments = stat_assignments,
   x, y
 )
+pf$set_nperms(2000)
+
+tic()
+pf$set_point_estimate()
+time_without_parallel <- toc()$callback_msg
+```
+
+```{r}
+time_without_parallel
+```
 
-pf$set_point_estimate(mean(y) - mean(x), overwrite = TRUE)
+```{r, eval=FALSE}
 pf$set_parameter_bounds(
   point_estimate = pf$point_estimate, 
   conf_level = pf$max_conf_level
 )
 pf$set_grid(
   parameters = pf$parameters, 
-  npoints = 50L
+  npoints = 100L
 )
 
 tictoc::tic()
 pf$evaluate_grid(grid = pf$grid)
-time_without_parallelization <- tictoc::toc()
+time_without_future <- tictoc::toc()$callback_msg
 ```
 
 ```{r}
-time_without_parallelization 
+time_without_future
 ```
 
 ## Computation with parallel processing
@@ -91,8 +103,8 @@ following code with the `progressr::handlers()` function. After these settings,
 in this example.
 
 ```{r, eval=FALSE}
-ncores <- 4
-future::plan(multisession, workers = ncores)
+ncores <- 3
+future::plan(future::multisession, workers = ncores)
 cl <- parallel::makeCluster(ncores)
 parallel::setDefaultCluster(cl)
 progressr::handlers(global = TRUE)
@@ -113,20 +125,30 @@ pf <- PlausibilityFunction$new(
   stat_assignments = stat_assignments,
   x, y
 )
+pf$set_nperms(2000)
 
-pf$set_point_estimate(mean(y) - mean(x), overwrite = TRUE)
+tic()
+pf$set_point_estimate()
+time_with_parallel <- toc()$callback_msg
+```
+
+```{r}
+time_with_parallel
+```
+
+```{r, eval=FALSE}
 pf$set_parameter_bounds(
   point_estimate = pf$point_estimate, 
   conf_level = pf$max_conf_level
 )
 pf$set_grid(
   parameters = pf$parameters, 
-  npoints = 50L
+  npoints = 100L
 )
 
 tictoc::tic()
 pf$evaluate_grid(grid = pf$grid)
-time_with_parallelization <- tictoc::toc()
+time_with_future <- tictoc::toc()$callback_msg
 
 parallel::stopCluster(cl)
 ```
@@ -135,18 +157,19 @@ It is good practice to shut down the workers with the `parallel::stopCluster()`
 function at the end of the code.
 
 ```{r}
-time_with_parallelization
+time_with_future
 ```
 
 This experiment proves that we can save a lot of computation time when using
-parallel processing, as we gained approximately 33 seconds in this example to
-evaluate the plausibility function.
+parallel processing. Indeed, by setting 3 cores for each of the parallel 
+processing tools, we reduced approximately by 3 times the computation time for 
+both computing a point estimation and evaluating the plausibility function.
 
 Finally, to return to a sequential plan with no progress updates, the following
-code can be used.
+code can be used. It also allows to shut down the workers used in [**future**](https://future.futureverse.org/index.html).
 
 ```{r, eval=FALSE}
-future::plan(sequential)
+future::plan(future::sequential)
 parallel::setDefaultCluster(NULL)
 progressr::handlers(global = FALSE)
 ```