Invokes a standard web request against all the sites in SharePoint. http://uglygizmo.blogspot.com.au/

Initialize-SPWeb.ps1

<#
.Synopsis
   Invokes a standard web request against all the sites in SharePoint.

.Description
   There are some dependencies to run this script as follows;

   The account used to run this script will need read access to all
   sites for the initialization to fully succeed.

   This script is required to be run on a SharePoint server or a
   machine with the Microsoft.SharePoint.PowerShell dll registered.
   
   Requires PowerShell v3.0.
#>
[CmdletBinding()]
Param
()

"Importing the PowerShell cmdlets."
$ver = $host | select version
if ($ver.Version.Major -gt 1)  {$Host.Runspace.ThreadOptions = "ReuseThread"}
Add-PsSnapin Microsoft.SharePoint.PowerShell

"Creating an Internet Explorer object for navigation."
$ie = New-Object -ComObject InternetExplorer.Application

"Enumerate the SharePoint sites."
$Webs = Get-SPWebApplication -IncludeCentralAdministration | Get-SPSite | Get-SPWeb

"Navigating to all $($Webs.Count) sites."
foreach ($Web in $Webs)
{
    "Initializing $($Web.URL)"
    try
    {
        $ie.Navigate($Web.URL)
        while ($ie.Busy)
        {
            Start-Sleep -Seconds 1
        }
        "Title: $($ie.Document.title)"
    }
    catch
    {
        Write-Host -Object ("Status: " + $Error[0].Exception.Message) -ForegroundColor Yellow
    }
}
$ie.Quit()

"SPWeb Initialization Complete."

This script has been updated to use an InternetExplorer.application COM object rather than the Invoke-WebRequest cmdlet. I found the Invoke-WebRequest was not doing a complete job at warming up the pages in SharePoint because it would not request embedded elements on the returned pages.
To see the old version, click the revisions link at the top of this page.