Clear FormAPI file field in Drupal 7

Question

I've put together a simple FormAPI form including a file field.

The user should be able to upload a file and save the form, then come back later having the option to "undo" the upload - making the previously set file field blank again.

What is the best way to allow the user to clear the field?

Note: I'm looking for a way to empty / delete the myfile variable. The actual file does not necessarily need to be deleted from the filesystem, although this would be a bonus.

function mymodule_form($form, &$form_state) {

  $form['myfile'] = array(
    '#name' => 'files[myfile]',
    '#type' => 'file',
    '#title' => t('Select a file to upload'),
    '#field_prefix' => variable_get('myfile'),  //show the filename if saved previously
  );

  $form['submit'] = array(
    '#type' => 'submit',
    '#value' => t('Save changes'),
  );

  $form['#validate'][] = 'mymodule_validate';
  $form['#submit'][] = 'system_settings_form_submit';
  $form['#theme'] = 'system_settings_form';

  return $form;
}

function mymodule_validate(&$form, &$form_state) {

  if ($tmp_file = file_save_upload('myfile')) {
    $destination = 'public://' . uniqid() . '_' . $tmp_file->filename;

    if ( $file = file_copy($tmp_file, $destination, FILE_EXISTS_RENAME) ) {
      $file->status = FILE_STATUS_PERMANENT;
      file_save($file);
      $form_state['values']['myfile'] = file_create_url($file->uri);
    } else {
      form_set_error('myfile', t("Failed to upload."));
    }
  }
}