List all posts from past week grouped by sub-category

I would recommend that you use WP_Query for this one:

$categories =  get_categories( 'child_of=83' );  
foreach ( $categories as $category ) {
    echo '<h3>' . $category -> name . '</h3>';
    echo '<ul>';

    // create a WP_Query that retreives all posts from the specified
    // category which is older then 1 week
    $args = array(
      'cat' => $category->cat_ID,
      'date_query' => array(
         array(
          'column' => 'post_date',
          'after' => '1 week ago'
         )
       ),
      'posts_per_page' => -1,
      'post_type' => 'post',
      'post_status' => 'publish'
    );
    $query = new WP_Query($args);

    // check so there is some posts in the resultset
    if($query->have_posts()) {
      // loop through the result
      while($query->have_posts()) {
        $query->the_post();
        // output data here
      }
    }

    echo '</ul>';
}