Can I refresh the thumbnails programmatically?

You may want to look at the plugin Regenerate Thumbnails by Viper007Bond.

Basically, this is how to do it:

function regenerateThumbnails() {

    $images = $wpdb->get_results( "SELECT ID FROM $wpdb->posts WHERE post_type="attachment" AND post_mime_type LIKE 'image/%'" );

    foreach ( $images as $image ) {
        $id = $image->ID;
        $fullsizepath = get_attached_file( $id );

        if ( false === $fullsizepath || !file_exists($fullsizepath) )
            return;

        if ( wp_update_attachment_metadata( $id, wp_generate_attachment_metadata( $id, $fullsizepath ) ) )
            return true;
        else
            return false;
    }
}

Note: This function is not very scalable. It will loop through all the images and regenerate thumbnails one by one, which may consume a large amount of memory. So, you may want to enhance it.