You may have answered your own question. Noise absorption tiles may
work. They are effective in commercial applications, but do your
homework first to learn what is available in the current market. Many
are not particularly attractive, but upgraded styles are available. I
am also not sure if velcro is the answer. You may consider doing a
small sample installation first because the elephants living above may
jog them loose. It would help tremendously if they had wall-to-wall
carpeting with thick padding and/or you had a sympathetic land lord,
but alas...Good luck.