The REU program in Computational and Applied Math (including the NSF funded UCLA REU site in Mathematical Modeling) at UCLA provides funded 8 week summer research opportunities for students to work on frontier level problems in Applied Math with UCLA faculty. Applications for the 2018 Computational and Applied Math REU are now open. Apply through mathprograms.org at this URL.
You can read about some of the projects being offered this summer here: https://www.marcusroper.org/2018/
There is also a FAQ page to answer some common questions about the program: https://www.marcusroper.org/faq