You might have read about OVDI or OVDP process but IRS introduced this new process called: Streamlined Filing Compliance Procedures
This was kind of introduced last year but was for US citizens residing abroad. This new process now includes permanent residents i.e. Green card holders as well.
Here are the quick links for your reference.
If you are already under OVDI/OVDP program check these FAQs: Transition Rules
I couldn’t find much time to read all through and do much analysis. I will update my thoughts as I understand it better but feel free to post your questions and comments.
Hope this helps.