Can a death row inmate expedite his/her execution?

I was watching a tv show about a killer who was caught and sentenced to death. I then began to wonder what would happen if the killer felt remorse and pain for what he did and didn't want to burden the system with court cases and such but simply asked to be executed within a short amount of time, would we allow it? Or would he simply rot in jail for 20 years racking up a bill before they gave him the injection? Has this ever happened in the U.S. Penal system?