I met a friend at Union Station when I was hopping off the Sunset Limited and boarding the Coastal Starlight. I was very impressed with how the station looked like a classic building. It was, and is, a work of art. Some of the Amtrak line-ups for trains could be improved but the station itself was great. One huge improvement would be if they could find a way to increase the speed limits on the last 6 or 8 miles outside Union Station. I made the mistake of thinking that 8 miles would take 15 minutes. Just like New Orleans, Amtrak has a huge problem in LA with speed limits. I talked to regulars on both trains and they said both trains always slow way down, in NO around 12 miles from the station and in LA around 6 or 8 miles from the station. The frustrating thing is that the train does 20 mph when going even 45 mpg would make a huge difference in time spent. I know the speed limits probably won't change, but it makes little difference if you are doing 95 mph for 10% of the trip instead of 79 mph when you spend half an hour or more covering the last 6 miles.
This is partly due to ridiculously tight curves and a roundabout route into the station. Look at Google maps and trace out the approach path from the east. The Sunset Limited and Southwest Chief are coming north up the LA River, then northeast, and then have to do a U-turn north of Union Station to enter the station.
There's also a sharp turn at the LA river which causes a slowdown. There's not much point in speeding up again for the intervening 3 miles between that and the U-turn. East of the LA river, there's a freight yard, which may cause a slowdown as well, but I think they're planning to speed that bit up.
There are plans to build "run-through tracks", which will allow trains to enter Union Station from the south, cutting off about a mile of U-turn track. This is mainly for the benefit of Metrolink and the Pacific Surfliner -- being shorter trips, the time savings matters more -- but it might take time off the Sunset Limited and Southwest Chief too.