Is there a canonical source explaining the cause of Anakin's recurrent dream about Padme's death? I don't know if Anakin/Vader had foresight abilities (according to Legends); certainly he had no such powers, if we are based just upon the movies. Because of that I am considering the possibility that Darth Sidious was tampering with Anakin's mind, for obvious purposes.
This doesn't really need a canon Force-based explanation. Basic human psych - Anakin had a major fear of losing people he cared about - resulting from losing Qui-Gon in childhood, to his mother in Ep2. Never mind the extra PTSD from Clone Wars combat.
This means that he was afraid of losing Padme even regardless of Force precognition. So having a nightmare on the topic is perfectly typical of a human in such a situation.
Add on to that the fact that Anakin DID have canonical precognitive abilities - from micro (his driving in Pod races) to macro (sensing his mother's danger). For the latter, from the ROTS script:
ANAKIN: It was a dream.
ANAKIN: Like the ones I used to have about my mother just before she died.
I'm not aware of any canon notion that the nightmares were influenced by Sidious, and therefore, given that from the above info we can show perfectly valid reasons for them outside Sidious, Occam's Razor dictates that this specific item wasn't a nefarious Sith plot.
From the Wookiepedia: For the most part, the Expanded Universe is considered canonical by Lucasfilm Ltd.—though subordinate to film canon. However, the issue is one of the most hotly debated topics among fans. link
So the films take precedence: Anakin has the ability to see the future (precognition) like most Jedi, he saw Padme's death and Palpatine used that (and his anger and fear) to manipulate Anakin.
Was Palpatine using mental domination on Ani? I don't think so, it looked like plain old manipulation in the movies.
To confirm this, you would probably have to read the novelization of the movies to see if Palpatine was using his Sith mental domination on Ani.
In Star Wars: Revenge of the Sith, Palpatine presumably laid the nightmares in Anakin's mind, so as Supreme Leader Snoke somehow used the force to bring Kylo and Rey together and maybe he was a reincarnation of Darth Plagues who was Palpatine's Mentor and Palpatine stated
I learned everything from my master
So maybe Palpatine learned how to mess with minds as well. Which means he could have messed with young Anakin's mind and could have been responsible for Anakin's nightmares, which soon lead to the birth of Darth Vader.
On a side note, Palpatine could have taken the life from Padme to build Anakin's anger. Right before Padme was about to deliver two children the medic said
She is perfectly healthy but we're losing her for reasons we can't explain.
This could explain Palpatine's method of saving the people he loved from dying, he could have not mentioned to Anakin that to save someone from dying you have to put the life force from another struggling life into that persons body. Also Palpatine supposedly said that Anakin killed Padme in his anger, but how else would Palpatine have known that Padame was dying unless he was the one who was responsible for it?