This is somewhat late to the party, but three years later I still haven’t seen this argument made anywhere else, so here goes.
Many critics have noted that Daniel Day-Lewis‘s performance in There Will Be Blood (2007) drew heavily from his fellow Irishman John Huston‘s turn in Chinatown (1974). See, for instance, here, here, here, and here. Or just compare for yourself:
…But that is only one level of mimicry. Paul Thomas Anderson’s feature itself is loosely based, structurally, on Stanley Kubrick’s 2001: A Space Odyssey (1968).
Now, I’m not claiming that Anderson consciously aped Kubrick’s masterpiece. 
1. The Dawn of Daniel Plainview
2001 opens with a lengthy dialogue-free section, “The Dawn of Man,” during which we witness a tribe of timid, herbivorous apes encounter a large black monolith. This encounter pushes the apes forward in their evolution, prompting them toward tool use and hunting. (My reading of this section is that encountering the very unnatural monolith in the middle of nature reveals to the apes the concept of artificiality.)
There Will Be Blood similarly opens (albeit in 1902) with a lengthy dialogue-free section, during which we watch the film’s protagonist, Daniel Plainview, discover the technology that will fuel the entirety of the film (in this case, Oil!).
The prologues of both films feature numerous shots of the sun rising and setting over arid, barren landscapes, and haunting, drone-based music:
2001: György Ligeti: Requiem (“Kyrie” section)
TWBB: Johnny Greenwood: Popcorn Superhet Receiver (excerpts):
On this point, at least one critic has noted:
[T]he starkly ominous orchestral score, by Radiohead guitarist Jonny Greenwood, […] at no point seems conventionally to gel with the action: Blood is as daringly counter-intuitive as Stanley Kubrick was in his use of composers such as Penderecki in The Shining. Indeed, having made several Altman-esque works, Anderson has now made one that in some ways feels as close to a posthumous Kubrick epic as is imaginable. “We’re all children of Kubrick, aren’t we?” Anderson admits over the phone. “Is there anything you can do that he hasn’t done?”
Another critic made this connection, regarding the respective scores, more explicitly:
[ENTERTAINMENT WEEKLY:] Paul, you have a dedication at the end of this movie to one of your heroes, Robert Altman. But this is one of your least Altmanesque films. A lot of it is one character out in the desert, with long silences suddenly giving way to screeching strings. It reminded me of 2001: A Space Odyssey, where Stanley Kubrick had the silence of space and then suddenly ”The Blue Danube” or one of the more dissonant pieces he used.ANDERSON: Well, it’s so hard to do anything that doesn’t owe some kind of debt to what Stanley Kubrick did with music in movies. Inevitably, you’re going to end up doing something that he’s probably already done before. It can all seem like we’re falling behind whatever he came up with. ”Singin’ in the Rain” in Clockwork Orange — that was the first time I became so aware of music in movies. So no matter how hard you try to do something new, you’re always following behind. The whole opening 20 minutes was meant to be silent. I always had a dream about trying to make a movie that had no dialogue in it, that was just music and pictures. I still haven’t done it yet, but I tried to get close in the beginning.
(Anderson met Kubrick on the set of Eyes Wide Shut, introduced by their mutual acquaintance Tom Cruise. And for another comparison of Anderson’s work with Kubrick’s, check out this analysis of some of the inspirations behind Punch-Drunk Love.)
2. Sunday Magnetic Anomaly One
2001 then jumps millennia ahead (the largest time elision in cinema), to show us how humans have developed into a space-faring people. Dr. Heywood Floyd leads a team of scientists out onto the surface of the moon, where another black monolith has been excavated. The monolith emits a piercing tone, deafening the team.
There Will Be Blood similarly jumps ahead (albeit only nine years) into a dialogue-based section. Daniel Plainview and his adopted son H.W. travel about buying up land so they can begin oil drilling (excavation). Plainvew’s empire starts to grow. Toward the end of this section, an accidental explosion deafens H.W.
3. Drilling Mission
2001 next picks up 18 months later. We join doctors David Bowman and Frank Poole en route to Jupiter, accompanied by other scientists (in suspended hibernation) and the sophisticated HAL 9000 computer. HAL, reasoning that his human companions are incapable of accomplishing their mission, murders all but Dave, who in turn disconnects HAL.
There Will Be Blood resumes at some indeterminate time after the drilling accident. Plainview encounters his half-brother, Henry, and much of this section involves the growing tensions between him, Plainview, and H.W. Plainview (after sending H.W. away) eventually discovers that Henry is an impostor, and murders him.
4. Milkshake and Beyond the Infinite
2001: Arriving alone at Jupiter, Dave travels through the Star Gate, eventually ending up in an ornately decorated bedroom. There he ages rapidly, becoming a decrepit old man. Lying on his deathbed, he encounters another black monolith, which allows him to reborn as the Star-Child.
There Will Be Blood jumps from 1911–2 to 1927. Daniel Plainview is now an alcoholic old man, living a solitary life in an ornate mansion. (As in 2001, a jarring effect is achieved by the drastic change in time and location.) This section of the film presents two main plot points: Plainview discredits and disowns H.W., sending him away; he then settles a personal score with his longtime nemesis, the preacher Eli Sunday, whom he murders with a bowling pin.
Again, I don’t know whether Paul Thomas Anderson thought consciously about Kubrick while adapting Upton Sinclair’s source novel to the screen, but I think the influence is clearly there.
I would also argue that the comparison enriches Anderson’s film, and not only because it shows off his awareness of great cinema. (I’m not a huge fan of his work, but I like There Will Be Blood well enough, and think that it benefits from less overtly imitating other films the way that Anderson’s previous films tended to.) And it’s smart to steal overarching structures. Anderson surely faced many challenges in only partially and loosely adapting Sinclair’s massive epic. It makes sense that he would base his film on a different, reliable structure.
Furthermore, whereas 2001 is focused on humanity’s millennia-long improvement (not for nothing is Thus Spake Zarathustra repeatedly invoked)—celebrating how the development of advanced technology may prove our species’ eventual salvation—There Will Be Blood takes a much smaller, more pessimistic approach. Daniel Plainview is the antithesis of 2001‘s naive but earnest scientists—HAL made flesh, perhaps (but lacking in even HAL’s humanity). Like Milton’s Lucifer, who desperately mines the earth in his struggle against Heaven, the advancement of technology enables Plainview only to cheat, murder, and despoil the landscape. As he confesses to Henry:
I have a competition in me. I want no one else to succeed. I hate most people. […] There are times when I look at people and I see nothing worth liking. I want to earn enough money that I can get away from everyone.
(It’s no coincidence that Anderson was compelled to adapt Oil! in the midst of the Iraq War. And he was sadly very right to have his protagonist bellow in triumph, at the very end of the film: “I am the Third Revelation! I am who the Lord has chosen!”)
There Will Be Blood, then, held in this plain view, becomes a dull, darkening mirror to 2001‘s ennobling, burning star core.
(Happy re-viewing!)