I've come up with about 20 different suggestions on how to do it, they either:
- Are so ludicrous not even DR Evil would attempt them (A giant electromagnet in space to pull the core towards to crust). Or
- Would destroy humanity on their own (eg 3 nukes per volcano!). Or
- Probably wont work (Eg Find asteroid made of dry ice. Crash into erupting volcano at high speed. Instant carbonated magma planet wide)
However there is one that I think will work:
In 2003 the Air Force studied a space weapon, they decided it wasn't that practical for anything they needed. But for what you need, I reckon it'll do the trick. With a little extra refinements in version 2.
Basically launch a spacecraft with 1500 of these as high as possible. At least 4000km up.
At carefully timed positions, release each rod with a slight push, so that it gradually shifts into the correct attack position for each volcano.
Then, simultaneously, each ones rocket motor fires and it hurtles towards its chosen volcano.
The 2003 airforce study estimated with a launch from LEO and only a few seconds of thrust they'd impact at Mach 10. Since this is humanities last ever project, we can afford to go all out on it. We put them much higher than LEO, like 10 times as high. We've put a bigger rocket motor on them, and they've a lot more atmosphere-free gravity to help with the acceleration. Ideally the rocket motor should burn all the way to impact. Basically you're slamming 9 tonnes into the earth at hypersonic speeds.
There's some handwaving going on here, and problems conveniently overlooked, but a rocket motor providing ~10g of acceleration for 5 minutes from 4000km should burn all the way to the ground. It'll impact at ~28km/s (mach 80).
The DU towards the front (but behind tracking and heatshield) is to help the rod penetrate through the rock. The DU on a standard bomb can get it through meters of reinforced concrete, and they don't travel at hypersonic speeds.
Basically this is equivalent to a small nuke, with all its energy targeted straight down into a 30cm circle, with one of the densest materials known on the front.
If you can target the magma chamber - I reckon this'll cut a nice small channel to it, forcing a small erruption and weaken the structure, forcing a bigger eruption.
There is a lot of handwaving going on here, I recall seeing that the penetration depth formula broke down at hypersonic speeds, but I can't find that reference. I'm assuming its true. If not - put handwavium on the front instead of DU (it's a newly discovered alloy that's 50 times as dense as DU!), (or make it longer) But I think this is your best bet for near future tech.