Special devices: @imsodin is right in suggesting a GriGri.
For the method: The common trick is to have two "belayers."
One attaches the GriGri on their harness as usual.
The second person stands facing the primary belayer and pulls hand over hand on the rope (essentially pulling away from the primary belayer, through the GriGri).
The primary belayer can help by pulling slack down towards the GriGri from above.
This should let you belay most climbers quite quickly.
But, as always, if at some point there is a lot of slack between the belay device and the climber, the climber must slow down to allow the belayer to catch up.
Taking big falls on static rope with slack is dangerous for everyone.
Some background:
The two-belayer-one-GriGri method was pretty common at local and regional speed climbing competitions in the US until 2018.
It was even used at IFSC comps until 2016.
In fact, USA Climbing Rule 7.2.1 stated that all routes must be belayed from below and this was the accepted practice.
However:
Following an incident in 2018 in which a speed climber took a bad fall that was attributed to this belaying style, USA Climbing removed this rule , allowing auto-belays to be used.
Similarly, the IFSC and USA Climbing both use special speed auto-belays for speed climbing competition.
I can't definitively speak to which is safer, and I'm not sure anyone can.
Hopefully this doesn't need saying, but this should only be done indoors with express permission of the gym staff.
Note that some gyms double-wrap their ropes at the top of the climb--
don't speed climb with this setup as the friction is both annoying and creates more heat, which is bad for the ropes.
outdoors.stackexchange.com/users/18067/jhch
I'd very much recommend using a GriGri or one of the newer device with the same mechanism.
Reason being, the braking mechanism is not dependent on the position of the braking hand.
Thus you can pull in rope in whatever way you want, as long as you have the braking strand in any of your hands at any point it's safe.
This removes a lot of the stress as compared to e.g. belaying devices of the "tuber style" (ATC, Reverso, ...), where you need to make sure that your braking hand stays below the height of the device at all time.
With a fitting rope you can pull in rope really fast with a GriGri.
outdoors.stackexchange.com/users/3602/imsodin
Ask them to slow down since their speed is affecting your ability to provide a belay.
Don't deviate from your standard belay method
outdoors.stackexchange.com/users/18276/Rob Jarvis
If they're top-roping (or when you need to pull in a LOT of slack lead belaying)
you can take your hand off the load side and put both on the brake side.
Then you can pull hard and straight away from your hips while thrusting your hips backwards, which pulls in a solid 4+ ft of rope at once.
Then, you reset your hips and slide your top hand back up to the belay device, then follow with the other.
A cycle takes about half a second, and is very intuitive once you've done it a couple times.
This technique allows you to take in a lot of slack without bringing your brake hand over the belay device.
It does look kinda goofy, but it's better to look foolish than drop your partner.
I should also add, this is a little awkward to do at first, especially on lead.
Without your top hand, the belay device will want to drop down and hang at groin level unless the load side is already taut.
This makes it kinda weird to pull slack, especially during the reset section.
It's not super difficult to do
but it's worth yarding out some rope with both climbers on the ground to practice.
outdoors.stackexchange.com/users/14346/Monica Apologists Get Out
If someone is going quickly I prefer to alternate my brake hand.
This way I don't need as much time to "reset".
It also allows me to grab large amounts of slack at once.
You just need to be comfortable enough with your non-dominant hand on the brake side, which isn't an issue for most.
Steps: Pull slack with left hand and brake with right hand; right hand islow on the brake side.
(standard motion) Put left hand over right hand on the brake side, high up, just below belay device Put right hand on climber's end of rope high above belay device Repeat but opposite.
So pull slack with right hand a brake with left hand.
outdoors.stackexchange.com/users/18120/noah
I am going to reiterate and expand on a previous deleted answer from a different user.
A hip belay can be incredibly fast and is a common technique in situations where falls are unlikely.
Many/most gyms do not allow you to hip belay, but many of those gyms also do not allow you to speed climb.
Outdoors you can obviously do whatever you want.
Another alternative is to progressively walk/run further away from the wall/cliff.
For lead climbing this is dangerous because the fall will pull you back towards the cliff face and there is a high likelihood of you losing control of the rope.
For top roping the angle of pull is different and the forces are reduced.
For climbs where there is no risk of hitting an obstacle/ledge on the way down (i.e., a climb that is well suited for speed climbing) this can be a reasonably option.
Again, most gyms will frown upon this (but not necessarily more so than speed climbing).
outdoors.stackexchange.com/users/5836/StrongBad
Some of my climber friends are pretty fast climbers with a speed of about 20s for a 10m long route. The really fast ones use auto-belaying-systems seen on speed climbing competitions.   Those auto-belaying-systems are not available everywhere, so sometimes it can be done manually with a conventional belay device.   The belaying procedure can become pretty hectic and exhaustive to pull in this much rope in such a short time. Are there special techniques or portable belay devices that relieve the belaying person?
How to belay top-rope climbers who are ascending quickly?
outdoors.stackexchange.com
<climbing><belaying><top-roping>
outdoors.stackexchange.com/questions/22639
outdoors.stackexchange.com/users/18211/Andreas Schwab
Don't deviate from your standard belay method. You could ask them to slow down, if necesary. A common approach is to have two "belayers." If someone is going quickly, you could alternate your brake hand or use a hip belay.  Using a GriGri or one of the newer devices with the same mechanism is recommended, allowing the rope to be pulled in very fast. Using your standard method with two belayers, alternating your brake hand and using hip belays can help. GriGri and similar devices can be useful as well.
Using your standard method and having two belayers helps. Alternating your brake hand and a hip belay can help. GriGri and similar devices can be helpful.