Title to be determined - GPU Technology Conferenceon-demand.gputechconf.com/gtc/2016/presentation/s6651-mauro-calde… · What we want to do → Quantum Transport: electrons and structures

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Mauro Calderara, Sascha Brück, Mathieu Luisier

Using Today’s Fastest Chips to Design the Chips of

Tomorrow

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What we want to do

How we do it

Apr 08 2016 Mauro Calderara 2

Overview

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What we want to do → Quantum Transport: electrons and structures

How we do it → How GPUs saved the day


Overview

| | Apr 08 2016 Mauro Calderara 4

Probably you’re familiar with this


Zooming in


The future?

(link to video: http://iis.ee.ethz.ch/~mauro/movie_SC15.avi)


From a somewhat more abstract POV

Device



Device

? e



Device

? e e



Device e

? e e



Device e

e

e e

? e e

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How do electrons behave w.r.t the

device?


This is what we’re ultimately interested in!

Device

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device?

Change in parameters → change in

behavior?



Device

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device?


behavior?



Device

e

e

e

e

e e

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device?


behavior?



Device

e

e

e

e

e e

Gate voltage

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device?


behavior?



Device

e

e

e

e

e e

Gate voltage

Dimensions

Material

properties

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device?


behavior?

Applies not just to transistors

Batteries

Storage devices

...



Device

e

e

e

e

e e

Gate voltage

Dimensions

Material

properties


How would we do that? The ‘‘easy’’ case:


How would we do that? The ‘‘easy’’ case:

→ device behaves like bulk material


How would we do that? The ‘‘difficult’’ case:


How would we do that? The ‘‘difficult’’ case:

→ device behaves like atomic structure


The cost of going small

Why is this ‘‘easy’’ ... ... and this ‘‘difficult’’?



Can assume is ‘‘infinite’’ and

use semi empirical model.

Very finite! Need to do

it from first principles.





































run

tim

e

run

tim

e

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run

tim

e

run

tim

e



Semi-empirical → O(Hours) First principles → O(Months)

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run

tim

e

run

tim

e




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run

tim

e

run

tim

e




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What we want to do → Quantum Transport: electrons and structures

How we do it → How GPUs saved the day


Overview


Where does all that time go?

run

tim

e

~ 40x



run

tim

e

~ 40x

Solve an eigenvalue

problem (not discussed

here).



run

tim

e

~ 40x Invert the matrix from

before (selectively!) using

a recursive algorithm.

Solve an eigenvalue

problem (not discussed

here).

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Instead of trying to invert selectively,

solve system using generic sparse

solver package


Avoiding the inversion, use a sparse solver instead ru

nti

me

~ 40x

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solver package

Gain: speed, parallelism, capacity for

somewhat larger systems



nti

me

~ 40x

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solver package



Cost: code now mem-bw bound

And: not such a good fit for GPUs ...



nti

me

~ 40x

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solver package



Cost: code now mem-bw bound

And: not such a good fit for GPUs ...



nti

me

~ 40x

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run

tim

e

We’ve been able to solve that one


Tackling the eigenvalue problem

run

tim

e

~ 200x

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Good speedup so far

(now: O(Days), still not

quite there...)


Now what?

run

tim

e

~ 70x overall

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Good speedup so far


quite there...)

But


Now what?

run

tim

e

~ 70x overall

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Good speedup so far


quite there...)

But


Now what?

run

tim

e

~ 70x overall

Mem-BW bound by sparse solver

?

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Good speedup so far


quite there...)

But


Now what?

run

tim

e

~ 70x overall


?

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Good speedup so far


quite there...)

But


Now what?

run

tim

e

~ 70x overall


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Good speedup so far


quite there...)

But


Now what?

run

tim

e

~ 70x overall


? Advisor PhD student

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Inverting sparse system not feasible


A Sparse Solver for Transport Problems running on GPUs

-1

=

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In our case: also not neccessary



-1

=

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Need first and last block rows only



-1

=

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Need first and last block rows only

If we can compute this fast, we can

interleave the solving step with the BC

computation

obtain the full solution very efficiently



-1

=

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Recursive algorithm based on the

Schwinger-Dyson equation


Obtaining the first and last block columns of the inverse

for i = N:1 𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1𝑋𝑖+1) \ 𝐴𝑖,𝑖−1

for i = 2:N

𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1

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for i = 2:N


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for i = 2:N


N

N+1

N-1

N-2

𝑋 𝐴

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for i = 2:N


N

N+1

N-1

N-2

𝑋 𝐴

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xGEMM + xGESV + xGEMM




for i = 2:N


N

N+1

N-1

N-2

𝑋 𝐴

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Very fast on accelerators




for i = 2:N


N

N+1

N-1

N-2

𝑋 𝐴

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Very fast on accelerators

Parallelizable




for i = 2:N


N

N+1

N-1

N-2

𝑋 𝐴

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Runs on GPUs, compute bound



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Arithmetic Intensity [log(FLOPS/Byte)]

Performance [log(FLOPS)]

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Interleaves with EV computation





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Memory efficient





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Memory efficient

Much faster than sparse solvers





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Memory efficient


Whole simulation: O(Hours)





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Memory efficient


Whole simulation: O(Hours)





~ 10x

/ 80x

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Summary

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Transforming a sparse problem to a dense one can be a good thing


Summary

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Large speedup over state of the art (15x - 150x)


Summary

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Significant increase in capacity (100’000 atoms → 10x - 100x)


Summary

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Uses hybrid ressources very efficiently (15 PF sustained)


Summary

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Uses hybrid ressources very efficiently (15 PF sustained)

Made ballistic ab-initio QT simulations for realistic structures a reality


Summary

| | Apr 08 2016 Mauro Calderara 23 (link to video: http://iis.ee.ethz.ch/~mauro/movie_Ag_Switch.avi)


Questions?

Documents

Title to be determined - GPU Technology Conferenceon-demand.gputechconf.com/gtc/2016/presentation/s6651-mauro-calde… · What we want to do → Quantum Transport: electrons and structures