Silent error resilience in numerical time-stepping schemes

1

Silent error resilience innumerical time-stepping schemes

Austin Bensonarbenson@stanford.eduStanford UniversityICME Colloquium, Jan. 26 2015

Joint work withSven Schmit, StanfordRob Schreiber, HP Labs

code + data: http://stanford.edu/~arbenson/silent.htmlpaper: Intl. J. of High Performance Computing Applications, 2014

2

Computer systems are getting bigger and more complicated. Software systems are getting bigger and more complicated. Pushing energy limits. Things break.

3

What breaks?

Hardware wears out Bit flips from cosmic rays Data races and other software bugs Firmware bugs

Silent errors are errors in application state that have escaped low-level error detection.

4

What can we do?

Checkpoint/restart: Occasionally save state of system. If error is detected, restart.

Does not scale. How to detect errors?

Other ABFT: Clever algorithms that address these issues for particular algorithms.

This work: Error detection for iterative computation in general, numerical time-stepping schemes in particular.

5

Spot the error!

6

At time step 120, multiplied single entry in right-hand-side of Crank-Nicolson and Backward Euler linear solves by 0.995.

7

General algorithm: “Base method” generates sequence B1, B2, … “Auxiliary method” generates sequence A1, A2, … If Di = ||Bi – Ai|| is abnormal, possible error

8

Base method: high-order numerical integration scheme: Runge-Kutta 5

Auxiliary method: lower-order scheme: Runge-Kutta 4

Difference: Di = |Bi – Ai|

Re-purposing an old idea for step-size control[Fehlberg, 1969], [Dormand and Prince, 1980]

9

Key idea: re-use data

RK 1/2 scheme for u’ = f(t, u)

Second-order scheme has error O(h^3)

No extra function evaluations.Provides O(h^2) check.

10

Key idea: re-use data

Implicit solve that is stable

Explicit solve checks.

It is OK that the explicit solve may be unstable. (Why?)

e.g., Backward Euler

e.g., Forward Euler

11

Backward/Forward Euler Richardson/Crank-Nicolson Runge-Kutta 1/2, 2/3, 4/5 Adams-Bashforth linear multistep method 2/3, 4/5 Explicit check on implicit scheme Extrapolation

Lots of these checks fornumerical time-stepping algorithms…

12

Exercise in step detection (change point detection)Algorithmic details in the paper. Main parameters:

Relative jump

Variance change

13

Experimental setup:

Solve heat equation for T time steps and artificially inject error at one time step.

Do this many times with differenttypes of errors.

True positive rate: #(real errors detected) / #(trials)

False positive rate: #(non-errors “detected”) / #(time steps)

14

Are large errors easier to detect?

Local truncation error (LTE)-normalized error

Output when no fault is injected.

Output when fault is injected.

15

Error injection:Multiply single entry of RHS in linear solves byz ~ N(1, 5e-5) at a single time step

16

Error injection:Multiply q(x, t) at one discrete x by z ~ N(1, 0.1)at a single time step

17

Takeaways

We have a general framework for detecting silent errors. Numerical integration is our central application. We detect large errors more easily. Not too many false positives.

18

How many silent errors are there? How worried should we be? Do we need systems solutions or algorithmic solutions? Both? “Defense in depth” is good. But how easy are ABFT methods to

incorporate into existing solvers?

Resilience: what do we need to discuss?

19

Silent error resilience innumerical time-stepping schemes

Austin Bensonarbenson@stanford.eduStanford UniversityICME Colloquium, Jan. 26 2015

Joint work withSven Schmit, StanfordRob Schreiber, HP Labs

code + data: http://stanford.edu/~arbenson/silent.htmlpaper: Intl. J. of High Performance Computing Applications, 2014

20

Tardy error detection