Talk on silent error resilience at the ICME Colloquium on January 26, 2015.

1. Silent error resilience in
numerical time-stepping schemes
Austin Benson
arbenson@stanford.edu
Stanford University
ICME Colloquium, Jan. 26 2015
Joint work with
Sven Schmit, Stanford
Rob Schreiber, HP Labs
code + data: http://stanford.edu/~arbenson/silent.html
paper: Intl. J. of High Performance Computing Applications, 2014
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2.  Computer systems are getting bigger and more complicated.
 Software systems are getting bigger and more complicated.
 Pushing energy limits.
 Things break. 2

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.
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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.
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5. Spot the error!
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6. At time step 120, multiplied single entry in
right-hand-side of Crank-Nicolson and
Backward Euler linear solves by 0.995. 6

7. General algorithm:
 “Base method” generates sequence B1, B2, …
 “Auxiliary method” generates sequence A1, A2, …
 If Di = ||Bi – Ai|| is abnormal, possible error
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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]
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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.
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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?) 10
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 for
numerical time-stepping algorithms…
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12. Exercise in step detection (change point detection)
Algorithmic details in the paper. Main parameters:
Relative jump
Variance change
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13. Experimental setup:
 Solve heat equation for T time steps and
artificially inject error at one time step.
 Do this many times with different
types of errors.
 True positive rate:
#(real errors detected) / #(trials)
 False positive rate:
#(non-errors “detected”) / #(time steps)
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14. Are large errors easier to detect?
Local truncation error (LTE)-normalized error
Output when no fault is injected.
Output when fault is injected.
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15. Error injection:
Multiply single entry of RHS
in linear solves by
z ~ N(1, 5e-5) at a single
time step
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16. Error injection:
Multiply q(x, t) at one
discrete x by z ~ N(1, 0.1)
at a single time step
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17. Takeaways
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 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?
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19. Silent error resilience in
numerical time-stepping schemes
Austin Benson
arbenson@stanford.edu
Stanford University
ICME Colloquium, Jan. 26 2015
Joint work with
Sven Schmit, Stanford
Rob Schreiber, HP Labs
code + data: http://stanford.edu/~arbenson/silent.html
paper: Intl. J. of High Performance Computing Applications, 2014
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20. Tardy error detection
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