AlgorithmBased Fault Tolerance for Matrix Operations

1
Algorithm-Based Fault Tolerance for Matrix
Operations

• Proposed by
• Kuang-Hua Huang
• Jacob A. Abraham

2
Problem Description

• Achieving a fault tolerant model that is
  algorithm based rather than hardware based
• Existing techniques require high overhead cost
• Error Masking (hardware redundancy)
• Error Detection and Recovery (hardware/time
  redundancy)

3
Existing Techniques

• Error Masking
• Triple Module Redundancy 200
• Quadded Logic 300
• Error Detection and Recovery
• TSC 73-83 hardware
• Alternating Logic 100 time 85 hardware
• RESO 100 time
• Watchdog processors

4
Algorithm-Based Fault Tolerance

• Pros
• Detects and corrects errors
• Extremely low overhead
• Cons
• Not generally applicable (mostly useful for MPP
  systems)
• Undetectable patterns if more than one error

5
Approach

• Encoding of data
• Redesign of Algorithm
• Information must be easy to recover
• Time overhead must not be low
• Distribution of computation steps
• All errors can be detected and corrected

6
Checksum Matrices

• Definitions
• Column checksum matrix
• Row checksum matrix
• Full checksum matrix

7
Theorems

• Matrix Multiplication
• A B C ? Ac Br Cf
• LU Decomposition
• C L U ? Cf Lc Ur
• Addition
• A B C ? Af Bf Cf
• Scalar Multiplication
• c Af (c A)f
• Transpose
• AfT (AT)f

8
Error Detection and Correction

• Detection
• Compute the sum (S1) of information in each
  row/column and compare to the corresponding
  checksum (S2)
• Location
• Intersection of the inconsistent row and column
  (S1 ? S2)
• Correction
• Correction of the error E E (S2 S1)
• Correct the error in checksum S1 ? S2

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Mesh Connected Processor Arrays
In a mesh connected system, each processor
individually handles a calculation in the
resultant matrix. In a systolic array, an array
of processes handles a row of values
Array B
Array A
10
Overhead for MPP systems
Mesh Connected Arrays
Systolic Arrays
11
Undetectable Loop Patterns
X
X

• Certain Patterns of error mask the errors
• Caused by faulty processors
• Requires a minimum number of processors to detect
  error

X
X
X
X
X
X
X
X
X
X
X
X
12
Uniprocessor Systems

• In uniprocessor system, a faulty processor can
  cause all elements to be incorrect

13
Conclusion

• Algorithm-based fault tolerance applied to matrix
  operations
• Low ratio of redundancy
• Ability to detect and correct errors
• Ongoing research