<p>Mixed-precision arithmetic, widely adopted for performance, creates an overlooked opportunity for fault tolerance. Our mixed-precision panel factorization (<Emphasis FontCategory="SansSerif">MPF</Emphasis>) algorithm already computes each LU panel in both <Emphasis FontCategory="SansSerif">FP16</Emphasis> and <Emphasis FontCategory="SansSerif">FP64</Emphasis> ; we show that this inherent redundancy enables soft-error detection at negligible additional cost. We propose four complementary detectors that compare checksum sketches of the two factorizations: <Emphasis FontCategory="SansSerif">Relative</Emphasis> (thresholded checksum discrepancy), <Emphasis FontCategory="SansSerif">Ratio</Emphasis> (outlier analysis of per-row differences), <Emphasis FontCategory="SansSerif">Hybrid</Emphasis> (a randomized-probe combination of both with separate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$U$</EquationSource> <EquationSource Format="MATHML"><math> <mi>U</mi> </math></EquationSource> </InlineEquation>-factor checks), and <Emphasis FontCategory="SansSerif">CrossCheck</Emphasis> (a lightweight cross-check inspired by algorithm-based fault tolerance). Experimental evaluation shows that <Emphasis FontCategory="SansSerif">Relative</Emphasis> and <Emphasis FontCategory="SansSerif">CrossCheck</Emphasis> achieve high sensitivity with low false-positive rates, while <Emphasis FontCategory="SansSerif">Hybrid</Emphasis> maintains near-zero false positives with moderate sensitivity and complementary coverage for challenging significand-bit faults. Bit-level analysis confirms reliable detection of exponent- and sign-bit errors, with sensitivity decreasing gracefully for low-order significand bits whose perturbations approach the <Emphasis FontCategory="SansSerif">FP16</Emphasis> rounding floor. A lightweight panel-fingerprint mechanism extends protection beyond the factorization loop, closing the temporal gap before the triangular solve with guaranteed detection of any single-element corruption and zero false positives. The approach requires only linear work and constant storage per panel, preserving the cubic scaling of standard LU decomposition.</p>

Exploiting mixed-precision redundancy for soft-error detection in LU decomposition

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2026年4月9日

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Nima Sahraneshinsamani,
Sandra Catalán,
José R. Herrero

摘要

Mixed-precision arithmetic, widely adopted for performance, creates an overlooked opportunity for fault tolerance. Our mixed-precision panel factorization (MPF) algorithm already computes each LU panel in both FP16 and FP64 ; we show that this inherent redundancy enables soft-error detection at negligible additional cost. We propose four complementary detectors that compare checksum sketches of the two factorizations: Relative (thresholded checksum discrepancy), Ratio (outlier analysis of per-row differences), Hybrid (a randomized-probe combination of both with separate $L$ $L$ - and $U$ $U$ -factor checks), and CrossCheck (a lightweight cross-check inspired by algorithm-based fault tolerance). Experimental evaluation shows that Relative and CrossCheck achieve high sensitivity with low false-positive rates, while Hybrid maintains near-zero false positives with moderate sensitivity and complementary coverage for challenging significand-bit faults. Bit-level analysis confirms reliable detection of exponent- and sign-bit errors, with sensitivity decreasing gracefully for low-order significand bits whose perturbations approach the FP16 rounding floor. A lightweight panel-fingerprint mechanism extends protection beyond the factorization loop, closing the temporal gap before the triangular solve with guaranteed detection of any single-element corruption and zero false positives. The approach requires only linear work and constant storage per panel, preserving the cubic scaling of standard LU decomposition.