Dynamic Voltage and Frequency Scaling (DVFS) exhibits fundamental limitations as a method to reduce energy consumption in computing systems. In the HPC domain, where performance is of highest priority and codes are heavily optimized to minimize idle time, DVFS has limited opportunity to achieve substantial energy savings. This paper explores if operating processors Near the transistor Threshold Volt- age (NTV) is a better alternative to DVFS for break- ing the power wall in HPC. NTV presents challenges, since it compromises both performance and reliability to reduce power consumption. We present a first of its kind study of a significance-driven execution paradigm that selectively uses NTV and algorithmic error tolerance to reduce energy consumption in performance- constrained HPC environments. Using an iterative algorithm as a use case, we present an adaptive execution scheme that switches between near-threshold execution on many cores and above-threshold execution on one core, as the computational significance of iterations in the algorithm evolves over time. Using this scheme on state-of-the-art hardware, we demonstrate energy savings ranging between 35% to 67%, while compromising neither correctness nor performance.