Inexact differential

Summary

An inexact differential or imperfect differential is a differential whose integral is path dependent. It is most often used in thermodynamics to express changes in path dependent quantities such as heat and work, but is defined more generally within mathematics as a type of differential form. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential. I.e., its value cannot be inferred just by looking at the initial and final states of a given system. Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions. Definition An inexact differential \delta u is a differential for which the integral over some two paths with the same end points is different. Specifically, there exis

Official source

https://en.wikipedia.org/wiki/Inexact_differential

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Opportunities for energy efficient computing: a study of inexact general purpose processors for high-performance and big-data applications

Christian Enz, Jérémy Lucien Maurice Schlachter

In this paper, we demonstrate that disproportionate gains are possible through a simple devise for injecting inexactness or approximation into the hardware architecture of a computing system with a general purpose template including a complete memory hierarchy. The focus of the study is on energy savings possible through this approach in the context of large and challenging applications. We choose two such from different ends of the computing spectrum---the IGCM model for weather and climate modeling which embodies significant features of a high-performance computing workload, and the ubiquitous PageRank algorithm used in Internet search. In both cases, we are able to show in the affirmative that an inexact system outperforms its exact counterpart in terms of its efficiency quantified through the relative metric of operations per virtual Joule (OPVJ)---a relative metric that is not tied to particular hardware technology. As one example, the IGCM application can be used to achieve savings through inexactness of (almost) a factor of 3 in energy without compromising the quality of the forecast, quantified through the forecast error metric, in a noticeable manner. As another example finding, we show that in the case of PageRank, an inexact system is able to outperform its exact counterpart by close to a factor of 1.5 using the OPVJ metric.

2015

Automatic Generation of Inexact Digital Circuits by Gate-level Pruning

Vincent Frédéric Camus, Christian Enz, Jérémy Lucien Maurice Schlachter

Inexact or approximate circuits show great ability to reduce power consumption at the cost of occasional errors in comparison to their conventional counterparts. Even though the benefits of such circuits have been proven for many applications, they are not wide spread owing to the absence of a clear design methodology and the required CAD tools. In this regard, this paper presents a methodology to automatically generate inexact circuits starting from a conventional design by adding only one small step in the digital design flow. Further, this paper also demonstrates that achieving pruning at gate-level can lead to substantial savings in terms of power consumption, critical path delay and silicon area. An order of magnitude area and power savings is demonstrated for a 64-bit gate level pruned high-speed adder for a 10% relative error magnitude.

2015

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Inexact and approximate circuit design is a promising approach to improve performance and energy efficiency in technology-scaled and low-power digital systems. Such strategy is suitable for error-tolerant applications involving perceptive or statistical outputs. This paper presents a novel architecture of an Inexact Speculative Adder with optimized hardware efficiency and advanced compensation technique with either error correction or error reduction. This general topology of speculative adders improves performance and enables precise accuracy control. A brief design methodology and comparative study of this speculative adder are also presented herein, demonstrating power savings up to 26 % and energy-delay-area reductions up to 60% at equivalent accuracy compared to the state-of-the-art.

2015