Power minimization under variability is formulated as a rigorous statistical robust optimization program with a guarantee of power and timing yields. Both power and timing metrics are treated probabilistically. Power reduction is performed by simultaneous sizing and dual threshold voltage assignment. An extremely fast run-time is achieved by casting the problem as a second-order conic problem and solving it using efficient interior-point optimization methods. When compared to the deterministic optimization, the new algorithm, on average, reduces static power by 31 % and total power by 17 % without the loss of parametric yield. The run time on a variety of public and industrial benchmarks is 30X faster than other known statistical power minimization algorithms.

informs ® doi 10.1287/opre.1050.0254 © 2005 INFORMS This paper concerns a method for digital circuit optimization based on formulating the problem as a geometric program (GP) or generalized geometric program (GGP), which can be transformed to a convex optimization problem and then very efficiently solved. We start with a basic gate scaling problem, with delay modeled as a simple resistor-capacitor (RC) time constant, and then add various layers of complexity and modeling accuracy, such as accounting for differing signal fall and rise times, and the effects of signal transition times. We then consider more complex formulations such as robust design over corners, multimode design, statistical design, and problems in which threshold and power supply voltage are also variables to be chosen. Finally, we look at the detailed design of gates and interconnect wires, again using a formulation that is compatible with GP or GGP.

As technology continues to scale beyond 100nm, there is a significant increase in performance uncertainty of CMOS logic due to process and environmental variations. Traditional circuit optimization methods assuming deterministic gate delays produce a flat “wall ” of equally critical paths, resulting in variation-sensitive designs. This paper describes a new method for sizing of digital circuits, with uncertain gate delays, to minimize their performance variation leading to a higher parametric yield. The method is based on adding margins on each gate delay to account for variations and using a new “soft maximum ” function to combine path delays at converging nodes. PSfrag Using replacements analytic models to predict the means and standard deterministic deviations method of gate delays as posynomial functions of the device sizes, PDF we create a simple, computationally efficient heuristic for uncertainty-aware sizing of digital circuits via Geometric Programming. Monte-Carlo simulations on custom 32bit adders and ISCAS’85 benchmarks show that about 10 % to 20 % delay reduction over deterministic sizing methods can be achieved, without any additional cost in area. 1.

Abstract—We consider the problem of choosing the gate sizes or scale factors in a combinational logic circuit in order to minimize the total area, subject to simple RC timing constraints, and a minimum-allowed gate size. This problem is well known to be a geometric program (GP), and can be solved by using standard interiorpoint methods for small- and medium-size problems with up to several thousand gates. In this paper, we describe a new method for solving this problem that handles far larger circuits, up to a million gates, and is far faster. Numerical experiments show that our method can compute an adequately accurate solution within around 200 iterations; each iteration, in turn, consists of a few passes over the circuit. In particular, the complexity of our method, with a fixed number of iterations, is linear in the number of gates. A simple implementation of our algorithm can size a 10 000 gate circuit in 25 s, a 100 000 gate circuit in 4 min, and a million gate circuit in 40 min, approximately. For the million gate circuit, the associated GP has three million variables and more than six million monomial terms in its constraints; as far as we know, these are the largest GPs ever solved. Index Terms—Gate sizing, geometric programming (GP), largescale optimization. I.

In this paper we give a brief overview of a heuristic method for approximately solving a statistical digital circuit sizing problem, by reducing it to a related deterministic sizing problem that includes extra margins in each of the gate delays to account for the variation. Since the method is based on solving a deterministic sizing problem, it readily handles large-scale problems. Numerical experiments show that the resulting designs are often substantially better than one in which the variation in delay is ignored, and often quite close to the global optimum. Moreover, the designs seem to be good despite the simplicity of the statistical model (which ignores gate distribution shape, correlations, and so on). We illustrate the method on a 32-bit Ladner-Fischer adder, with a simple resistor-capacitor (RC) delay model, and a Pelgrom model of delay variation.

Unlike their hard realtime counterparts, soft realtime applications are only expected to guarantee their "expected delay" over input data space. This paradigm shift calls for customized statistical design techniques to replace the conventional pessimistic worst case analysis methodologies. We present a novel statistical time budgeting algorithm to translate the application expected delay constraint into its components' local delay constraints. We utilize the mathematical properties of the problem to quickly calculate the system expected delay, and incrementally estimate the component utility variation with its timing relaxation. Our algorithm determines the optimal maximum weighted timing relaxation of an application under expected delay constraint. Experimental results on core-based synthesis of several multi-media applications targeting FPGAs, show that our technique always improves the design area. Furthermore, it consistently outperforms optimal time budgeting under hard realtime constraint, which is the best existing competitor. Design area improvements were up to 26% and averaged about 17%, on several MediaBench applications.

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