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Energy and power consumption are major limitations to continued scaling of computing systems. Inexactness, where the quality of the solution can be traded for energy savings, has been proposed as an approach to overcoming those limitations. In the past, however, inexactness necessitated the need for highly customized or specialized hardware. The current evolution of commercial off-the-shelf(COTS) processors facilitates the use of lower-precision arithmetic in ways that reduce energy consumption. We study these new opportunities in this paper, using the example of an inexact Newton algorithm for solving nonlinear equations. Moreover, we have begun developing a set of techniques we call reinvestment that, paradoxically, use reduced precision to improve the quality of the computed result: They do so by reinvesting the energy saved by reduced precision., Comment: 9 pages, 12 figures, PDFLaTeX. 12 Oct 2016: Corrected author Hal Finkel's affiliation to show ALCF/Argonne

Gradient-based optimization requires accurate derivatives of the objective function and constraints. These gradients may have previously been obtained by manual differentiation of analysis codes, symbolic manipulators, finite-difference approximations, or existing automatic differentiation (AD) tools such as ADIFOR (Automatic Differentiation in FORTRAN). Each of these methods has certain deficiencies, particularly when applied to complex, coupled analyses with many design variables. Recently, a new AD tool called ADJIFOR (Automatic Adjoint Generation in FORTRAN), based upon ADIFOR, was developed and demonstrated. Whereas ADIFOR implements forward-mode (direct) differentiation throughout an analysis program to obtain exact derivatives via the chain rule of calculus, ADJIFOR implements the reverse-mode counterpart of the chain rule to obtain exact adjoint form derivatives from FORTRAN code. Automatically-generated adjoint versions of the widely-used CFL3D computational fluid dynamics (CFD) code and an algebraic wing grid generation code were obtained with just a few hours processing time using the ADJIFOR tool. The codes were verified for accuracy and were shown to compute the exact gradient of the wing lift-to-drag ratio, with respect to any number of shape parameters, in about the time required for 7 to 20 function evaluations. The codes have now been executed on various computers with typical memory and disk space for problems with up to 129 x 65 x 33 grid points, and for hundreds to thousands of independent variables. These adjoint codes are now used in a gradient-based aerodynamic shape optimization problem for a swept, tapered wing. For each design iteration, the optimization package constructs an approximate, linear optimization problem, based upon the current objective function, constraints, and gradient values. The optimizer subroutines are called within a design loop employing the approximate linear problem until an optimum shape is found, the design loop limit is reached, or no further design improvement is possible due to active design variable bounds and/or constraints. The resulting shape parameters are then used by the grid generation code to define a new wing surface and computational grid. The lift-to-drag ratio and its gradient are computed for the new design by the automatically-generated adjoint codes. Several optimization iterations may be required to find an optimum wing shape. Results from two sample cases will be discussed. The reader should note that this work primarily represents a demonstration of use of automatically- generated adjoint code within an aerodynamic shape optimization. As such, little significance is placed upon the actual optimization results, relative to the method for obtaining the results.

This report describes preliminary results obtained using an automated adjoint code generator for Fortran to augment a widely-used computational fluid dynamics flow solver to compute derivatives. These preliminary results with this augmented code suggest that, even in its infancy, the automated adjoint code generator can accurately and efficiently deliver derivatives for use in transonic Euler-based aerodynamic shape optimization problems with hundreds to thousands of independent design variables.