Numerical optimization informed by computational cost for simulation tuning

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Here’s another short post that refers interested people to a recently accepted paper for publication.

M. A. Abramson, T. J. Asaki, J. E. Dennis, Jr., R. Magallanez, and M. J. Sottile. “Efficently solving computationally expensive optimization problems with CPU time-related functions”, to appear in Structural and Multidisciplinary Optimization.

A preprint lives here. The abstract is:

In this paper, we characterize a new class of computationally expensive optimization problems and introduce an approach for solving them. In this class of problems, objective function values may be directly related to the computational time required to obtain them, so that, as the optimal solution is approached, the computational time required to evaluate the objective is significantly less than at points farther away from the solution. This is motivated by an application in which each objective function evaluation requires both a numerical fluid dynamics simulation and an image registration process, and the goal is to find the parameter values of a predetermined reference image by comparing the flow dynamics from the numerical simulation and the reference image through the image comparison process. In designing an approach to numerically solve the more general class of problems in an efficient way, we make use of surrogates based on CPU times of previously evaluated points, rather than their function values, all within the search step framework of mesh adaptive direct search algorithms. Because of the expected positive correlation between function values and their CPU times, a time cutoff parameter is added to the objective function evaluation to allow its termination during the comparison process if the computational time exceeds a specified threshold. The approach was tested using the NOMADm and DACE MATLAB software packages, and results are presented.

Snapshot of a lid-driven cavity problem
Snapshot of a lid-driven cavity problem

The basic problem that this paper is related to is parameter optimization for simulations, specifically those that involve fluid dynamics models. The figure above shows a snapshot of a fluid model for the simple lid-driven cavity problem that we used in the paper. In a number of problems that I encountered in my work at LANL, we were faced with the question of finding the optimal set of parameters for a simulation to match data that was experimentally obtained. Often this included algorithmic parameters, like mesh and grid resolutions, along with solver parameters (e.g., convergence criteria), and physical parameters. What is interesting is that a number of parameters have a direct impact on compute time. For example, resolving a grid to a finer resolution will result in a higher per-iteration compute time. The question that we posed was, what can one do if attacking the problem of both tuning parameters based on observed experimental data and identifying parameters that minimize compute time. That’s what this paper is all about.