Delaunay-based optimization in CFD leveraging multivariate adaptive polyharmonic splines (MAPS)

A derivative-free optimization algorithm for the efficient minimization of functions obtained via statistical averaging

“…This result is reasonable, since the underlying function in the shifted parabolic case is much simpler. are applied to a representative optimization problem, based on infinite-timeaveraged statistics, of the type considered in this paper [see (1)]. The specific problem considered here is derived from the well-known 3-state Lorenz model [21,41], the dynamics of which exhibit a familiar chaotic behavior that roughly characterizes the bulk flow of a fluid within a hollow torus that is heated from below and cooled from above [10], and is governed by…”

“…In this section, Algorithm 1 (∆-DOGS), Algorithm 2 (α-DOGS), and the Surrogate Management Framework (SMF) developed in [15] and implemented in [25] are applied to a representative optimization problem, based on infinite-timeaveraged statistics, of the type considered in this paper [see (1)].…”

“…However, this class of method scales poorly with the dimension of the problem. The surrogate management framework [15,35,42] and Bayesian algorithms [30-32, 36, 39] are amongst the best derivative-free methods available today, and are implemented for minimizing a problem of the form in (1) in [25,26,40,43]. In this class, the method developed in [39] develops a promising Bayesian approach, in a manner which increases the sampling of new measurements as convergence is approached.…”

“…In this paper, the Delaunay-based derivative-free optimization algorithm developed in [1,2,[12][13][14]47], dubbed ∆-DOGS, is modified to minimize an objective function, f (x) : R n → R, which cannot be evaluated directly, but can be approximated to a tuneable degree of precision. As the precision of any function evaluation is increased, the computational (or experimental) cost of that function evaluation is increased accordingly.…”

“…In this paper, a provably globally convergent (under the appropriate assumptions) new optimization approach is developed for problems of the form given in (1). The structure of the remainder of the paper is as follows: Section 2 briefly reviews the key features of the ∆-DOGS(Z) algorithm developed in [12], upon which the present paper is built.…”