Bayes approach to solving T.E.A.M. benchmark problems 22 and 25 and its comparison with other optimization techniques
Introduction
Nowadays, optimization techniques represent a very important and desired tool in the domain of designing particular elements or even complex systems in many (and not only) technical branches [1], [2]. Optimization is expected to improve their parameters, characteristics and other properties, bring savings in production costs, increase of their efficiency, safety, and so forth. In the last five decades, a great number of both deterministic and heuristic algorithms have been suggested and successfully applied in solution of various problems, as far as these problems were not too much complicated [3]. Anyway, it is known that the time necessary for finding the optimum strongly increases with the number of parameters to be optimized, which is even more expressed in nonlinear or multiphysics problems. Other complications may occur in defining the objective functions and their eventual weighting.
Despite of availability of up to now developed advanced optimization techniques, the solution of complex optimization problems is still rather slow. This is related with searching the sample space in case of the heuristic algorithms or slow convergence of the deterministic algorithms. Another danger consists in the possibility that the algorithm stops at some local extreme instead of the global one. That is why all over the world further efficient techniques are permanently proposed and explored.
Many of the above techniques are also used for solving complex technical problems. But their reliability and speed may strongly differ from one another, which depends on the type of the task solved. The most prospective and versatile algorithms with a wide spectrum of applicability were also implemented into existing professional codes. Nevertheless, the development in the area is not yet finished from far, which is confirmed by a great number of relevant recent papers and books.
The Bayes optimization technique is not new [4], but due to its implementation complexity it was used rather for solving some artificial mathematical problems defined on simple domains (such as unit squares, circles, etc.) [5], [6]. As it has not been implemented in any professional FEM-based code so far, the authors built its algorithms to their own application Agros Suite with the aim to carry out a number of tests and then to optimize several tasks whose results are known. The choice fell on two T.E.A.M. benchmarks 22 and 25. The details of the work are described in the following sections.
Section snippets
Benchmark problems 22 and 25
The first solved example, T.E.A.M. benchmark problem 22 [7], [8], [9] is an interesting optimization problem of electromagnetic design, which deals with the optimization of a superconducting magnetic energy storage (SMES) configuration, see Fig. 3. It can be formulated as a multiobjective problem [10] with two objectives:
- •
to minimize the stray field evaluated along lines (Fig. 3), while not violating the quench condition assuring the superconductivity state,
- •
to minimize the deviation from the
Optimization algorithms
The derivative-free optimization has been used to evaluate the given benchmarks. These type of algorithms are effective in case of discontinuous objective functions and constraints (without analytical derivatives). We used three of the methods available in the COMSOL Optimization Module: the Bound Optimization by Quadratic Approximation (BOBYQA), Nelder–Mead, and Monte Carlo methods and four methods available in the Agros Suite: the BOBYQA, Nelder–Mead, Non-dominated Sorting Genetic Algorithm
Agros Suite
Agros Suite [28] is a group of applications for solving complex non-linear and nonstationary coupled problems. The application 2, that has been developed for years in our group, tightly cooperates with the libraries Hermes and deal.II [29]. The library contains the most recent algorithms for hard-coupled and fully adaptive solution of systems of generally non-linear and non-stationary (PDEs) that is based on the finite element method of higher order of accuracy. The codes written in C++ is
T.E.A.M. Workshop Problem 22
Consider two concentric coils (Fig. 3) carrying current with opposite directions and operating in superconducting conditions. They offer an opportunity of storing a significant amount of energy in their magnetic field while keeping the stray field within certain limits. An optimal design of the system should, therefore, couple the desired value of energy to be stored with a minimal stray field.
Distribution of magnetic field in the system is described by the equation for magnetic vector
Conclusion
The Bayes optimization technique was tested on two typical technical problems - T.E.A.M. benchmarks 22 and 25 (due to its complexity, this technique was mostly used for optimizing purely mathematical artificial problems defined on simple definition areas so far). The results are very promising and in a lot of aspects (speed, convergence rate, accuracy) the technique seems to be superior to other standard methods, which holds mostly in the case of flat or only slightly wavy objective functions
Acknowledgment
This research has been supported by the Ministry of Education, Youth and Sports of the Czech Republic under the RICE – New Technologies and Concepts for Smart Industrial Systems, project No. LO1607.
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