Weightless Regularised Identification Using Multi-objective Optimisation Method

Although the regularisation increased the popularity of inverse analysis due to its capability of deriving a stable solution, the significant problem is that the solution depends upon the regularisation parameters chosen. This paper presents a technique for deriving solutions without the use of the parameters, and further an optimisation method, which can work efficiently for problems of concern. Numerical examples show that the technique can efficiently search for appropriate solutions. INTRODUCTION It often becomes difficult to solve inverse problems if measurement data are not sufficiently available and/or if measurement data and/or the direct model contains large errors [1]. One of the approaches for overcoming this problem is to introduce a regularisation term to a functional to be minimised [2,3], which normally consists of a function multiplied with weighting factors. The term makes the functional smooth, so that a conventional calculus-based optimisation can obtain an appropriate parameter set without divergence or vibration. The problem of the regularisation is however the selection of its weighting factors as the solution obtained depends upon the selection. Most of the research work thereby shows results with a couple of selections and leaves the selection for further studies. Finding the best value of the weighting factors has not yet been much studied and can been found only in several papers to the best of the authors’ knowledge [4-6]. In some techniques, the best weighting factors are found after a single solution is obtained. In this case, additional parameters are however introduced to find it, and the solution is again dependent on these parameters. The other techniques find solutions each with a different set of weighting factors by a step size before finding a single solution by some criteria, thereby the solutions not depending on the weighting factors. Deriving a number of solutions with a single optimisor is however time-consuming, and, in addition, the solutions are governed by the step size of each weighting factor. On the other hand, multi-objective optimisation methods, which optimise a vector functional thereby giving a set of admissible solutions rather than a single solution, have been proposed, mostly by the evolutionary computation community, and have received remarkable attention [7-9]. The most popular evolutionary algorithm (EA) is the genetic algorithm (GA) [10], which incorporates binary strings and their reproduction. Despite, GA is too inefficient for the minimisation of continuous functions with continuous parameters, which is a typical inverse problem and the problem of the authors’ concern. In this paper, a technique for solving a regularised inverse problem without weighting factors is first proposed. In this technique, regularisation terms are each formulated as another objective function, and the multi-objective optimisation