13 New Approaches to Inverse Structural Modification Theory Using Random Projections 131 1 5 10 15 20 1 5 10 15 20 Dimensionality 100 200 300 400 a b | 1 | % Error p magnitude: 10 p magnitude: 1 p magnitude: 1/10 p magnitude: 1/100 Dimensionality 5 10 15 20 25 | 1 | % Error p magnitude: 1/10 p magnitude: 1/100 Fig. 13.2 Plots describing how the average magnitudes of the elements (defined by Eq. (13.14)) inside the Mand Kmatrices affect the δλ1 term, defined in Eq. (13.13). (a) Different pmagnitudes plotted against | δλ1|μ term as a percentage error. (b) A zoomed-in version of subplot (a) to more clearly show the cases of p=1/10 and p=1/100 In order to calculate the size of terms inside the matrices, Eq. (13.14) is used, whereUrefers to the uniform distribution, the i,j = 1, . . . ,d indices refer to each term in the matrix, and d = 1, . . . ,20 defines the dimensionality (size) of the matrices. Through this definition there are d 2 degrees of freedom inside the matrix at any time. i,j ∼p· U[0,1], (13.14) The value of d changes because we are considering the effect of dimensionality on the quality of the linear step size, δλ. Moreover the scalar p ∈ {1/100,1/10,1,10} defines the magnitude of the terms in the matrices. Thus p in a practical sense (that is, in reference to a gradient-based optimisation algorithm) can be interpreted as the step size of the algorithm. From Fig. 13.2 it can be seen that as the average magnitude of the elements inside the matrices increase, the absolute difference between the theoretical δλ1 value, and those calculated via Eq. (13.6), that is, δλ † 1, becomes larger. Even when the average step size takes value p = 1/100, there appears to be a bias in the magnitude of the error, which is made clear in the zoomed in subplot of Fig. 13.2b. Moreover as the dimensionality of the matrices increase, the | δλ1| errors appear to increase slightly. Hence in summary as this experimental analysis of Eq. (13.6) seems to suggest, gradient based methods are potentially difficult to implement. In particular, it would appear that we would require p <1/100 at a minimum, and that this value would need to continually decrease as dimensionality increases. For this reason, we opt to explore the viability of particle swarm optimisation as a means for optimisation since it is a non-gradient based approach, and gradient-based approaches seem to require very small step sizes for accurate gradients. 13.2.2 Particle Swarm Optimisation Particle Swarm Optimisation (PSO) is a stochastic, evolutionary optimisation first proposed by Kenedy and Eberhart[9]. The algorithm works by generating an array of candidate particles (the swarm) across an objective space. Within this space each particle searches for the global optimum through the sharing of information within the swarm in a classic explorationexploitation trade-off. This is clarified in Eqs. (13.15) and (13.16). vk+1 i =ωvk i +c1r1(pi −x k i ) +c2r2(pG−x k i ) (13.15) xk+1 i =xk i +αv k+1 i (13.16) where α denotes step size, ωcontrols the particle’s inertia, c1 and c2 (known as the acceleration coefficients) are constants which control the degree of the exploration-exploitationtrade-off, pi and pG are the local optima, and global optimum, for each, and across all particles respectively (that is each particle stores their own local optimum, but shares knowledge of the current global optimum), andr1,r2 ∼U(0,1). In this way, every particle is made aware of the current global optimum, and explores the objective space accordingly (as specified through the c1 andc2 constants).
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