Convergence analysis of two-level response and parameter mapping-based methods

Figures & data

Figure 1. Diagram of the mapping performed in the RPM algorithm. The time expensive evaluations in the fine model are replaced by evaluations in the surrogate model (4). X_c and X_f are the input parameter spaces of the coarse and fine model, respectively. p is the input mapping function while S is the output mapping function.

Table 1. Quality factors for test functions (52) and (53).

αRPM, x		βRPM	γRPM
0.037	0.6	2.625	1.504

Figure 2. Speed of convergence on the basis of the quadratic test function using different parameter mapping constants: slower speed of convergence for larger constants L_c(p)L_e. For parameter mapped coarse models, slower convergence histories can be obtained.

Figure 3. Comparison of with factors. For convergence towards correct solution, needs to be smaller than .

Table A1. Glossary of symbols.

xc ∈ Xc	input parameter vector of coarse model with feasibility region Xc
c(xc)	coarse model evaluation of parameters xc
xf ∈ Xf	input parameter vector of fine model with feasibility region Xf
f (xf)	fine model evaluation of parameters xf
y	objective
	k-th iterate of fine model input parameter vector
	k-th iterate of the parameter extraction vector
p^(k)(xf)	k-th iterate of the input mapping function
B^(k)	k-th iterate of the affine input mapping matrix
S^(k)(·)	k-th iterate of the RPM output mapping function
D^(k)	k-th iterate of the affine output mapping matrix
J	Jacobian
γ(·)	objective function
s^(k)	surrogate model
L	Lipschitz constant
αRPM, x, , βRPM, γRPM	quality factors
	set of solutions of the parameter extraction procedure
	set of solutions of the response corrected coarse model