Multiparametric response surface construction by means of proper generalized decomposition: An extension of the PARAFAC procedure

Abstract

Multidimensional problems are found in almost every scientific field. In particular, this is standard in parametric design, inverse analysis, in optimization and in metamodeling analysis, whereby statistical or deterministic approximations of multiparametric solutions are built from the results of experimental campaigns or computer simulations. Multidimensional fitting or approximation of response functions exponentially increase their complexity and computational cost with the number of dimensions responding to the well-known “curse of dimensionality”. To reduce the order of complexity and make the solution of many-parameter problems affordable, we propose to combine the model reduction technique known as Proper Generalized Decomposition (PGD) and the response surface (RSM) methodology. As a proof of concept we have used a simple fitting procedure as it is least squares, although other more complex fitting procedures may be easily included. The combined algorithm is presented and its capabilities discussed in a set of multidimensional examples. The data samples to be fit in each of these examples are obtained by means of appropriate discretizing the interval of interest for each design factor and then generating the output values (exact or stochastically modified) by means of virtual experiments. To have an idea of the number of discretization points needed along each direction, the Taguchi’s design of experiments is used. The obtained results show evident improvements in computer time and accuracy when compared to other traditional multiparametric approximation techniques based on polynomial functions and the standard Levenberg–Marquardt algorithm, especially in problems with non-linear behavior and with high number of design parameters. Further comparison was done with the PARAFAC-ALS algorithm. The combination of PGD and RSM seems to be an appealing tool for accurately modelling multiparametric problems in almost real-time if a sufficient set of previous off-line results is available despite the intrinsic complexity of the problem and of the number of parameters involved.