  Bayesian analysis enables combining prior knowledge with measurement data to
learn model parameters. Commonly, one resorts to computing the maximum a
posteriori (MAP) estimate, when only a point estimate of the parameters is of
interest. We apply MAP estimation in the context of structural dynamic models,
where the system response can be described by the frequency response function.
To alleviate high computational demands from repeated expensive model calls, we
utilize a rational polynomial chaos expansion (RPCE) surrogate model that
expresses the system frequency response as a rational of two polynomials with
complex coefficients. We propose an extension to an existing sparse Bayesian
learning approach for RPCE based on Laplace's approximation for the posterior
distribution of the denominator coefficients. Furthermore, we introduce a
Bayesian optimization approach, which allows to adaptively enrich the
experimental design throughout the optimization process of MAP estimation.
Thereby, we utilize the expected improvement acquisition function as a means to
identify sample points in the input space that are possibly associated with
large objective function values. The acquisition function is estimated through
Monte Carlo sampling based on the posterior distribution of the expansion
coefficients identified in the sparse Bayesian learning process. By combining
the sparsity-inducing learning procedure with the sequential experimental
design, we effectively reduce the number of model evaluations in the MAP
estimation problem. We demonstrate the applicability of the presented methods
on the parameter updating problem of an algebraic two-degree-of-freedom system
and the finite element model of a cross-laminated timber plate.
