## 1 Introduction
Particularly for parametric nonlinear HDMs, many approaches which can also be applied to parametric linear HDMs have been proposed, including hyper-reduced ROM, which guarantees feasibility [49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61]. With these methods, the parameter admissible space must be sampled at a few points using a greedy sampling algorithm (e.g., [62]), and a set of problems must be solved, yielding a set of parametric solution snapshots. This generally results from a proper orthogonal decomposition (POD), which are compressed using, for example, singular value decomposition (SVD) to construct a global reduced-order basis (ROB). For linear vibroacoustic HDMs, the most common choice for the parametric ROB consists of taking the modes of the different parts that constitute the vibroacoustic system (e.g., the elastic modes of the structure and the acoustic modes of the internal acoustic fluid)[41; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74]. The design of a vibroacoustic system is used to manufacture a real system and to construct a nominal computational model with the methodologies listed above, and which will be presented in the next sections. In practice, the real system can exhibit variabilities in its responses due to fluctuations in the manufacturing process and due to small variations of the configuration around a nominal configuration associated with the design. The vibroacoustic computational model has parameters such as geometry, mechanical properties, and boundary conditions, which can be uncertain, inducing uncertainties in the computational model parameters. On the other hand, the modeling process induces some modeling errors defined as the model uncertainties. It is important to consider both the model-parameters uncertainties and the modeling uncertainties in order to improve the predictions. Two main types of approaches can be used to model uncertainties in the framework of the probability theory [75]. The first is the parametric probabilistic approach, which consists of modeling the uncertain parameters by random variables (e.g., [75; 76; 77; 78; 79; 80; 81; 82; 83; 84]), but which does not have the capability to take modeling uncertainties into account. Two main methods can then be used to take them into account. The first is the output-prediction-error method, which requires experimental data [85,86] and uses the Bayesian method [87,88], but for which the computational model cannot be updated using experimental data, which constitutes a lock for robust design and optimization. The second one is the nonparametric probabilistic approach of modeling uncertainties induced by modeling errors proposed in [75,89,90,91], based on the maximum entropy principle [92] in the context of Information Theory [93] and on the random matrix theory [94], and extended for nonlinear dynamical systems in [95]. The outline of this paper is the following:  
* Statement of the problem in the frequency domain. * External inviscid acoustic fluid equations in the frequency domain and acoustic impedance boundary operator. * Internal dissipative acoustic fluid equations in the frequency domain. * Structure equations with frequency-dependent constitutive equation. * Boundary value problem in terms of the structural displacement and the internal pressure field. * Reduced-order vibroacoustic computational model. * Uncertainty quantification for the vibroacoustic computational model. * Experimental validation with a complex computational vibroacoustic model of an automobile.