Proper generalized decompositions of parameterized Helmholtz problems in the heterogeneous and unbounded domains

Solving the Helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge. This is due to the particular nature of the Helmholtz operator and the sensibility of the solution to small variations of the data. Here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency. Moreover, this is applied to a real engineering problem: water agitation inside real harbors for low to mid-high frequencies. The Proper Generalized Decomposition (PGD) model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency. Here, its applicability to such a problem is discussed and demonstrated. More precisely, the separability of the operator is addressed taking into account both the non-constant coefficients in the domain and the use of the Perfectly Matched Layers in order to model the unbounded domain. Then, the performance of the PGD in this framework is discussed and improved using a higher-order projection and a Petrov-Galerkin approach to construct the separated basis. Moreover, the PGD is also discussed as an efficient higher-order projection scheme and compared with the higher-order singular value decomposition.