Parametric domain decomposition for accurate reduced order models: Applications of MP-LROM methodology

The multivariate predictions of local reduced-order-models (MP-LROM) methodology, recently proposed by the authors Moosavi et al. (0000), uses machine learning based regression methods to predict the errors of reduced-order models. This study considers two applications of MP-LROM. First, the error model is used in conjunction with a greedy sampling algorithm to generate decompositions of one dimensional parametric domains with overlapping regions, such that the associated local reduced-order models meet the prescribed accuracy requirements. Once a parametric domain decomposition is constructed, any parametric configuration belongs to (at least) one of the partitions; the local reduced-order model associated with that partition approximates the full order model at the given parameters within an accuracy level that is estimated a-priori. The parameter domain decomposition creates a database of the available local bases, local reduced-order, and high-fidelity models, and identifies the most accurate solutions for an arbitrary parametric configuration. Next, this database is used to enhance the accuracy of the reduced-order models using: (1) Lagrange interpolation of reduced bases in the matrix space; (2) Lagrange interpolation of reduced bases in the tangent space of the Grassmann manifold; (3) concatenation of reduced bases followed by a Gram-Schmidt orthogonalization process; and (4) Lagrange interpolation of high-fidelity model solutions. Numerical results with a viscous Burgers model illustrate the potential of the MP-LROM methodology to improve the design of parametric reduced-order models.