Domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) nonlinear model reduction
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While reduced-order models (ROMs) have demonstrated success in many applications across computational science, challenges remain when applied both to extreme-scale models due to the prohibitive cost of generating requisite training data, and to decomposable systems due to many-query problems often requiring repeated reconfigurations of system components. Therefore, we propose the domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) model-reduction method applicable to parameterized systems of nonlinear algebraic equations. In contrast with previous works, we adopt an algebraically non-overlapping decomposition strategy rather than a spatial-decomposition strategy, which facilitates application to different spatial-discretization schemes. Rather than constructing a low-dimensional subspace for the entire state space in a monolithic fashion—which would be infeasible for extreme-scale systems and decomposable models—the methodology constructs separate subspaces for the different subdomains/components characterizing the original model. In the offline stage, the method constructs low-dimensional bases for the interior and interface of components. In the online stage, the approach constructs an LSPG ROM for each component and enforces strong or weak compatibility on the 'ports' connecting them. We propose four different ways to construct reduced bases on the interface/ports of subdomains and several ways to enforce compatibility across connecting ports. We derive a posteriori and a priori error bounds for the DD-LSPG solutions. Numerical results performed on nonlinear benchmark problems in heat transfer and fluid dynamics demonstrate that the proposed method performs well in terms of both accuracy and computational cost, with different choices of basis and compatibility constraints yielding different performance profiles.
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