Publications
Domain-decomposition least-squares Petrov–Galerkin (DD-LSPG) nonlinear model reduction
A novel domain-decomposition least-squares Petrov–Galerkin (DD-LSPG) model-reduction method applicable to parameterized systems of nonlinear algebraic equations (e.g., arising from discretizing a parameterized partial-differential-equations problem) is proposed. 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, the methodology constructs separate subspaces for the different subdomains/components characterizing the original model. During the offline stage, the method constructs low-dimensional bases for the interior and interface of subdomains/components. During the online stage, the approach constructs an LSPG reduced-order model for each subdomain/component (equipped with hyper-reduction in the case of nonlinear operators), and enforces strong or weak compatibility on the ‘ports’ connecting them. We propose several different strategies for defining the ingredients characterizing the methodology: (i) four different ways to construct reduced bases on the interface/ports of subdomains, and (ii) different ways to enforce compatibility across connecting ports. In particular, we show that the appropriate compatibility-constraint strategy depends strongly on the basis choice. In addition, 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 that employ both finite-element and finite-difference spatial discretizations demonstrate that the proposed method performs well in terms of both accuracy and (parallel) computational cost, with different choices of basis and compatibility constraints yielding different performance profiles.