Domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) nonlinear model reduction

Abstract

While reduced-order models have demonstrated success in many applications across computational science and engineering, they encounter challenges when applied both to truly extreme-scale models due to the prohibitive cost of generating requisite training data, and to decomposable systems due to many-query problems (e.g., design) often requiring repeated reconfigurations of system components. To address these challenges, we propose the 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). In contrast with previous works, we adopt an algebraically non-overlapping decomposition strategy rather than a spatialdecomposition 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. 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.

Projection-based model reduction of dynamical systems using space-time subspace and machine learning

“…• We show that while our proposed approach is less accurate than the least-square Petrov-Galerkin (LSPG) model reduction (without hyper-reduction) [44,45,46], it is significantly faster.…”

Reduced order models for Lagrangian hydrodynamics

“…directly applying snapshot SVD and using the SNS relation [13]. To avoid complicating the results with respect to different ROM parameters, we fix the reduced basis dimensions as (n v , n e , n x , n F 1 , n F tv ) = (28,44,10,28,44). We also compare different algorithms for obtaining the sampling indices, namely the over-sampling DEIM (see Algorithm 3 of [81] and Algorithm 5 of [82]) and QDEIM (see [79]).…”

“…These approaches take advantage of both the known governing equation and the solution data generated from the corresponding FOM simulations to form linear subspace reduced order models (LS-ROM). Example applications include, but are not limited to, the nonlinear diffusion equations [10,11], the Burgers equation and the Euler equations in small-scale [12][13][14], the convection-diffusion equations [15,16], the Navier-Stokes equations [17,18], rocket nozzle shape design [19], flutter avoidance wing shape optimization [20], topology optimization of wind turbine blades [21], lattice structure design [22], porous media flow/reservoir simulations [23][24][25][26], computational electro-cardiology [27], inverse problems [28], shallow water equations [29,30], Boltzmann transport problems [31], computing electromyography [32], spatio-temporal dynamics of a predator-prey system [33], acoustic wave-driven microfluidic biochips [34], and Schrödinger equation [35]. Survey papers for the projection-based LS-ROM techniques can be found in [36,37].…”

Windowed space-time least-squares Petrov-Galerkin method for nonlinear model order reduction

“…[34,35] employ a distance-based algorithm to select the local basis at a given time instance. Lastly, the present work displays commonalities with domain decomposition ROMs (DD-ROMs) [36][37][38][39]. In these DD-ROMs, the spatial domain is broken down into subdomains and bases tailored specifically to each subdomain are then employed.…”