Registration-based model reduction of parameterized two-dimensional conservation laws

Ferrero, Andrea; Taddei, Tommaso; Zhang, Lei

doi:10.48550/arxiv.2105.02024

Cited by 1 publication

(2 citation statements)

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“…Several model reduction methods have been proposed that leverage nonlinear trial manifolds to overcome the reduction limitation set by the Kolmogorov n-width. One popular approach to construct such manifolds, and the approach adopted in this work, use a nonlinear parametrization of an affine space, usually a transformation of the underlying spatial domain [43,67,52,54,53,12,41,6,58,68,62,38,55,4,19,61], as described in Section 1.2. Many of these methods limit the trial manifold by decoupling the search for the transformation and subspace approximation, whereas the proposed approach simultaneously determines the subspace approximation and mapping that minimize the HDM residual (Section 3.4 highlights the significance of this distinction).…”

Section: Connection To Previous Workmentioning

confidence: 99%

“…For each problem, we will compare the accuracy of fixed-domain model reduction and model reduction with implicit feature tracking; the accuracy of each method will be measured by the relative L 2 pΩ 0 q error with respect to the corresponding HDM solution (or exact solution, if available). For any µ P D and G P G, the fixed-domain ROM solution (dimension k) U k p ¨; G, µq, defined in (19) and either (23) (Galerkin) or (24) (minimum-residual), approximates the HDM solution U h p ¨; G, µq, defined in ( 9)- (10). For all cases considered, the domain mapping is frozen at the nominal map G " Ḡ and the reduced basis is constructed by applying POD to the HDM snapshots tU h p ¨; Ḡ, µq | µ P D tr u, where D tr Ă D is a finite training set.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking

Mirhoseini¹,

Zahr²

2021

Preprint

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This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions: model reduction with implicit feature tracking. Traditional model reduction techniques use an affine subspace to reduce the dimensionality of the solution manifold and, as a result, yield limited reduction and require extensive training due to the slowly decaying Kolmogorov n-width of convection-dominated problems. The proposed approach circumvents the slowly decaying n-width limitation by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with a space of bijections of the underlying domain. Central to the implicit feature tracking approach is a residual minimization problem over the reduced nonlinear manifold that simultaneously determines the reduced coordinates in the affine space and the domain mapping that minimize the residual of the unreduced PDE discretization. This is analogous to standard minimum-residual reduced-order models, except instead of only minimizing the residual over the affine subspace of PDE states, our method enriches the optimization space to also include admissible domain mappings. The nonlinear trial manifold is constructed by using the proposed residual minimization formulation to determine domain mappings that cause parametrized features to align in a reference domain for a set of training parameters. Because the feature is stationary in the reference domain, i.e., the convective nature of solution removed, the snapshots are effectively compressed to define an affine subspace. The space of domain mappings, originally constructed using high-order finite elements, are also compressed in a way that ensures the boundaries of the original domain are maintained. Several numerical experiments are provided, including transonic and supersonic, inviscid, compressible flows, to demonstrate the potential of the method to yield accurate approximations to convection-dominated problems with limited training.

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