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

Understand

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.