Accelerated solutions of convection-dominated partial differential equations using implicit feature tracking and empirical quadrature

2023/05/25 by Mirhoseini, Marzieh Alireza, Zahr, Matthew J.
#FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2305.15661

Abstract

This work introduces an empirical quadrature-based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection-dominated problems with limited training. The proposed approach circumvents the slowly decaying n-width limitation of linear model reduction techniques applied to convection-dominated problems by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with bijections of the underlying domain. The reduced-order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online-efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh-independent operations. The proposed reduced-order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock-dominated computational fluid dynamics benchmarks.

Related

• Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking
• An adaptive, training-free reduced-order model for convection-dominated problems based on hybrid snapshots
• Model reduction of convection-dominated viscous conservation laws using implicit feature tracking and landmark image registration
• Accelerated construction of projection-based reduced-order models via incremental approaches
• Adaptive greedy algorithms based on parameter-domain decomposition and reconstruction for the reduced basis method