This contribution introduces a model order reduction approach for an advection-reaction problem with a parametrized reaction function. The underlying discretization uses an ultraweak formulation with an $L^2$-like trial space and an 'optimal' test space as introduced by Demkowicz et al. This ensures the stability of the discretization and in addition allows for a symmetric reformulation of the problem in terms of a dual solution which can also be interpreted as the normal equations of an adjoint least-squares problem. Classic model order reduction techniques can then be applied to the space of dual solutions which also immediately gives a reduced primal space. We show that the necessary computations do not require the reconstruction of any primal solutions and can instead be performed entirely on the space of dual solutions. We prove exponential convergence of the Kolmogorov $N$-width and show that a greedy algorithm produces quasi-optimal approximation spaces for both the primal and the dual solution space. Numerical experiments based on the benchmark problem of a catalytic filter confirm the applicability of the proposed method.

翻译：本文针对具有参数化反应函数的对流-反应问题，提出了一种模型降阶方法。其基础离散化采用由Demkowicz等人提出的超弱变分公式，该公式使用类$L^2$试探空间与'最优'检验空间。这保证了离散化的稳定性，并允许将问题对称地重构为对偶解形式，该形式亦可解释为伴随最小二乘问题的正规方程。随后，经典模型降阶技术可应用于对偶解空间，该空间也直接给出了一个简化的原始解空间。我们证明，必要的计算无需重构任何原始解，而可完全在对偶解空间上执行。我们证明了Kolmogorov $N$-宽度的指数收敛性，并表明贪婪算法能为原始解空间与对偶解空间生成拟最优逼近空间。基于催化过滤器基准问题的数值实验验证了所提方法的适用性。