This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin projection onto a linear low-dimensional subspace. In unsteady applications, space-time reduced basis (ST-RB) methods have been developed to achieve a dimension reduction both in space and time, eliminating the computational burden of time marching schemes. However, nonaffine parameterizations dilute any computational speedup achievable by traditional ROMs. Computational efficiency can be recovered by linearizing the nonaffine operators via hyper-reduction, such as the empirical interpolation method in matrix form. In this work, we implement new hyper-reduction techniques explicitly tailored to deal with unsteady problems and embed them in a ST-RB framework. For each of the proposed methods, we develop a posteriori error bounds. We run numerical tests to compare the performance of the proposed ROMs against high-fidelity simulations, in which we combine the finite element method for space discretization on 3D geometries and the Backward Euler time integrator. In particular, we consider a heat equation and an unsteady Stokes equation. The numerical experiments demonstrate the accuracy and computational efficiency our methods retain with respect to the high-fidelity simulations.

翻译：本文针对参数化非定常偏微分方程的高效数值模拟，提出了若干创新技术。基于投影的降阶模型（如降基方法）采用（佩特罗夫-）伽辽金投影到线性低维子空间。在非定常应用中，时空降基方法已被开发以实现空间和时间两个维度的降阶，从而消除时间推进格式的计算负担。然而，非仿射参数化会削弱传统降阶模型的计算加速效果。通过超降阶方法（如矩阵形式的经验插值方法）对非仿射算子进行线性化，可恢复计算效率。本文专门针对非定常问题设计并实现了新型超降阶技术，并将其嵌入时空降基框架中。针对所提出的每种方法，我们推导了后验误差界。通过数值测试，我们将所提降阶模型与高保真模拟进行性能对比——高保真模拟采用三维几何空间离散的有限元法配合后向欧拉时间积分器。具体算例涵盖热传导方程与非定常斯托克斯方程。数值实验表明，本文方法相对于高保真模拟保持了良好的精度与计算效率。