In this work, the dual-weighted residual (DWR) method is applied to obtain a certified incremental proper orthogonal decomposition (POD) based reduced order model. A novel approach called MORe DWR (Model Order Rduction with Dual-Weighted Residual error estimates) is being introduced. It marries tensor-product space-time reduced-order modeling with time slabbing and an incremental POD basis generation with goal-oriented error control based on dual-weighted residual estimates. The error in the goal functional is being estimated during the simulation and the POD basis is being updated if the estimate exceeds a given threshold. This allows an adaptive enrichment of the POD basis in case of unforeseen changes in the solution behavior which is of high interest in many real-world applications. Consequently, the offline phase can be skipped, the reduced-order model is being solved directly with the POD basis extracted from the solution on the first time slab and -- if necessary -- the POD basis is being enriched on-the-fly during the simulation with high-fidelity finite element solutions. Therefore, the full-order model solves can be reduced to a minimum, which is demonstrated on numerical tests for the heat equation and elastodynamics.

翻译：本文应用对偶加权残差（DWR）方法构建了具有认证能力的增量式本征正交分解（POD）降阶模型。提出了一种名为MORe DWR（基于对偶加权残差误差估计的模型降阶）的新方法。该方法将张量积时空降阶建模与时间分片技术、增量式POD基生成以及基于对偶加权残差估计的目标导向误差控制相结合。在模拟过程中对目标泛函误差进行实时估计，当误差估计超过预设阈值时，动态更新POD基。这使得POD基能够自适应地丰富，以应对实际应用中常见且备受关注的解行为不可预见变化。因此，离线阶段可被省略，降阶模型直接从首个时间分片解中提取POD基进行求解，并在模拟过程中根据需求利用高保真有限元解实时丰富POD基。由此，全阶模型求解次数可降至最低，该特性通过热传导方程与弹性动力学数值测试得到验证。