Space-time adaptive hierarchical model reduction for parabolic equations.

Area of Science:

• Computational Science and Engineering
• Numerical Analysis
• Scientific Computing

Background:

• Surrogate models are crucial for balancing reliability and computational efficiency in complex scientific and engineering problems.
• Hierarchical model (HiMod) reduction is an effective surrogate modeling technique for phenomena with dominant directional dynamics.
• HiMod naturally incorporates transverse dynamics effects into reduced-order 1D models.

Purpose of the Study:

• To formalize and extend the pointwise HiMod reduction strategy to an unsteady (time-dependent) framework.
• To develop an adaptive procedure for automatically selecting optimal spatial and temporal discretizations for HiMod.
• To validate the proposed unsteady HiMod approach in a practical engineering context.

Main Methods:

• Coupling finite element approximation along the main direction with locally tunable modal representation for transverse dynamics.
• Implementing a pointwise HiMod reduction strategy with modal tuning at each finite element node.
• Employing an adaptive procedure with a posteriori error analysis to determine the HiMod lookup diagram (time partition, mesh, and modal distribution).

Main Results:

• Numerical verification demonstrates the robustness of the adaptive procedure in terms of accuracy, goal quantity sensitivity, and computational savings.
• The adaptive procedure effectively manages accuracy and computational cost for unsteady problems.
• Validation in a groundwater experimental setting yields promising results, indicating the method's practical applicability.

Conclusions:

• Extending HiMod reduction to unsteady problems is a significant advancement for practical engineering applications.
• The HiMod approximation is confirmed as a viable and effective approach through validation.
• The developed adaptive procedure provides an automated way to generate efficient and accurate reduced-order models.