An optimization-based coupling of reduced order models with efficient reduced adjoint basis generation approach

TL;DR

This work advances optimization-based coupling (OBC) by integrating reduced-order models (ROMs) in a time-dependent, non-overlapping domain decomposition for an advection–diffusion transmission problem. It adopts an optimize-then-reduce strategy and introduces Modified Gradient Descent for the Reduced Adjoint (MGD$m$RA) to efficiently generate adjoint bases from forward-state snapshots, enabling parallel-in-time computation. Numerical results show that state-based adjoint bases can be prohibitively large, while MGD1RA matches full-order accuracy with far fewer adjoint snapshots, offering substantial speedups over FOM-FOM coupling. The study provides guidance on selecting state and adjoint ROM sizes and demonstrates substantial practical benefits for ROM-ROM couplings in time-dependent PDEs.

Abstract

Optimization-based coupling (OBC) is an attractive alternative to traditional Lagrange multiplier approaches in multiple modeling and simulation contexts. However, application of OBC to time-dependent problems has been hindered by the computational cost of finding the stationary points of the associated Lagrangian, which requires primal and adjoint solves. This issue can be mitigated by using OBC in conjunction with computationally efficient reduced order models (ROM). To demonstrate the potential of this combination, in this paper we develop an optimization-based ROM-ROM coupling for a transient advection-diffusion transmission problem. We pursue the ``optimize-then-reduce'' path towards solving the minimization problem at each timestep and solve reduced-space adjoint system of equations, where the main challenge in this formulation is the generation of adjoint snapshots and reduced bases for the adjoint systems required by the optimizer. One of the main contributions of the paper is a new technique for efficient adjoint snapshot collection for gradient-based optimizers in the context of optimization-based ROM-ROM couplings. We present numerical studies demonstrating the accuracy of the approach along with comparison between various approaches for selecting a reduced order basis for the adjoint systems, including decay of snapshot energy, average iteration counts, and timings.

Figures (5)

• Figure 1: Example of two non-overlapping domains sharing an interface
• Figure 1: Left: initial conditions and right: monolithic FEM solution at the final time step.
• Figure 2: $u_m$ and $u_c$ on the interface and $u_c$ on the entire domain.
• Figure 3: Comparison of snapshot energy for state solutions in $\Omega_1$ (left) and $\Omega_2$ (right), for $\nu=\{1e-3,1e-5\}$, as a function of the number of modes retained.
• Figure 4: Comparison of snapshot energy for adjoint solutions in $\Omega_1$ (left) and $\Omega_2$ (right), for values of $\nu$ as a function of the number of modes retained.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5