Location
M3 4001
Candidate
Adam Vieno | Applied Mathematics, University of Waterloo
Title
A Reduced-Order Modeling Approach to Shape Optimization Via Time-Dependent Bases
Abstract
Partial differential equation (PDE)-constrained optimization plays a central role in engineering applications such as fluid flow control, thermal analysis, and aerodynamic
shape optimization. However, its practical use is limited by the need to repeatedly evaluate high-fidelity full-order models (FOMs), which involves solving large-scale nonlinear PDE systems that can become computationally prohibitive, especially in high-dimensional design spaces.
Reduced-order modeling (ROM) techniques mitigate this cost by constructing low-dimensional surrogates of the FOM. However, traditional static approaches often require extensive pre-processing and can struggle to adapt to dynamic nonlinear
changes within the design space.
To address these challenges, we employ the time-dependent basis-CUR (TDB-CUR) framework, which constructs reduced bases dynamically using a time-discrete variational principle. This approach yields a low-rank approximation that remains well-conditioned while naturally adapting to evolving nonlinear solution structures, without relying on a fixed global basis.
Hence, the main contribution of this work is the integration of the TDB-CUR algorithm within a surrogate-inspired shape optimization framework that progressively refines the design space. We propose an iterative strategy that shrinks the design domain toward promising configurations while maintaining a low, static rank during ROM evaluations. This enables improved local approximation quality near the optimum without increasing the reduced dimension. We further introduce heuristics for adaptive rank selection and shrink factor control to balance accuracy and computational cost.
The approach is tested on shape optimization problems governed by the quasi-one-dimensional compressible Euler equations. Numerical experiments with up to ten design variables show that the proposed framework accurately reconstructs the target solutions while substantially reducing computational cost relative to the FOM. Overall, the proposed method represents a promising first step toward more efficient PDE-constrained shape optimization through the integration of dynamic low-rank modeling and iterative design space refinement.