[Paper Review] Accelerating design optimization using reduced order models

Youngsoo Choi, Geoffrey Oxberry+2 more|arXiv (Cornell University)|Sep 25, 2019

Topology Optimization in Engineering57 references34 citations

TL;DR

The paper proposes-ROM-based acceleration for PDE-constrained design optimization by using inexact ROM solves early and ROM-recycling Krylov solves near convergence, demonstrated on topology optimization problems.

ABSTRACT

Although design optimization has shown its great power of automatizing the whole design process and providing an optimal design, using sophisticated computational models, its process can be formidable due to a computationally expensive large-scale linear system of equations to solve, associated with underlying physics models. We introduce a general reduced order model-based design optimization acceleration approach that is applicable not only to design optimization problems, but also to any PDE-constrained optimization problems. The acceleration is achieved by two techniques: i) allowing an inexact linear solve and ii) reducing the number of iterations in Krylov subspace iterative methods. The choice between two techniques are made, based on how close a current design point to an optimal point. The advantage of the acceleration approach is demonstrated in topology optimization examples, including both compliance minimization and stress-constrained problems, where it achieves a tremendous reduction and speed-up when a traditional preconditioner fails to achieve a considerable reduction in the number of linear solve iterations.

Motivation & Objective

• Motivate and develop a ROM-based acceleration framework for design optimization and other PDE-constrained problems.
• Replace expensive full-order model solves with reduced-order models or ROM-enhanced iterative solves to speed up iterations.
• Determine when to switch between inexact ROM solves and precise Krylov solves using KKT-condition norms.
• Demonstrate effectiveness on topology optimization examples including compliance and stress constraints.
• Show applicability to both structured and unstructured meshes and discuss implementation in parallel C++ code.

Proposed method

• Use incremental SVD and Gram-Schmidt to build reduced bases on-the-fly from solution snapshots.
• Apply projection-based ROMs and ROM-recycling in Krylov solvers with AMG preconditioning to reduce linear solve iterations.
• Define a KKT-condition-norm threshold to decide when to replace FOM solves with ROMs during optimization.
• Employ an interior-point method (IPM) framework to establish stopping criteria via KKT conditions and to guide ROM switching.
• Construct an optimal reduced basis efficiently with incremental QR or incremental SVD, updating as new snapshots arrive.

Experimental results

Research questions

• RQ1 How can reduced order models (ROMs) accelerate a sequence of large-scale linear solves in gradient-based design optimization?
• RQ2 When should ROMs replace full-order solves during optimization to maintain convergence toward a KKT point?
• RQ3 Can incremental basis updates (QR/SVD) provide an efficient, on-the-fly ROM basis for PDE-constrained optimization?
• RQ4 What is the impact of ROM-based acceleration on topology optimization problems with compliance and stress constraints?
• RQ5 Do ROM-based methods preserve performance across structured and unstructured meshes?

Key findings

• Replacing exact linear solves with ROMs in early optimization stages yields substantial speed-ups (e.g., reducing linear solve time from 1.7 hours to 0.48 hours for a wind turbine blade problem).
• A hybrid strategy using inexact ROM solves initially and ROM-recycling Krylov methods near convergence can accelerate the overall topology optimization process.
• The norm of the KKT conditions is used to decide when to switch between ROM and FOM solves during optimization.
• Incremental QR and incremental SVD enable on-the-fly construction of reduced bases with manageable computational cost.
• The method is demonstrated on topology optimization problems including compliance minimization and stress-constrained scenarios on both structured and unstructured meshes.
• A fully parallel C++ production code implementation at LLNL supports the approach.