Online Spectral Deflation for State Constrained Optimal Control Problems

2026-06-16

AI Summary

This work addresses the challenge posed by state constraints in parametric PDE-constrained optimal control, where parameter-dependent nonsmooth active sets induce frequent structural and spectral changes in the linear systems, hindering reuse of solver information. The authors propose an online spectral scaling strategy based on a full-domain reference Schur complement: by dynamically restricting precomputed low-frequency reference eigenmodes to the current inactive set, they construct an A-DEF2 scaling basis that accelerates the Jacobi-preconditioned conjugate gradient method. This approach is the first to enable online adaptation of reference-domain spectral information to evolving inactive sets, facilitating efficient reuse within Krylov subspace and multigrid solvers while remaining compatible with POD enhancement and Rayleigh–Ritz reselection for mode construction. Numerical experiments on diffusion, convection–diffusion, and conjugate heat transfer problems demonstrate 55%–98% reductions in iteration counts, with GPU implementations significantly outperforming CPU-based sparse direct solvers and algebraic multigrid methods.

Abstract

Parametric PDE-constrained optimal control with pointwise state constraints requires repeated solution of restricted Schur-complement systems on parameter-dependent inactive sets. In a primal active-set method, each inactive-set system is symmetric positive definite, but the active set can change nonsmoothly with the parameter. The resulting operator may vary in dimension, sparsity pattern, and spectrum, limiting reuse of sparse factorizations, multigrid hierarchies, and Krylov information. We propose a reusable spectral-deflation strategy anchored to one full-domain reference Schur complement. Low reference eigenmodes are computed once, restricted online to each inactive set, and used as an A-DEF2 deflation basis for Jacobi-preconditioned CG. The framework also supports POD enrichment, Rayleigh-Ritz reselection, coarse-grid or analytical reference modes, and conditioning safeguards. Given the active set, the method preserves the high-fidelity inactive-set system and solves it to the prescribed CG tolerance; it accelerates the linear algebra rather than replacing the optimal-control solve with a surrogate. We explain the method through a spectral-coherence view, motivated by interlacing and perturbation arguments and assessed with principal-angle diagnostics. Across diffusion, convection-diffusion, nonlinear thermal, and conjugate-heat-transfer benchmarks, deflation reduces CG iterations by about 55 to 98 percent. GPU deployments also show wall-time gains over CPU sparse-direct and algebraic-multigrid baselines, because the reference basis is built once whereas competing solver structures are rebuilt per instance. Coarse-grid or analytical modes amortize the offline cost within a single parameter sweep; fine-grid eigensolves remain more precompute-limited. Timings isolate the inactive-set linear-solve kernel; reducing the active-set outer loop is outside the present scope.

Problem

• state-constrained optimal control
• parametric PDE-constrained optimization
• inactive-set systems
• spectral variability
• solver reusability

Innovation

• spectral deflation
• parameterized PDE-constrained optimization
• inactive-set methods
• reference eigenmodes
• accelerated Krylov solvers