A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints

Journal Article · Sun Nov 30 00:00:00 EST 2025 · Numerical Linear Algebra with Applications

DOI:https://doi.org/10.1002/nla.70040· OSTI ID:3014007

Hartland, Tucker A. ^[1]; Petra, Cosmin G. ^[1]; Petra, Noémi ^[2]; Wang, Jingyi ^[1]

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1. Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
2. Univ. of California, Merced, CA (United States)

Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.

• Research Organization: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
• Sponsoring Organization: National Science Foundation (NSF); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC52-07NA27344
• OSTI ID: 3014007
• Report Number(s): LLNL--JRNL-858226
• Journal Information: Numerical Linear Algebra with Applications, Journal Name: Numerical Linear Algebra with Applications Journal Issue: 6 Vol. 32; ISSN 1070-5325; ISSN 1099-1506
• Publisher: Wiley
• Country of Publication: United States
• Language: English

Related Subjects

PDE-constrained optimization
Schur complement
algebraic multigrid
finite element method
inequality-constrained optimization
interior-point method
saddle-point matrices
preconditioned Krylov subspace solvers