A Scalable Interior-Point Gauss-Newton Method for PDE-Constrained Optimization With Bound Constraints
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- University of California, Merced
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:
- 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
- Country of Publication:
- United States
- Language:
- English
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