We present a scalable approach to solve a class of elliptic 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, 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 propose a regularization and log-barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)-based solution of the related IP-Gauss-Newton-Schur complement linear systems. The scalability of the approach is demonstrated on an example problem with bound and nonlinear elliptic PDE constraints. The numerical solution of the optimization problem 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 preconditioner and linear system matrix applies, achieved with algebraic multigrid based solvers, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE- and bound-constrained problems.

翻译：本文提出了一种可扩展的方法，用于求解一类带边界约束的椭圆型偏微分方程（PDE）约束优化问题。该方法采用了一种鲁棒的全空间内点（IP）-高斯牛顿优化方法。为了应对每个优化步需要求解的病态内点-高斯牛顿鞍点线性系统，我们提出了两个谱相关的预条件子。这些预条件子利用了正则化PDE约束优化问题中数据的有限信息性。针对基于GMRES的内点-高斯牛顿线性系统求解，提出了一种块高斯-赛德尔预条件子。研究表明，对于一大类PDE和边界约束优化问题，块高斯-赛德尔预条件处理后的内点-高斯牛顿矩阵的谱渐近地独立于离散化，并且不受内点法众所周知的病态性影响。针对基于预条件共轭梯度（PCG）的相关内点-高斯牛顿-舒尔补线性系统求解，我们提出了一种正则化与对数障碍Hessian预条件子。该方法在具有边界和非线性椭圆PDE约束的示例问题上展示了可扩展性。优化问题的数值求解被证明需要独立于离散化次数的内点-高斯牛顿线性求解。此外，线性系统在独立于离散化和内点病态性的预条件Krylov子空间迭代次数内得以求解。预条件子和线性系统矩阵的并行可扩展性应用，通过基于代数多重网格的求解器实现，上述算法可扩展性为一大类PDE和边界约束问题的求解提供了一种并行可扩展的手段。