Parallel Lagrange-newton-krylov-schur Methods for Pde-constrained Optimization Part I: the Kkt Preconditioner

1. Introduction. Optimization problems that are constrained by partial differential equations (PDEs) arise naturally in many areas of science and engineering. In the sciences, such problems often appear as inverse problems in which some of the parameters in a simulation are unavailable, and must be estimated by comparison with physical data. These parameters are typically boundary conditions, initial conditions, sources, or coefficients of a PDE. Examples include empirically-determined parameters in a complex constitutive law, and material properties of a medium that is not directly observable. In engineering, PDE-constrained optimization problems often take the form of optimal design or optimal control problems. We refer to the unknown PDE field quantities as the state variables; the PDE constraints as the state equations; solution of the PDE constraints as the forward problem; the inverse, design, or control variables as the decision variables; and the problem of determining the optimal values of the inverse, design, or control variables as the optimization problem. In contrast to the large body of work on parallel PDE solution, very little has been published on parallel algorithms for optimization of PDEs (but see [8], [19], [31], [33]). This is expected: it makes little sense to address the inverse problem until one has successfully tackled the forward problem. However, with the recent maturation of parallel PDE solvers for a number of problem classes, the time is ripe to begin focusing on parallel algorithms for large scale PDE-constrained optimization. Sequential quadratic programming (SQP) methods [11] appear to offer the best hope for smooth optimization of large-scale systems governed by PDEs. SQP methods interleave optimization with simulation, simultaneously improving the design (or control or inversion) while converging the state equations. Thus, unlike popular reduced gradient methods, they avoid complete solution of the state equations at each optimization iteration. Additionally, SQP methods can be made to exploit the structure of the simulation problem, thus building on the advances in parallel PDE solvers over the past 20 years. The current state-of-the-art for solving PDE-constrained optimization problems is reduced SQP (RSQP) methods. implementations of RSQP methods exhibiting high parallel efficiency and good scalability have been developed [19], [28]. Roughly speaking, RSQP methods project the optimization problem onto the space of decision variables (thereby eliminating the state variables), and then solve the resulting reduced system typically using a quasi-Newton method. The advantage of such an approach is that only two linearized forward problems 1 need to …