Large-scale PDE-constrained optimization: An introduction

Optimal design, optimal control, and parameter estimation of systems governed by partial differential equations (PDE) give rise to a class of problems known as PDE-constrained optimization. The size and complexity of the discretized PDEs often pose significant challenges for contemporary optimization methods. Recent advances in algorithms, software, and high performance computing systems have resulted in PDE simulations that can often scale to millions of variables, thousands of processors, and multiple physics interactions. As PDE solvers mature, there is increasing interest in industry and the national labs in solving optimization problems governed by such large-scale simulations. This article provides a brief introduction and overview to the Lecture Notes in Computational Science and Engineering volume entitled Large-Scale PDE-Constrained Optimization. This volume contains nineteen articles that were initially presented at the First Sandia Workshop on Large-Scale PDE-Constrained Optimization. The articles in this volume assess the state-of-the-art in PDE-constrained optimization, identify challenges to optimization presented by modern highly parallel PDE simulation codes and discuss promising algorithmic and software approaches to address them. These contributions represent current research of two strong scientific computing communities, in optimization and PDE simulation. This volume merges perspectives in these two different areas and identifies interesting open questions for further research. We hope that this volume leads to greater synergy and collaboration between these communities.