CDS&E: Space-Time Parallel Algorithms for Solving PDE-Constrained Optimization Problems

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

Many fields of science and engineering, from atmospheric science to aeronautics, and from material science to cosmology, rely on complex models built from "first principles" in order to study the phenomena of interest; the fundamental physical laws are often described by time-dependent partial differential equations (PDEs). These models are implemented in complex software that performs vast amount of computations and processes large data sets in order to simulate the physical reality. PDE-based models typically run long times on parallel computers, using large numbers of cores. A central problem in these fields of science and engineering is that of optimizing the system of interest according to specific design criteria. For example, in aeronautics, one wants not only to simulate the flight of an airplane, but also to design the best aircraft using shape optimization. In numerical weather prediction, one needs not only simulate the evolution of the atmosphere, but also to optimally utilize the information coming from satellite, aircraft, and ground based measurements in order to keep the forecasts accurate. All these applications seek to optimize systems governed by PDEs. This is an extremely challenging quest, since solving a PDE-constrained optimization problem is one-two orders of magnitude costlier than the underlying forward PDE simulation. There is considerable need for novel highly-parallel solution methodologies. This project develops the algorithmic infrastructure to support large-scale optimization of systems governed by time-dependent PDEs. New ideas will be used to unravel and exploit the inherent parallelism. First, we seek to parallelize the computations in both space and time. The space is divided in subdomains, the time in subintervals, and sub-models on each time subinterval and on each spatial subdomain are run concurrently on different sets of processors. Next, to further increase computational effectiveness, we will build surrogate models, i.e., inexpensive approximate models that capture the main dynamical characteristics of the full PDE-based models. Parallel construction of new surrogates is proposed using local-in-space-and-time information. The main idea is to perform optimization using the inexpensive surrogate models, transferring the improved design to the full PDE-model, re-computing a surrogate for the new configuration, and iterating. Enormous computational savings can be realized this way. Lastly, the new algorithms will be laid on solid theoretical foundations, and will be applied to speed up the incorporation of measurement data in a numerical weather prediction model. The tools developed in this project will enable leap developments in many fields in science and engineering where time-dependent PDE-constrained optimization problems are central. Important examples include aircraft shape optimization, seismic imaging, medical imaging, optimal control of fabrication processes, and inverse problems. The project will directly train one doctoral student and one postdoctoral researcher, will involve undergraduates in research, will develop graduate level educational materials, and will attract students from under-represented groups in parallel computing and large-scale simulations of the physical world. This project develops the algorithmic infrastructure to support large-scale optimization of systems governed by time-dependent partial differential equations (PDEs). PDE optimization problems are central to many fields in science and engineering. They are considerably more complex, and costlier to solve, than the underlying PDE simulations. There is considerable need for highly-parallel solution methodologies. In order to address this, the project proposes a space-time parallel formalism, and new reduced order modeling techniques, that have the potential to speed up the PDE-constrained optimization solution process by several orders of magnitude. (1) Intellectual merit: This work develops a space-time parallel formalism for the solution of large scale PDE-constrained optimization problems. The space is divided in subdomains, the time in subintervals, and the forward and adjoint models are run in parallel on each time subinterval and on each spatial subdomain. Solution continuity equations are imposed more stringently as the optimization process advances. This work formulates reduced-order PDE-constrained optimization problems using local-in-space-and-time reduced order models. Such models can represent the system dynamics much better than traditional global approaches. Moreover, both the off-line construction of local reduced order models and the on-line reduced order simulations can be carried out concurrently on each time subinterval and on each spatial subdomain, resulting in considerable speed-ups. A trust region framework is employed for provably convergent reduced order optimization algorithms. The new methodologies are demonstrated on large atmospheric data assimilation applications. (2) Broader impact: The tools developed in this project will enable leap developments in many fields in science and engineering where time-dependent PDE-constrained optimization problems are central. Important examples include aircraft shape optimization, seismic imaging, medical imaging, optimal control of fabrication processes, and inverse problems. One doctoral student and one postdoctoral researcher are directly trained in PDE-constrained optimization, reduced order modeling, high performance computing, and science applications. Graduate level educational materials are developed. This project is supported by the Office of Advanced Cyberinfrastructure in the Directorate for Computer & Information Science & Engineering and the Division of Mathematical Sciences in the Directorate of Mathematical and Physical Sciences.