ALLOWAP: Algorithms for Large-scale Optimization of Wave Propagation Problems

Description

The goal of the ALLOWAPP project is to design space-time parallel algorithms for optimization problems that arise when modelling wave phenomena. Such problems occur in data assimilation for geophysical applications, and in medical applications such as brain imaging. To make the optimization tractable, parallel computers must be used to cope with the large amounts of data and intensive computation inherent to these problems. In the last decade, parallel-in-time methods have made enormous progress: for parabolic problems, a near-optimal scaling with respect to the number of processors has been achieved (scalability). For wave propagation, there has been no such success.In this project, we will design innovative, space-time parallel methods for solving optimization problems with wave constraints. Our methods will be used to tackle two concrete problems, namely wave localization in complex geometries and data assimilation in geophysical and environmental systems. The former has important applications in telecommunications, cordless charging and medical imaging, while the latter will lead to improvements in air quality prediction and geophysical hazard management.We will consider three interrelated aspects. The first aspect is the direct simulation of hyperbolic systems, which must be done repeatedly during the optimization process. Here, we propose using an optimized Schwarz waveform relaxation method with many subdomains in space, and adaptive pipelining in time. This approach increases the scalability of the overall algorithm by solving the problem not only over many subdomains, but also many time steps. We will also consider parallelization in time by direct methods.The second aspect is the optimization over bounded time horizons. Here, our approach is to split the full optimality system into many subsystems in time and in space, and to use transmission conditions to ensure consistency with the global solution. We will then use the discrete Hilbert Uniqueness Method to derive optimal transmission conditions. Approximating these conditions by local, easy-to-implement conditions will then lead to highly efficient methods, when combined with specially designed schemes to handle high frequencies, such as bi-grid filtering or regularization.The third aspect concerns the assimilation of infinite streams of data, where adjoint techniques are inapplicable. We will tackle this problem by combining time-parallel simulation with observer approaches. Our idea is to group the observed data by time blocks, process them sequentially according to their order of arrival using a Luenberger-type observer that is parallelized in time. Our goal is to make the acceleration factor independent of the observer used.