Adaptive Lossy Trajectory Compression for Optimal Control of Parabolic PDEs
Optimal control problems governed by nonlinear, time dependent PDEs on three dimensional spatial domains are an important tool in many fields, ranging from engineering applications to medicine. For the solution of such optimization problems, methods working on the reduced objective functional are often employed to avoid a full spatio temporal discretization of the problem. The evaluation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, requiring the storage of a full 4D data set. If Newton CG methods are used, two additional trajectories have to be stored. To get numerical results that are accurate enough, in many cases very fine discretizations in time and space are necessary, leading to a significant amount of data to be stored and transmitted to mass storage. This thesis deals with the development and analysis of methods for lossy compression of such finite element solutions. The algorithms are based on a change of basis to reduce correlations in the data, combined with quantization. This is achieved by transforming the finite element coefficient vector from the nodal to the hierarchical basis, followed by rounding the coefficients to a prescribed precision. Due to the inexact reconstruction, and thus inexact data for the adjoint equation, the error induced in the reduced gradient, and reduced Hessian, has to be controlled, to not impede convergence of the optimization. Accuracy requirements of different optimization methods are analyzed, and computable error estimates for the influence of lossy trajectory storage are derived. These tools are used to adaptively control the accuracy of the compressed data. The efficiency of the algorithms is demonstrated on several numerical examples, ranging from a simple linear, scalar equation to a semi linear system of reaction diffusion equations. In all examples considerable reductions in storage space and bandwidth requirements are achieved, without significantly influencing the convergence behavior of the optimization methods. Finally, to go beyond pointwise error control, the hierarchical basis transform can be replaced by more sophisticated wavelet transforms. Numerical experiments indicate that choosing suitable norms for error control allows higher compression factors.