State Trajectory Compression for Optimal Control with Parabolic PDEs

Abstract

In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and Newton type methods working on the reduced functional are often employed. The computation 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, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression maintaining a certain error bound turns out to be sufficient.

State Trajectory Compression in Optimal Control

“…* E-mail: goetschel@zib.de * * E-mail: weiser@zib.de For the case of uniform refinement and sufficiently smooth solutions of the state equation, a theoretical estimate of the expected performance using hierarchical bases [5] yields a compression factor of about 30 compared to storing floating point values at 64bit per value, with an L ∞ quantization error for the state not exceeding the discretization's interpolation error. See [3] for the derivation of this result. Computational complexity of the spatial compression routine is about half a V-cycle of a multigrid algorithm.…”

“…As the data is used inside a discretized, iterative algorithm, lossy coding of the prediction errors is sufficient as long as the quantization error is below the discretization error. For details we refer to [3].…”

Compression Challenges in Large Scale Partial Differential Equation Solvers

“…This hierarchical basis transform allows an efficient in-place computation of optimal complexity and with low overhead, and is readily available in many finite element codes. The transform coefficients can then be quantized uniformly according to the required accuracy and entropy coded [21], e.g., using a range coder [66]. Typically, this transform coding scheme (TCUG) takes much less than 5% of the iterative solution time, see [21,34,46].…”

“…The transform coefficients can then be quantized uniformly according to the required accuracy and entropy coded [21], e.g., using a range coder [66]. Typically, this transform coding scheme (TCUG) takes much less than 5% of the iterative solution time, see [21,34,46]. A priori error estimates for compression factors and induced distortion can be derived for functions in Lebesgue or Sobolev spaces.…”

Wavefield compression for adjoint methods in full-waveform inversion

“…Related ideas have been proposed in previous work by Unat et al (2009) ;Weiser and Götschel (2012); Hanzich et al (2013) and Götschel and Weiser (2014), achieving compression factors between 8 and 213 for different applications. However, we can obtain significantly higher compression factors by tailoring the methods to the computation of sensitivity kernels in high-order finite-element methods for time-domain FWI.…”