Lossy Compression for PDE-constrained Optimization: Adaptive Error Control

Please always quote using this URN: urn:nbn:de:0297-zib-18575
  • For the solution of optimal control problems governed by nonlinear parabolic PDEs, 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 ad-joint 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 which are accurate enough, in many case very fine discretizations in time and space are necessary, which leads to a significant amount of data to be stored and transmitted to mass storage. Lossy compression methods were developed to overcome the storage problem by reducing the accuracy of the stored trajectories. The inexact data induces errors in the reduced gradient and reduced Hessian. In this paper, we analyze the influence of such a lossy trajectory compression method on Newton-CG methods for optimal control of parabolic PDEs and design an adaptive strategy for choosing appropriate quantization tolerances.

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Author:Sebastian GötschelORCiD, Martin WeiserORCiD
Document Type:ZIB-Report
Tag:Newton-CG; lossy compression; optimal control; semilinear parabolic PDEs; trajectory storage
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K58 Semilinear parabolic equations
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J20 Optimal control problems involving partial differential equations
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Mxx Numerical methods [See also 90Cxx, 65Kxx] / 49M15 Newton-type methods
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area) / 68Pxx Theory of data / 68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx]
94-XX INFORMATION AND COMMUNICATION, CIRCUITS / 94Axx Communication, information / 94A29 Source coding [See also 68P30]
Date of first Publication:2013/05/24
Series (Serial Number):ZIB-Report (13-27)
Published in:to be published in Comput. Optim. Appl.