68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx]
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Language
- English (4)
Keywords
- optimal control (3)
- trajectory storage (3)
- Newton-CG (2)
- compression (2)
- Context-Based Coding (1)
- defibrillation (1)
- lossy compression (1)
- mesh compression (1)
- monodomain model (1)
- semilinear parabolic PDEs (1)
Project
- F9 (4)
Application Area
- F (4)
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 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 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.
In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data.
This paper presents efficient computational techniques for solving an optimization problem in cardiac defibrillation governed by the monodomain equations. Time-dependent electrical currents injected at different spatial positions act as the control. Inexact Newton-CG methods are used, with reduced gradient computation by adjoint solves. In order to reduce the computational complexity, adaptive mesh refinement for state and adjoint equations is performed. To reduce the high storage and bandwidth demand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compression technique for storing trajectory data is applied. An adaptive choice of quantization tolerance based on error estimates is developed in order to ensure convergence. The efficiency of the proposed approach is demonstrated on numerical examples.
Multiresolution meshes provide an efficient and structured representation of geometric objects. To increase the
mesh resolution only at vital parts of the object, adaptive refinement is widely used. We propose a lossless compression
scheme for these adaptive structures that exploits the parent-child relationships inherent to the mesh
hierarchy. We use the rules that correspond to the adaptive refinement scheme and store bits only where some
freedom of choice is left, leading to compact codes that are free of redundancy. Moreover, we extend the coder to
sequences of meshes with varying refinement. The connectivity compression ratio of our method exceeds that of
state-of-the-art coders by a factor of 2 to 7.
For efficient compression of vertex positions we adapt popular wavelet-based coding schemes to the adaptive
triangular and quadrangular cases to demonstrate the compatibility with our method. Akin to state-of-the-art
coders, we use a zerotree to encode the resulting coefficients. Using improved context modeling we enhanced the
zerotree compression, cutting the overall geometry data rate by 7% below those of the successful Progressive
Geometry Compression. More importantly, by exploiting the existing refinement structure we achieve compression
factors that are 4 times greater than those of coders which can handle irregular meshes.