Reducing Memory Requirements in Scientific Computing and Optimal Control

Götschel, Sebastian; Tycowicz, Christoph von; Polthier, Konrad; Weiser, Martin

doi:10.1007/978-3-319-23321-5_10

Cited by 4 publications

(11 citation statements)

References 58 publications

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“…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.…”

Section: Multilevel Transform Coding On Unstructured Grids For Comprementioning

confidence: 99%

“…Using linear finite elements, spatial adaptivity is performed individually for state and adjoint using a hierarchical error estimator [82], with a restriction to at most 25, 000 vertices in space. The adaptively refined grids were stored using the methods from [83], which reduced the storage space for the mesh to less than 1 bit/vertex (see [34]). Lossy compression of state values, i.e., the finite element solutions v and w, at all time steps, affects the accuracy of the reduced gradient computed by adjoint methods, and results in inexact quasi-Newton updates.…”

Section: Pde-constrained Optimizationmentioning

confidence: 99%

“…For reconstruction the mesh has to be stored as well. Fortunately, knowledge about the mesh refinement algorithm can be used for extremely efficient compression of the mesh structure [34]. Table 1: Compression factors for adaptive mesh refinement and transform coding of the peak function shown in Figure 10.…”

Section: Adaptive Grid Refinementmentioning

confidence: 99%

“…The horizontal line shows the approximate discretization error of the reduced gradient. See also[34].…”

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confidence: 99%

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Compression Challenges in Large Scale Partial Differential Equation Solvers

Götschel

Weiser

2019

Algorithms

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Solvers for partial differential equations (PDE) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that needs to be stored and transmitted on all levels of the memory hierarchy. Often, bandwidth is the limiting factor due to relatively small arithmetic intensity, and increasingly so due to the growing disparity between computing power and bandwidth. Consequently, data compression techniques have been investigated and tailored towards the specific requirements of PDE solvers during the last decades. This paper surveys data compression challenges and discusses examples of corresponding solution approaches for PDE problems, covering all levels of the memory hierarchy from mass storage up to main memory. Exemplarily, we illustrate concepts at particular methods, and give references to alternatives.

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Section: Multilevel Transform Coding On Unstructured Grids For Comprementioning

confidence: 99%

Section: Pde-constrained Optimizationmentioning

confidence: 99%

Section: Adaptive Grid Refinementmentioning

confidence: 99%

“…The horizontal line shows the approximate discretization error of the reduced gradient. See also[34].…”

mentioning

confidence: 99%

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Compression Challenges in Large Scale Partial Differential Equation Solvers

Götschel

Weiser

2019

Algorithms

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“…When rðf; gÞ is close to −1 or +1, f and g are said to be inversely or directly associated, respectively. If rðf; gÞ is close to zero, this means less or no association between f and g. The NRMSE over the range of the observed data can be defined as [9]:…”

mentioning

confidence: 99%

Model-based quantization for perceptually weighted compressed video sensing

et al. 2016

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Exploiting perceptual-based weighting can improve the reconstruction quality for compressed video sensing (CVS). Nevertheless, practical implementation of the compressed sensing (CS) requires quantizing the measurements. We propose a simplified sampling rate model for the perceptual CVS to achieve compromise between the number of measurements and the quantization bit-depth, which are the main contributing factors in the CS rate-distortion (RD) performance. The proposed model can achieve near optimal RD-performance obtained through exhaustive simulations. In addition, simulation results show that the quantized perceptual CVS achieves remarkable RD-performance gain, with lower sampling rate, compared to applying the quantization model on the standard CS.

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Impact of mixed precision and storage layout on additive Schwarz smoothers

Schneck

Weiser

Wende

2021

Numerical Linear Algebra App

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The growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth-bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage.

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Reducing Memory Requirements in Scientific Computing and Optimal Control

Cited by 4 publications

References 58 publications

Compression Challenges in Large Scale Partial Differential Equation Solvers

Compression Challenges in Large Scale Partial Differential Equation Solvers

Model-based quantization for perceptually weighted compressed video sensing

Impact of mixed precision and storage layout on additive Schwarz smoothers

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