Hierarchical Tensor Approximation of Output Quantities of Parameter-Dependent PDEs

Ballani, Jonas; Grasedyck, Lars

doi:10.1137/140960980

Cited by 43 publications

(76 citation statements)

References 30 publications

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“…To find such a tensor, low‐rank techniques like the singular value decomposition or the cross approximation were generalized to be applicable to the

scriptH

‐tensor format, cf. Grasedyck et al…”

Section: Projection‐based Model Order Reductionmentioning

confidence: 99%

“…More precisely, when

scriptA

is given in the hierarchical Tucker decomposition

scriptO ((), d k^{3} + d k n)

operations are needed to compute the mean. See Ballani and Grasedyck for a detailed elaboration.…”

Section: Projection‐based Model Order Reductionmentioning

confidence: 99%

“…To find such a tensor, low-rank techniques like the singular value decomposition or the cross approximation [44] were generalized to be applicable to the -tensor format, cf. Grasedyck et al [45][46][47] Once an accurate approximationÃ ∈  k of the tensor A ∈ R  is found, the evaluation of a single entry requires only  ( dk 3 ) arithmetic operations.…”

Section: Figure 10mentioning

confidence: 99%

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Challenges of order reduction techniques for problems involving polymorphic uncertainty

et al. 2019

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Modeling of mechanical systems with uncertainties is extremely challenging and requires a careful analysis of a huge amount of data. Both, probabilistic modeling and nonprobabilistic modeling require either an extremely large ensemble of samples or the introduction of additional dimensions to the problem, thus, resulting also in an enormous computational cost growth. No matter whether the Monte‐Carlo sampling or Smolyak's sparse grids are used, which may theoretically overcome the curse of dimensionality, the system evaluation must be performed at least hundreds of times. This becomes possible only by using reduced order modeling and surrogate modeling. Moreover, special approximation techniques are needed to analyze the input data and to produce a parametric model of the system's uncertainties. In this paper, we describe the main challenges of approximation of uncertain data, order reduction, and surrogate modeling specifically for problems involving polymorphic uncertainty. Thereby some examples are presented to illustrate the challenges and solution methods.

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“…To find such a tensor, low‐rank techniques like the singular value decomposition or the cross approximation were generalized to be applicable to the

scriptH

‐tensor format, cf. Grasedyck et al…”

Section: Projection‐based Model Order Reductionmentioning

confidence: 99%

“…More precisely, when

scriptA

is given in the hierarchical Tucker decomposition

scriptO ((), d k^{3} + d k n)

operations are needed to compute the mean. See Ballani and Grasedyck for a detailed elaboration.…”

Section: Projection‐based Model Order Reductionmentioning

confidence: 99%

Section: Figure 10mentioning

confidence: 99%

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Challenges of order reduction techniques for problems involving polymorphic uncertainty

et al. 2019

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“…As an alternative, given sufficient properties of the parameter to solution map, surrogate models can be employed. Here we will exploit two properties, first the regularity will be exploited by a tensorized Chebyshef polynomial basis and second the low-rank approximability by the Hierarchical Tucker format, similar to [3].…”

Section: Introductionmentioning

confidence: 99%

Low rank surrogates for fuzzy‐stochastic partial differential equations

Gruhlke

Eigel

Moser

et al. 2019

Proc Appl Math and Mech

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We consider a particular fuzzy-stochastic PDE depending on the interaction of probabilistic and non-probabilistic (via fuzzy arithmetic in terms of possibility theory) influences. Such a combination is beneficial in an engineering context, where aleatoric and epistemic uncertainties appear simultaneously. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To alleviate this severe obstacle, a compressed low-rank approximation in form of Hierarchical Tucker representation of the desired parametric quantity of interest is derived. The performance of the proposed model order reduction approach is demonstrated.

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“…Compute I >k−1 by the maxvol algorithm and (optionally) truncate. This process can be also organized in a form of a binary tree, which gives rise to the so-called hierarchical Tucker cross algorithm [1]. In total, we need O(dnr 2 ) evaluations of ξ and O(dnr 3 ) additional operations in computations of the maximum volume matrices.…”

mentioning

confidence: 99%

Kriging in Tensor Train Data Format

Dolgov¹,

Litvinenko

Liu

2019

Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Combination of low-tensor rank techniques and the Fast Fourier transform (FFT) based methods had turned out to be prominent in accelerating various statistical operations such as Kriging, computing conditional covariance, geostatistical optimal design, and others. However, the approximation of a full tensor by its low-rank format can be computationally formidable. In this work, we incorporate the robust Tensor Train (TT) approximation of covariance matrices and the efficient TT-Cross algorithm into the FFT-based Kriging. It is shown that here the computational complexity of Kriging is reduced to O(dr 3 n), where n is the mode size of the estimation grid, d is the number of variables (the dimension), and r is the rank of the TT approximation of the covariance matrix. For many popular covariance functions the TT rank r remains stable for increasing n and d. The advantages of this approach against those using plain FFT are demonstrated in synthetic and real data examples.

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Hierarchical Tensor Approximation of Output Quantities of Parameter-Dependent PDEs

Cited by 43 publications

References 30 publications

Challenges of order reduction techniques for problems involving polymorphic uncertainty

Challenges of order reduction techniques for problems involving polymorphic uncertainty

Low rank surrogates for fuzzy‐stochastic partial differential equations

Kriging in Tensor Train Data Format

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