Hierarchical Approximate Proper Orthogonal Decomposition

Himpe, Christian; Leibner, Tobias; Rave, Stephan

doi:10.1137/16m1085413

Cited by 55 publications

(69 citation statements)

References 38 publications

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“…In the above examples, the POD consumed multiple hours of time. Hierarchical approximations, such as [42], might mitigate the effects by enabling parallel computations. Overall, the long-term perspective is to extend this RB framework efficiently to the context of dissipative materials.…”

Section: Advantages Compared To General Displacement-based Schemesmentioning

confidence: 99%

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Kunc

Fritzen

2019

MCA

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The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5-100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method.

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Section: Advantages Compared To General Displacement-based Schemesmentioning

confidence: 99%

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Kunc

Fritzen

2019

MCA

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“…Even for moderately sized systems, the computation of the (cross) Gramian's singular vectors may be a computationally challenging task 4 . To compute the dominant subspace projections from the cross Gramian, or the controllability and observability Gramians, the hierarchical approximate proper orthogonal decomposition (HAPOD) [18] is used. The HAPOD enables a swift computation of left singular vectors of arbitrary partitioned data sets, based on a selected projection error (on the input data) ε > 0 and a tree hierarchy with the data (Gramian) partitions as leafs.…”

Section: Fused Computationmentioning

confidence: 99%

Cross-Gramian-based dominant subspaces

Benner

2019

Adv Comput Math

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A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.

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“…In the following, we present a different approach which combines a randomized with an incremental SVD. Similar to the HAPOD method [12], the hybrid procedure is based on splitting A into n c different chunks with an approximately equal number of snapshots such that nc i=1 r i = n, where r i denotes the number of snapshots in chunk i. The SVD of A can then be computed by combining the RBs resulting from the randomized SVDs of its chunks in an incremental manner (figure 1) 1 .…”

Section: Singular Value Decomposition Algorithmsmentioning

confidence: 99%

Randomized nonlinear model order reduction methods for large dynamical problems with explicit time integration

Bach

Bartol

Ceglia

et al. 2018

Proc Appl Math and Mech

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Projection-based nonlinear model order reduction (MOR) often involves the computation of a truncated singular value decomposition (SVD) of a snapshot matrix A ∈ IR m×n , m ≥ n, computed from training simulations, where only the first k basis vectors are retained. A can however become very large in case of detailed models with a large number of degrees of freedom (DoFs), or when many snapshots are present. This is often the case for explicit FEM simulations of industrial problems. Computing the SVD can then become problematic, as most widely used SVD algorithms for MOR have an asymptotic complexity of O(mn 2 ) [1]. Randomized algorithms can alternatively be used to compute only the k most significant basis vectors. Such methods only scale with O(mnk) or even O(mn log(k)) [2]. We present a modified hybrid randomized incremental algorithm for efficient reduced basis (RB) computation with low memory requirements, and apply the method to a large-scale example problem. The results show that the hybrid algorithm is capable of efficiently computing a RB even for very large problems at a manageable computational effort. Projection based nonlinear model order reductionConsider the second-order nonlinear equation Mẍ = f (x,ẋ, t) describing the behavior of a mechanical system, where M ∈ IR m×m is the mass matrix, x ∈ IR m are the nodal displacements of a FEM discretization, and f is the vector of nonlinear nodal forces, depending on x, its first time derivativeẋ, and the time t. The principal assumption of projection based nonlinear MOR methods is that the solutions of physical problems described by partial differential equations (PDEs) typically reside on a low-dimensional manifold of dimension k, with k m [3]. The displacements x can thus be approximated bywhere x 0 is a constant (e.g. the initial conditions), V is the reduced basis, andx is the vector of reduced global DoFs. In practice, V is often computed using the method of snapshots [3], more precisely as the truncated left-singular vectors of an SVD of the resulting snapshot matrix A ∈ IR m×n . We refer to [3][4][5] for further details, including additional clustering and hyper-reduction methods. Singular value decomposition algorithmsThere are several methods for computing the (truncated) SVD of a given m × n snapshot matrix A. Perhaps the simplest one is to compute the full SVD following a bidiagonalization of A [1], and subsequently truncate it to a desired rank k or up to a specified tolerance. Methods based on first computing an eigenvalue decomposition of A T A can be more efficient; however both approaches have an asymptotic time complexity of O(mn 2 ). By contrast, efficient iterative (e.g. Arnoldi or Lanczos) and incremental algorithms [6] can compute a rank-k truncated SVD at an asymptotic complexity as low as O(mnk). Over the past decade, randomized low-rank approximations have emerged as an interesting alternative to conventional approaches. A randomized rank-k SVD can be computed at an asymptotic complexity of O(mn log(k)) using structured random matrice...

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Hierarchical Approximate Proper Orthogonal Decomposition

Cited by 55 publications

References 38 publications

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Cross-Gramian-based dominant subspaces

Randomized nonlinear model order reduction methods for large dynamical problems with explicit time integration

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