Incremental proper orthogonal decomposition for PDE simulation data

Fareed, Hiba; Singler, John R.; Zhang, Yangwen; Shen, Jiguang

doi:10.1016/j.camwa.2017.09.012

Cited by 31 publications

(62 citation statements)

References 20 publications

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“…In the following result, we give a basic property of the core SVD. The result is similar to Proposition 2.3 in [13]. The proof is also similar, and is omitted.…”

Section: The Matrix Svd and Weighted Inner Productssupporting

confidence: 74%

“…As we show in this note, for certain types of finite dimensional data the SVD required for the continuous time POD can be reduced to a matrix SVD with respect to weighted inner products. Below, we review this type of matrix SVD, mostly following [13]. First, we give notation.…”

Section: The Matrix Svd and Weighted Inner Productsmentioning

confidence: 99%

“…See [13] for more discussion about the core SVD. If both inner products are unweighted, we call the core SVD the standard core SVD for clarity.…”

Section: The Matrix Svd and Weighted Inner Productsmentioning

confidence: 99%

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A note on incremental POD algorithms for continuous time data

Fareed

Singler

2019

Applied Numerical Mathematics

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In our earlier work [13, Fareed et al., Comput. Math. Appl. 75 (2018), no. 6, 1942-1960, we developed an incremental approach to compute the proper orthogonal decomposition (POD) of PDE simulation data. Specifically, we developed an incremental algorithm for the SVD with respect to a weighted inner product for the discrete time POD computations. For continuous time data, we used an approximate approach to arrive at a discrete time POD problem and then applied the incremental SVD algorithm. In this note, we analyze the continuous time case with simulation data that is piecewise constant in time such that each data snapshot is expanded in a finite collection of basis elements of a Hilbert space. We first show that the POD is determined by the SVD of two different data matrices with respect to weighted inner products. Next, we develop incremental algorithms for approximating the two matrix SVDs with respect to the different weighted inner products. Finally, we show neither approximate SVD is more accurate than the other; specifically, we show the incremental algorithms return equivalent results.

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“…In the following result, we give a basic property of the core SVD. The result is similar to Proposition 2.3 in [13]. The proof is also similar, and is omitted.…”

Section: The Matrix Svd and Weighted Inner Productssupporting

confidence: 74%

Section: The Matrix Svd and Weighted Inner Productsmentioning

confidence: 99%

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A note on incremental POD algorithms for continuous time data

Fareed

Singler

2019

Applied Numerical Mathematics

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“…Since the original snapshot matrix S can not be stored, only an approximation of the actual truncated SVD in (33) can be computed. Methods to compute an incremental SVD are introduced in [34,35], with the latter referring to Brand's incremental algorithm [36] which is used in the present study with minor modifications. With the previously computed basis B at hand, the approximation of S is known…”

Section: Methods C: Incremental Svdmentioning

confidence: 99%

Data-Driven Microstructure Property Relations

Lißner

Fritzen

2019

MCA

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An image based prediction of the effective heat conductivity for highly heterogeneous microstructured materials is presented. The synthetic materials under consideration show different inclusion morphology, orientation, volume fraction and topology. The prediction of the effective property is made exclusively based on image data with the main emphasis being put on the 2-point spatial correlation function. This task is implemented using both unsupervised and supervised machine learning methods. First, a snapshot proper orthogonal decomposition (POD) is used to analyze big sets of random microstructures and thereafter compress significant characteristics of the microstructure into a low-dimensional feature vector. In order to manage the related amount of data and computations, three different incremental snapshot POD methods are proposed. In the second step, the obtained feature vector is used to predict the effective material property by using feed forward neural networks. Numerical examples regarding the incremental basis identification and the prediction accuracy of the approach are presented. A Python code illustrating the application of the surrogate is freely available.

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“…Most attention in the MOR community has so far focused on SVD methods based on Arnoldi or Lanczos iterations, as proposed in the work of Chaturantabut and Sorensen and on incremental SVD algorithms. () These are also suitable for reducing the asymptotic complexity to

scriptO false(m n k false)

. For an overview, we refer to the work of Bach et al…”

Section: State Of the Artmentioning

confidence: 99%

Fixed‐precision randomized low‐rank approximation methods for nonlinear model order reduction of large systems

Bach

Duddeck

Song

2019

Numerical Meth Engineering

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Summary Many model order reduction (MOR) methods employ a reduced basis boldV∈Rm×k to approximate the state variables. For nonlinear models, V is often computed using the snapshot method. The associated low‐rank approximation of the snapshot matrix boldA∈Rm×n can become very costly as m,n grow larger. Widely used conventional singular value decomposition methods have an asymptotic time complexity of scriptOfalse(minfalse(mn2,m2nfalse)false), which often makes them impractical for the reduction of large models with many snapshots. Different methods have been suggested to mitigate this problem, including iterative and incremental approaches. More recently, the use of fast and accurate randomized methods was proposed. However, most work so far has focused on fixed‐rank approximations, where rank k is assumed to be known a priori. In case of nonlinear MOR, stating a bound on the precision is usually more appropriate. We extend existing research on randomized fixed‐precision algorithms and propose a new heuristic for accelerating reduced basis computation by predicting the rank. Theoretical analysis and numerical results show a good performance of the new algorithms, which can be used for computing a reduced basis from large snapshot matrices, up to a given precision ε.

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Incremental proper orthogonal decomposition for PDE simulation data

Cited by 31 publications

References 20 publications

A note on incremental POD algorithms for continuous time data

A note on incremental POD algorithms for continuous time data

Data-Driven Microstructure Property Relations

Fixed‐precision randomized low‐rank approximation methods for nonlinear model order reduction of large systems

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