Sparse matrix multiplication package (SMMP)

Bank, Randolph E.; Douglas, Craig C.

doi:10.1007/bf02070824

Cited by 40 publications

(34 citation statements)

References 6 publications

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“…As a result, a standard algorithm [43] for sparse matrix multiplications can be employed to assemble Z which can be used to construct the sparsity pattern of the Jacobian matrix.…”

Section: Sparsity Patternmentioning

confidence: 99%

Efficient solution techniques for implicit finite element schemes with flux limiters

Möller

2007

Numerical Methods in Fluids

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SUMMARYThe algebraic flux correction (AFC) paradigm is equipped with efficient solution strategies for implicit time-stepping schemes. It is shown that Newton-like techniques can be applied to the nonlinear systems of equations resulting from the application of high-resolution flux limiting schemes. To this end, the Jacobian matrix is approximated by means of first-or second-order finite differences. The edge-based formulation of AFC schemes can be exploited to devise an efficient assembly procedure for the Jacobian. Each matrix entry is constructed from a differential and an average contribution edge by edge. The perturbation of solution values affects the nodal correction factors at neighbouring vertices so that the stencil for each individual node needs to be extended. Two alternative strategies for constructing the corresponding sparsity pattern of the resulting Jacobian are proposed. For nonlinear governing equations, the contribution to the Newton matrix which is associated with the discrete transport operator is approximated by means of divided differences and assembled edge by edge. Numerical examples for both linear and nonlinear benchmark problems are presented to illustrate the superiority of Newton methods as compared to the standard defect correction approach.

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“…As a result, a standard algorithm [43] for sparse matrix multiplications can be employed to assemble Z which can be used to construct the sparsity pattern of the Jacobian matrix.…”

Section: Sparsity Patternmentioning

confidence: 99%

Efficient solution techniques for implicit finite element schemes with flux limiters

Möller

2007

Numerical Methods in Fluids

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“…In particular, the performance of RP schemes that involve matrix multiplication can be improved by fast algorithms for sparse matrix multiplication, e.g. [22,23]. Some other RP schemes can also be optimized to take advantage of the data sparsity [15].…”

Section: Fixed-weight Random Projection Layermentioning

confidence: 99%

Training neural networks on high-dimensional data using random projection

Wójcik

Kurdziel

2018

Pattern Anal Applic

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Training deep neural networks (DNNs) on high-dimensional data with no spatial structure poses a major computational problem. It implies a network architecture with a huge input layer, which greatly increases the number of weights, often making the training infeasible. One solution to this problem is to reduce the dimensionality of the input space to a manageable size, and then train a deep network on a representation with fewer dimensions. Here, we focus on performing the dimensionality reduction step by randomly projecting the input data into a lower-dimensional space. Conceptually, this is equivalent to adding a random projection (RP) layer in front of the network. We study two variants of RP layers: one where the weights are fixed, and one where they are fine-tuned during network training. We evaluate the performance of DNNs with input layers constructed using several recently proposed RP schemes. These include: Gaussian, Achlioptas', Li's, subsampled randomized Hadamard transform (SRHT) and Count Sketch-based constructions. Our results demonstrate that DNNs with RP layer achieve competitive performance on high-dimensional real-world datasets. In particular, we show that SRHT and Count Sketch-based projections provide the best balance between the projection time and the network performance.

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“…For mold exterior surfaces in contact with the ambient air, the heat transfer coefficient over vertical surfaces where free convection takes place can be represented as [20] Nu ¼ 0:677Pr 0:25 a ð0:952 þ Pr a Þ À0:25 Gr 0:25 (21) with the Grashof number, Gr, and Prandtl number of air, Pr a , defined as…”

Section: Article In Pressmentioning

confidence: 99%