Fast Linear Interpolation

Zhang, Nathan; Canini, Kevin Robert; Silva, Sean; Gupta, Manisha

doi:10.1145/3423184

Cited by 14 publications

(5 citation statements)

References 11 publications

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“…Although, in our examples, the solution time of FEM was so much faster that continued solving the PDE with FEM on different adapted grids would likely still be considerably faster than solving and evaluating the PINN. In addition, we believe that the evaluation time in FEM could be significantly sped up using more appropriate interpolation methodology, such as Lin & Lee (2005) ; Zhang et al. (2021) .…”

Section: Discussionmentioning

confidence: 99%

Can physics-informed neural networks beat the finite element method?

Grossmann,

Komorowska,

Latz

et al. 2024

IMA Journal of Applied Mathematics

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Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen–Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.

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Section: Discussionmentioning

confidence: 99%

Can physics-informed neural networks beat the finite element method?

Grossmann,

Komorowska,

Latz

et al. 2024

IMA Journal of Applied Mathematics

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show abstract

“…(3) can then be detailed as Agreeably, [16] gather that the linear interpolation technique is simple but very useful. This is demonstrated by [17] when they used linear interpolation to reduce the training samples required in a Machine Learning algorithm and in [18] where linear interpolation was found to optimize the interpolation of Look-Up Tables (LTUs) on standard CPUs or more specifically, compute kernels. In [19], it was used alongside an artificial neural network to estimate the health of lithium-ion batteries in an online system.…”

Section: The Undetermined Cs Problemmentioning

confidence: 99%

An Enhanced Subsampling Technique in Compressive Sensing using Linear Interpolation and Random Measurement Matrix

Osei-Wusu,

Ahene,

Muntaka

2024

Preprint

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In Compressive Sensing, the incoherence of a measurement matrix during subsampling is a crucial requirement for the accurate reconstruction of a signal. However, such incoherence is only probable and not assured when subsampling is done with the widely used random measurement matrix. The study proposes an enhanced subsampling technique that integrates linear interpolation with the conventional random measurement matrix to provide assured incoherence during subsampling in Compressive Sensing. The experiments show that the proposed technique is less costly computationally and does a faster subsampling of an audio digital signal than when the traditional random measurement matrix is used solely. Additionally, the results demonstrated that the proposed technique outperformed state-of-the-art techniques with respect to the accuracy and speed of the signal reconstruction along with the L1 optimization. This was proven through the use of performance evaluation metrics such as computational complexity, execution time and Mean Square Error.

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“…This solves the problem of finding the necessary grid nodes surrounding any point inside an inhomogeneously spaced grid by projecting it to a homogeneous grid on which the localization is negligible in computational cost, i.e. it can only be left or right from the respectively localized node on the original grid [14]. The lookup table is pre-computed for each field during the import.…”

Section: δXmentioning

confidence: 99%

FemtoTrack: A space-charge tracking tool for few-electron ultrashort bunches

Lautenschläger¹,

Niedermayer²

2023

Preprint

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Ultrafast electron experiments usually work with low-emittance few-electron pulsed beams. The structures are usually much larger than the (average) electron pulse size posing challenges to the resolution of simulations. We present the tracking code FemtoTrack, which allows tackling these multi-scale challenges in a simple manner. The computationally most heavy interpolation of the fields is treated either by a moving window or by a supplemental grid, leading to a significant speedup. Space charge is treated by direct particle-particle interaction within each bunch, where however many bunches can be simulated in the same window simultaneously. This allows to obtain statistics similar to what is obtained on the screen in an experiment. FemtoTrack is applied to two examples: An ultrafast nanotip electron source and a length scalable laser-driven electron accelerator on a microchip. In these setups, previous results have been reproduced with tremendous speedup, allowing for parameter scans similar to the tuning of an experiment.

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Fast Linear Interpolation

Cited by 14 publications

References 11 publications

Can physics-informed neural networks beat the finite element method?

Can physics-informed neural networks beat the finite element method?

An Enhanced Subsampling Technique in Compressive Sensing using Linear Interpolation and Random Measurement Matrix

FemtoTrack: A space-charge tracking tool for few-electron ultrashort bunches

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