A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

Alekseenko, Alexander; Nguyen, Truong; Wood, Aihua W.

doi:10.3934/krm.2018047

Cited by 7 publications

(9 citation statements)

References 47 publications

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“…The post-collision velocities for the pair of particles will be v + ξ and v 1 + ξ, where v and v 1 are given by (3). We notice, in particular, that choices of θ and ε in (3) are not affected by ξ.…”

Section: Shift Invariance Property Of Kernelmentioning

confidence: 99%

“…Another essential attribute of an efficient numerical formulation of the collision operator consists in re-writing it in the form of a convolution [8,7,27,13,12,23,22]. A bilinear convolution form [3] follows for the Galerkin projection of the collision operator by exploring translational invariance of the collision operator [20]. We argue in this paper that this convolution form leads to development of efficient discretizations of the collision operator using structured locally supported bases.…”

Section: Introductionmentioning

confidence: 99%

“…In [1,2,17] a closely related translational invariance of the Galerkin projection of the collision operator was used to reduce the storage requirements of pre-computed collision kernels. In [3], the translational invariance was used to introduce a bilinear convolution form of the Galerkin projection of the collision operator. We will show that, in the case of uniform grids, this convolution form allows to re-write the collision operator as a convolution of multidimensional sequences.…”

Section: Introductionmentioning

confidence: 99%

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Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(N^2) $\end{document}</tex-math></inline-formula> operations using the discrete fourier transform

Alekseenko¹,

Limbacher²

2019

Kinetic &Amp; Related Models

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We present a numerical algorithm for evaluating the Boltzmann collision operator with O(N 2 ) operations based on high order discontinuous Galerkin discretizations in the velocity variable. To formulate the approach, Galerkin projection of the collision operator is written in the form of a bilinear circular convolution. An application of the discrete Fourier transform allows to rewrite the six fold convolution sum as a three fold weighted convolution sum in the frequency space. The new algorithm is implemented and tested in the spatially homogeneous case, and results in a considerable improvement in speed as compared to the direct evaluation. Simultaneous and separate evaluations of the gain and loss terms of the collision operator were considered. Less numerical error was observed in the conserved quantities with simultaneous evaluation.2000 Mathematics Subject Classification. 76P05,76M10,65M60.

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Section: Shift Invariance Property Of Kernelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(N^2) $\end{document}</tex-math></inline-formula> operations using the discrete fourier transform

Alekseenko¹,

Limbacher²

2019

Kinetic &Amp; Related Models

Self Cite

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“…Simulation of gas mixtures and gases with internal energies, as well as multidimensional models can be found in [14][15][16][17][18][19][20], and references therein. Other fast methods include representing the solution as a sum of homogeneous Gaussians [21,22], polynomial spectral discretization [23], utilizing non-uniform meshes [24], and a hyperbolic cross approximation [25]. Additional review of recent results can be found in [26,27].…”

Section: Introductionmentioning

confidence: 99%

Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

2021

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The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, there would be immediate value in an efficient simulation method. The collision integral component of the equation is the main contributor of the large complexity. A plethora of new mathematical and numerical approaches have been proposed in an effort to reduce the computational cost of solving the Boltzmann collision integral, yet it still remains prohibitively expensive for large problems. This paper aims to accelerate the computation of this integral via machine learning methods. In particular, we build a deep convolutional neural network to encode/decode the solution vector, and enforce conservation laws during post-processing of the collision integral before each time-step. Our preliminary results for the spatially homogeneous Boltzmann equation show a drastic reduction of computational cost. Specifically, our algorithm requires O(n3) operations, while asymptotically converging direct discretization algorithms require O(n6), where n is the number of discrete velocity points in one velocity dimension. Our method demonstrated a speed up of 270 times compared to these methods while still maintaining reasonable accuracy.

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“…T HE Gaussian mixture model (GMM) is a probabilistic model that assumes all the observed data points are generated from a mixture of a finite number of Gaussian (normal) distributions [1]- [4]. It has wide applications in pattern recognition and unsupervised machine learning [5], [6], big data analytics [7]- [10], and image segmentation and denoising [11]- [15], as well as recent applications in applied and computational physics, e.g., gas kinetic [16] and plasma kinetic algorithms [17], [18]. Some other applications of GMM can be found in [19]- [22].…”

Section: Introductionmentioning

confidence: 99%

An Adaptive EM Accelerator for Unsupervised Learning of Gaussian Mixture Models

Nguyen,

Chen,

Chacon

2020

Preprint

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We propose an Anderson Acceleration (AA) scheme for the adaptive Expectation-Maximization (EM) algorithm for unsupervised learning a finite mixture model from multivariate data (Figueiredo and Jain 2002). The proposed algorithm is able to determine the optimal number of mixture components autonomously, and converges to the optimal solution much faster than its non-accelerated version. The success of the AA-based algorithm stems from several developments rather than a single breakthrough (and without these, our tests demonstrate that AA fails catastrophically). To begin, we ensure the monotonicity of the likelihood function (a the key feature of the standard EM algorithm) with a recently proposed monotonicity-control algorithm (Henderson and Varahdan 2019), enhanced by a novel monotonicity test with little overhead. We propose nimble strategies for AA to preserve the positive definiteness of the Gaussian weights and covariance matrices strictly, and to conserve up to the second moments of the observed data set exactly. Finally, we employ a K-means clustering algorithm using the gap statistic to avoid excessively overestimating the initial number of components, thereby maximizing performance. We demonstrate the accuracy and efficiency of the algorithm with several synthetic data sets that are mixtures of Gaussians distributions of known number of components, as well as data sets generated from particle-in-cell simulations. Our numerical results demonstrate speed-ups with respect to nonaccelerated EM of up to 60× when the exact number of mixture components is known, and between a few and more than an order of magnitude with component adaptivity.

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A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

Cited by 7 publications

References 47 publications

Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(N^2) $\end{document}</tex-math></inline-formula> operations using the discrete fourier transform

Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(N^2) $\end{document}</tex-math></inline-formula> operations using the discrete fourier transform

Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

An Adaptive EM Accelerator for Unsupervised Learning of Gaussian Mixture Models

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