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Numerical Methods for High-Dimensional Kinetic Equations

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Uncertainty Quantification for Hyperbolic and Kinetic Equations

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 14))

Abstract

High-dimensionality is one of the major challenges in kinetic modeling and simulation of realistic physical systems. The most appropriate numerical scheme needs to balance accuracy and computational complexity, and it also needs to address issues such as multiple scales, lack of regularity, and long-term integration. In this chapter, we review state-of-the-art numerical techniques for high-dimensional kinetic equations, including low-rank tensor approximation, sparse grid collocation, and ANOVA decomposition.

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Notes

  1. 1.

    For instance, if we represent p j (z j ) in terms of an interpolant

    $$\displaystyle \begin{aligned}p_j(z_j) = \sum_{k=1}^{q_z} \text{p}_{j,k} \phi_{j,k}(z_j),\end{aligned}$$

    then \(\mathbf {p}_j = (\text{p}_{j,1},\cdots ,\text{p}_{j,q_z})\).

  2. 2.

    The residual W(z) incorporates both the truncation error arising from the time discretization as well as the error arising from the finite-dimensional expansion (5).

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Acknowledgements

We gratefully acknowledge support from DARPA grant N66001-15-2-4055, ARO grant W991NF-14-1-0425, and AFOSR grant FA9550-16-1-0092.

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Authors and Affiliations

  1. University of Maryland, College Park, MD, USA

    Heyrim Cho

  2. University of California, Santa Cruz, CA, USA

    Daniele Venturi

  3. Brown University, Providence, RI, USA

    George Em Karniadakis

Authors
  1. Heyrim Cho
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  2. Daniele Venturi
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  3. George Em Karniadakis
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Correspondence to Heyrim Cho .

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Editors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, Wisconsin, USA

    Shi Jin

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Ferrara, Italy

    Lorenzo Pareschi

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Cho, H., Venturi, D., Karniadakis, G.E. (2017). Numerical Methods for High-Dimensional Kinetic Equations. In: Jin, S., Pareschi, L. (eds) Uncertainty Quantification for Hyperbolic and Kinetic Equations. SEMA SIMAI Springer Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-67110-9_3

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