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Design of IMEXRK time integration schemes via Delaunay-based derivative-free optimization with nonconvex constraints and grid-based acceleration

Author

Listed:
  • Ryan Alimo

    (UC San Diego
    California Institute of Technology)

  • Daniele Cavaglieri

    (UC San Diego)

  • Pooriya Beyhaghi

    (UC San Diego)

  • Thomas R. Bewley

    (UC San Diego)

Abstract

This paper develops a powerful new variant, dubbed $$\varDelta $$ Δ -DOGS( $$\varOmega _Z$$ Ω Z ), of our Delaunay-based Derivative-free Optimization via Global Surrogates family of algorithms, and uses it to identify a new, low-storage, high-accuracy, Implicit/Explicit Runge–Kutta (IMEXRK) time integration scheme for the stiff ODEs arising in high performance computing applications, like the simulation of turbulence. The $$\varDelta $$ Δ -DOGS( $$\varOmega _Z$$ Ω Z ) algorithm, which we prove to be globally convergent under the appropriate assumptions, combines (a) the essential ideas of our $$\varDelta $$ Δ -DOGS( $$\varOmega $$ Ω ) algorithm, which is designed to efficiently optimize a nonconvex objective function f(x) within a nonconvex feasible domain $$\varOmega $$ Ω defined by a number of constraint functions $$c_\kappa (x)$$ c κ ( x ) , with (b) our $$\varDelta $$ Δ -DOGS(Z) algorithm, which reduces the number of function evaluations on the boundary of the search domain via the restriction that all function evaluations lie on a Cartesian grid, which is successively refined as the iterations proceed. The optimization of the parameters of low-storage IMEXRK schemes involves a complicated set of nonconvex constraints, which leads to a challenging disconnected feasible domain, and a highly nonconvex objective function; our simulations indicate significantly faster convergence using $$\varDelta $$ Δ -DOGS( $$\varOmega _Z$$ Ω Z ) as compared with the original $$\varDelta $$ Δ -DOGS( $$\varOmega $$ Ω ) optimization algorithm on the problem of tuning the parameters of such schemes. A low-storage third-order IMEXRK scheme with remarkably good stability and accuracy properties is ultimately identified using this approach, and is briefly tested on Burgers’ equation.

Suggested Citation

  • Ryan Alimo & Daniele Cavaglieri & Pooriya Beyhaghi & Thomas R. Bewley, 2021. "Design of IMEXRK time integration schemes via Delaunay-based derivative-free optimization with nonconvex constraints and grid-based acceleration," Journal of Global Optimization, Springer, vol. 79(3), pages 567-591, March.
  • Handle: RePEc:spr:jglopt:v:79:y:2021:i:3:d:10.1007_s10898-019-00855-1
    DOI: 10.1007/s10898-019-00855-1
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    References listed on IDEAS

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    1. Pooriya Beyhaghi & Thomas R. Bewley, 2016. "Delaunay-based derivative-free optimization via global surrogates, part II: convex constraints," Journal of Global Optimization, Springer, vol. 66(3), pages 383-415, November.
    2. Pooriya Beyhaghi & Daniele Cavaglieri & Thomas Bewley, 2016. "Delaunay-based derivative-free optimization via global surrogates, part I: linear constraints," Journal of Global Optimization, Springer, vol. 66(3), pages 331-382, November.
    3. Pooriya Beyhaghi & Thomas Bewley, 2017. "Implementation of Cartesian grids to accelerate Delaunay-based derivative-free optimization," Journal of Global Optimization, Springer, vol. 69(4), pages 927-949, December.
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