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

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Abstract

This paper develops a powerful new variant, dubbed \(\varDelta \)-DOGS(\(\varOmega _Z\)), 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 \(\varDelta \)-DOGS(\(\varOmega _Z\)) algorithm, which we prove to be globally convergent under the appropriate assumptions, combines (a) the essential ideas of our \(\varDelta \)-DOGS(\(\varOmega \)) algorithm, which is designed to efficiently optimize a nonconvex objective function f(x) within a nonconvex feasible domain \(\varOmega \) defined by a number of constraint functions \(c_\kappa (x)\), with (b) our \(\varDelta \)-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 \(\varDelta \)-DOGS(\(\varOmega _Z\)) as compared with the original \(\varDelta \)-DOGS(\(\varOmega \)) 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.

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Notes

  1. Proofs of these properties are provided in §3 of [9].

  2. A time marching scheme is said to be L-stable if its stability region contains the entire left-half plane (LHP), and \(\sigma (\infty ) = \lim _{z \rightarrow \infty } \sigma (z) = 0\).

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Acknowledgements

The authors gratefully acknowledge Fred Y. Hadaegh, Rebecca Castano, David Hanks, Navid Dehghani, and Firouz M. Naderi for their support, and Sebastien Le Digabel for for his constructive feedback. The authors gratefully acknowledge funding from AFOSR FA 9550-12-1-0046, from the Cymer Center for Control Systems & Dynamics, from the Leidos corporation in support of this work. Also, the research was supported by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

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Authors and Affiliations

  1. Flow Control and Coordinated Robotics Labs, UC San Diego, San Diego, USA

    Ryan Alimo, Daniele Cavaglieri, Pooriya Beyhaghi & Thomas R. Bewley

  2. Jet Propulsion Laboratory, California Institute of Technology, Pasadensa, USA

    Ryan Alimo

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  1. Ryan Alimo
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  2. Daniele Cavaglieri
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  4. Thomas R. Bewley
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Alimo, R., Cavaglieri, D., Beyhaghi, P. et al. Design of IMEXRK time integration schemes via Delaunay-based derivative-free optimization with nonconvex constraints and grid-based acceleration. J Glob Optim 79, 567–591 (2021). https://doi.org/10.1007/s10898-019-00855-1

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