Optimization with Gradient and Hessian Information Calculated Using Hyper-Dual Numbers

Fike, Jeffrey A.; Jongsma, S.H.; Alonso, Juan J.; Weide, Edwin Theodorus Antonius van der

doi:10.2514/6.2011-3807

Cited by 28 publications

(14 citation statements)

References 15 publications

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“…To find the optimal value is not easy and the optimal value may not be accurate enough for fast convergence to steady state. In [vdWS12], dual numbers are used (see [FJAvW11]) to approximate the Jacobian numerically. This is an algebraic trick that rids the problem with cancellation and fast convergence is obtained.…”

Section: Applicationsmentioning

confidence: 99%

Review of summation-by-parts schemes for initial–boundary-value problems

Svärd

Nordström

2014

Journal of Computational Physics

395

366

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Abstract. High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-byparts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area.

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Section: Applicationsmentioning

confidence: 99%

Review of summation-by-parts schemes for initial–boundary-value problems

Svärd

Nordström

2014

Journal of Computational Physics

395

366

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“…Because of Julia's type inference capability and the JIT compiler, when the evaluation function is called with complex-perturbed solution variables, every function is compiled with datatypes of all the variables known to the compiler. If a different AD datatype is used, such as hyper-dual numbers [33], all the numerical routines are recompiled, generating efficient machine code to operate on dual numbers. The result is a highly efficient mechanism for algorithmic differentiation, with the full range of compiler optimization available and no source code duplication.…”

Section: Solver Abstraction and Implementationmentioning

confidence: 99%

Investigation of Stabilization Methods for Multi-Dimensional Summation-by-parts Discretizations of the Euler Equations

Crean

Panda

Ashley

et al. 2016

54th AIAA Aerospace Sciences Meeting

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We present an extensible Julia-based solver for the Euler equations that uses a summationby-parts (SBP) discretization on unstructured triangular grids. While SBP operators have been used for tensor-product discretizations for some time, they have only recently been extended to simplices. Here we investigate the accuracy and stability properties of simplexbased SBP discretizations of the Euler equations. Non-linear stabilization is a particular concern in this context, because SBP operators are nearly skew-symmetric. We consider an edge-based stabilization method, which has previously been used for advection-diffusionreaction problems and the Oseen equations, and apply it to the Euler equations. Additionally, we discuss how the development of our software has been facilitated by the use of Julia, a new, fast, dynamic programming language designed for technical computing. By taking advantage of Julia's unique capabilities, code that is both efficient and generic can be written, enhancing the extensibility of the solver.

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“…In these situations it may still be possible to use hyper-dual numbers, if the effect of computing the derivatives can be achieved. One example of this is the solution of a linear system, Ay = b, where derivatives can be computed by several calls to the real-valued routine [8]. First derivatives of the solution of a linear system, Ay = b, can be computed by solving…”

Section: Chapter 3 Parameterization Using Hyper-dual Numbersmentioning

confidence: 99%

Parameterized reduced-order models using hyper-dual numbers.

Fike¹,

Brake²

2013

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The goal of most computational simulations is to accurately predict the behavior of a real, physical system. Accurate predictions often require very computationally expensive analyses and so reduced order models (ROMs) are commonly used. ROMs aim to reduce the computational cost of the simulations while still providing accurate results by including all of the salient physics of the real system in the ROM. However, real, physical systems often deviate from the idealized models used in simulations due to variations in manufacturing or other factors. One approach to this issue is to create a parameterized model in order to characterize the effect of perturbations from the nominal model on the behavior of the system. This report presents a methodology for developing parameterized ROMs, which is based on Craig-Bampton component mode synthesis and the use of hyper-dual numbers to calculate the derivatives necessary for the parameterization.

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Optimization with Gradient and Hessian Information Calculated Using Hyper-Dual Numbers

Cited by 28 publications

References 15 publications

Review of summation-by-parts schemes for initial–boundary-value problems

Review of summation-by-parts schemes for initial–boundary-value problems

Investigation of Stabilization Methods for Multi-Dimensional Summation-by-parts Discretizations of the Euler Equations

Parameterized reduced-order models using hyper-dual numbers.

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