Simultaneous single-step one-shot optimization with unsteady PDEs

Günther, Stefanie; Gauger, Nicolas R.; Wang, Qiqi

doi:10.1016/j.cam.2015.07.033

Cited by 23 publications

(8 citation statements)

References 23 publications

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“…Through this method they show better sensitivity behavior and successful optimizations; they also applied this method to error estimation and adaptive mesh refinement [17]. This method inherits some of the characteristics shown in the work on one-shot optimzation methods [5] including the guaranteed convergence of the tangent and adjoint problems, due to their similarity to the black box approach [21]. Padway and Mavriplis [16] showed by numerical experiment that for an approximately linearized quasi-Newton fixed point iteration, convergence of the non-linear problem led to a decrease in error from the approximate linearization.…”

Section: Introductionmentioning

confidence: 91%

Approximate Linearization of Fixed Point Iterations: Error Analysis of Tangent and Adjoint Problems Linearized about Non-Stationary Points

Padway,

Mavriplis

2021

Preprint

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Previous papers have shown the impact of partial convergence of discretized PDE on the accuracy of tangent and adjoint linearizations. A series of papers suggested linearization of the fixed point iteration used in the solution process as a means of computing the sensitivities rather than linearizing the discretized PDE, as the lack of convergence of the nonlinear problem indicates that the discretized form of the governing equations has not been satisfied. These works showed that the accuracy of an approximate linearization depends in part on the convergence of the nonlinear system. This work shows an error analysis of the impact of the approximate linearization and the convergence of the nonlinear problem for both the tangent and adjoint modes and provides a series of results for an exact Newton solver, an inexact Newton solver, and a low storage explicit Runge-Kutta scheme to confirm the error analyses.

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Section: Introductionmentioning

confidence: 91%

Approximate Linearization of Fixed Point Iterations: Error Analysis of Tangent and Adjoint Problems Linearized about Non-Stationary Points

Padway,

Mavriplis

2021

Preprint

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“…Since we focused on preconditioning in‐domain Navier–Stokes control problem, we used a laminar flow model in IFISS to study the performance of the MSSS preconditioning technique. The next step to extend this research shall focus on applying the turbulent flow model for the real‐world wind farm control applications, while some recent development in optimal control of unsteady Reynolds‐averaged Navier–Stokes equations (RANS) 74 is a good starting point to extend our preconditioning technique.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

Qiu

Gijzen

Wingerden

et al. 2020

Numerical Linear Algebra App

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In this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle‐point system. The Schur complement for the generalized saddle‐point system is very difficult or even impossible to approximate, which prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle‐point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simplified wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare the performance of the MSSS preconditioner with the performance of the state‐of‐the‐art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for the control of the Navier–Stokes equation.

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“…In brief, the one-shot method consists of simultaneously timestepping an underlying integrator, an adjoint solver, and a parameter update process. The use of approximate gradients and warm-starting lead to much faster algorithms [13,11] but these features also complicate the convergence analysis of optimization algorithms. In order to prove convergence, the relative time-scales of the auxiliary system (state and adjoint integration) and the parameter process (approximate gradient descent) needs to be considered in analyzing these schemes.…”

Section: Related Workmentioning

confidence: 99%

“…In order to prove convergence, the relative time-scales of the auxiliary system (state and adjoint integration) and the parameter process (approximate gradient descent) needs to be considered in analyzing these schemes. Several authors have described practical implementations of one-shot methods with dynamic time-scaling [11,16]. A convergence proof for a class of one-shot methods using adaptive time-scaling was given in [12].…”

Section: Related Workmentioning

confidence: 99%

A persistent adjoint method with dynamic time-scaling and an application to mass action kinetics

Flynn¹

2020

Preprint

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In this article we consider an optimization problem where the objective function is evaluated at the fixed-point of a contraction mapping parameterized by a control variable, and optimization takes place over this control variable. Since the derivative of the fixed-point with respect to the parameter can usually not be evaluated exactly, one approach is to introduce an adjoint dynamical system to estimate gradients. Using this estimation procedure, the optimization algorithm alternates between derivative estimation and an approximate gradient descent step. We analyze a variant of this approach involving dynamic time-scaling, where after each parameter update the adjoint system is iterated until a convergence threshold is passed. We prove that, under certain conditions, the algorithm can find approximate stationary points of the objective function. We demonstrate the approach in the settings of an inverse problem in chemical kinetics, and learning in attractor networks.

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Simultaneous single-step one-shot optimization with unsteady PDEs

Cited by 23 publications

References 23 publications

Approximate Linearization of Fixed Point Iterations: Error Analysis of Tangent and Adjoint Problems Linearized about Non-Stationary Points

Approximate Linearization of Fixed Point Iterations: Error Analysis of Tangent and Adjoint Problems Linearized about Non-Stationary Points

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

A persistent adjoint method with dynamic time-scaling and an application to mass action kinetics

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