Sensitivity analysis for a class of evolution equations

Gibson, Jay R.; Clark, L. G.

doi:10.1016/0022-247x(77)90224-4

Cited by 15 publications

(9 citation statements)

References 2 publications

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“…That is, for each he P, there are constants K Although the assumptions (H1)-(H13) are rather technical, we shall see that they can easily be verified for delay systems even in the case that the unknown parameter is the delay itself. Therefore, the results presented here remove the limitations placed on the perturbation B(q) in papers [13,16].…”

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confidence: 56%

Untitled

1993

Quart. Appl. Math.

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A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasilinearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations. 1. Introduction. During the past fifteen years considerable effort has been devoted to the problem of estimating unknown parameters in distributed parameter systems. The recent book by Banks and Kunisch [9] provides an excellent account of the progress made in the field. Many parameter estimation problems are best formulated as optimization problems (sometimes over infinite-dimensional "parameter spaces"), and algorithms are developed to minimize an appropriate cost function. Although there are several approaches to these problems, their infinite-dimensional nature requires that numerical approximations be introduced at some point in the analysis. Consequently, there are two basic classes of algorithms for optimization-based parameter estimation. The first type of algorithm, and the most frequently used for dynamic problems, is indirect and proceeds by initially approximating the dynamic equations (e.g., finite elements, finite differences, etc.) and then using optimization algorithms on the finite-dimensional problem. This approach is typified by the papers [1-6, 8, 10, 18].

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confidence: 56%

Untitled

1993

Quart. Appl. Math.

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“…For ordinary differential equations, this type of result is classical [41]. Parameter differentiability theory for various infinite dimensional evolution equations has been obtained more recently [42][43][44][45][46][47][48]. These theoretical results are important since they can provide insight for choosing appropriate numerical methods to approximate the sensitivities.…”

Section: Sensitivity Analysis For Semilinear Parabolic Equationsmentioning

confidence: 97%

Transition to turbulence, small disturbances, and sensitivity analysis I: A motivating problem

Singler

2008

Journal of Mathematical Analysis and Applications

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For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows this approach fails to match experimental results. Recently, new scenarios for transition have been proposed that are based on the interaction of the linearized equations of motion with small disturbances to the flow system. These new "mostly linear" theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored in detail. This paper is the first of a two part work in which sensitivity analysis is used to study the effects of small disturbances on transition to turbulence. In this part, we study a highly sensitive one-dimensional Burgers' equation as a motivating problem. Sensitivity analysis is used to predict the large changes in solutions in the presence of a small disturbance. Also, sensitivity analysis is shown to provide more information about the disturbed nonlinear problem than a purely linear analysis of the problem. In the second part of this work, this analysis will be extended to the three-dimensional Navier-Stokes equations to show that small disturbances have great potential to trigger transition to turbulence.

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“…Sensitivity for partial differential equations has been studied formally in many contexts (see, e.g., [2,10,12,17,26,32,34,43,44,56,57,58,60,64,67].…”

Section: Error Of Continuous Data Assimilation To Viscositymentioning

confidence: 99%

“…In [64], it was shown formally that the sensitivity equations for the steady-state 2D Navier-Stokes equations are globally well-posed. Additionally, rigorous results on the existence of derivatives of solutions to generic linear and nonlinear differential equations with respect to parameters were proven in [12,32] via semigroup theory. After the preparation of this manuscript, it also came to our attention that some analysis for the sensitivity equations has been carried out in the slightly more general context of a large eddy simulation (LES) model of the 2D Navier-Stokes equations in an unpublished PhD thesis [57].…”

Section: Introductionmentioning

confidence: 99%

Parameter Recovery for the 2 Dimensional Navier--Stokes Equations via Continuous Data Assimilation

Carlson¹,

Hudson²,

Larios³

2020

SIAM J. Sci. Comput.

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We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number (or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.

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Sensitivity analysis for a class of evolution equations

Cited by 15 publications

References 2 publications

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Transition to turbulence, small disturbances, and sensitivity analysis I: A motivating problem

Parameter Recovery for the 2 Dimensional Navier--Stokes Equations via Continuous Data Assimilation

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