Derivation of the Stochastic Helmholtz Equation for Sound Propagation in a Turbulent Fluid

Neubert, J. A.; Lumley, John L.

doi:10.1121/1.1912263

Cited by 14 publications

(5 citation statements)

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“…An important objective in DFAT is to accurately shape sound pressure fields in a region of interest by using acoustic sources, such as loudspeakers, located away from and directed at the region of interest. When the refractive index of the medium in the region of interest is random, we may assume that the governing physics are modeled by the stochastic Helmholtz equation, as derived, for example, in [44,31,42]. The stochasticity is often represented by a Karhunen-Loéve (KL) expansion of the refractive index.…”

Section: Optimal Control Of Stochastic Helmholtz Equationmentioning

confidence: 99%

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A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

108

116

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Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

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Section: Optimal Control Of Stochastic Helmholtz Equationmentioning

confidence: 99%

“…In (5.14), u is the wave pressure, k > 0 is the wave number, and (1 + σ ) denotes the stochastic refractive index [44,31,42]. Assuming that the physical domain is sufficiently large, the Sommerfeld radiation condition holds.…”

Section: Optimal Control Of Stochastic Helmholtz Equationmentioning

confidence: 99%

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

108

116

View full text Add to dashboard Cite

show abstract

“…We assume that the refractive index of the medium in the region of interest D R is random and use the stochastic Helmholtz equation to model the governing physics, as derived, for example, in [23,13,21]. Specifically, in D R we set K(y, x) = k 2 (1 + σ (y, x)) 2 , with stochastic refractive index 1 + σ (y, x) given by σ = 0.1, and…”

Section: Optimal Control Of Stochastic Helmholtz Equationmentioning

confidence: 99%

Inexact Objective Function Evaluations in a Trust-Region Algorithm for PDE-Constrained Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2014

SIAM J. Sci. Comput.

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This paper improves the trust-region algorithm with adaptive sparse grids introduced in [SIAM J. Sci. Comput., 35 (2013), pp. A1847-A1879] for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain coefficients. The previous algorithm used adaptive sparse-grid discretizations to generate models that are applied in a trust-region framework to generate a trial step. The decision whether to accept this trial step as the new iterate, however, required relatively high-fidelity adaptive discretizations of the objective function. In this paper, we extend the algorithm and convergence theory to allow the use of low-fidelity adaptive sparse-grid models in objective function evaluations. This is accomplished by extending conditions on inexact function evaluations used in previous trust-region frameworks. Our algorithm adaptively builds two separate sparse grids: one to generate optimization models for the step computation and one to approximate the objective function. These adapted sparse grids often contain significantly fewer points than the high-fidelity grids, which leads to a dramatic reduction in the computational cost. This is demonstrated numerically using two examples. Moreover, the numerical results indicate that the new algorithm rapidly identifies the stochastic variables that are relevant to obtaining an accurate optimal solution. When the number of such variables is independent of the dimension of the stochastic space, the algorithm exhibits near dimension-independent behavior. Introduction.The solution of large-scale optimization problems in science and engineering must accommodate model uncertainties, such as unknown material properties and boundary conditions. The coupling of traditional optimization methods with uncertainty quantification faces significant computational challenges due to the potentially large number of stochastic variables. To address this issue, we have developed an algorithm that shows promising results for optimization problems governed by partial differential equations (PDEs) with random coefficients [18]. This algorithm uses a trust-region framework to manage models that are based on sparse grids. In [18] we use adaptive sparse grids to compute the optimization step and a fixed high-fidelity sparse grid to determine whether to accept the step. As dis-

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“…1] if the acoustic wave length is less than a typical length scale of the turbulent field depending on its Prandtl number. 30…”

Section: A Geometrical Optics Equations In a Inhomogeneous Fluid Mediummentioning

confidence: 99%

Stochastic simulation of the high-frequency wave propagation in a random medium

Lü

Darmon

Potel

2012

Journal of Applied Physics

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A stochastic model is proposed to simulate the propagation of an acoustic wave in a random medium characterized by weak velocity fluctuations. After the acoustic wave propagation through a random velocity field, the propagation field becomes itself a random field. In the developed stochastic approach, the wave field in such random medium is modeled by the combination of the wave field in a mean homogeneous medium and of fluctuation corrections. These corrections are provided by a random field generator whose inputs are the statistical moments of the travel times. Using this stochastic modeling, the propagation of both an incident plane wave and an acoustic realistic beam generated by a real transducer in a random velocity field is calculated and the corresponding simulations are validated by comparison with those obtained with a deterministic model based on geometrical optics. V

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Derivation of the Stochastic Helmholtz Equation for Sound Propagation in a Turbulent Fluid

Cited by 14 publications

References 0 publications

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Inexact Objective Function Evaluations in a Trust-Region Algorithm for PDE-Constrained Optimization under Uncertainty

Stochastic simulation of the high-frequency wave propagation in a random medium

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