Asymptotic analysis of P.D.E.s with wide–band noise disturbances, and expansion of the moments

Bouc, R.; Pardoux, Étienne

doi:10.1080/07362998408809044

Cited by 44 publications

(19 citation statements)

References 9 publications

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“…The extra term in (A.2) is the Stratonovich correction. The white noise limit for stochastic partial differential equations is analyzed in [10] and a rigrous theory of the Ito-Schrödinger equation is given in [11]. The ergodic theory of the ItoSchroedinger equation is explored in [17].…”

Section: Discussionmentioning

confidence: 99%

Statistical Stability in Time Reversal

Papanicolaou¹,

Sølna²,

Ryzhik³

2004

SIAM J. Appl. Math.

107

146

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Abstract. When a signal is emitted from a source, recorded by an array of transducers, time reversed and re-emitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency, remote sensing regime, and show that, because of multiple scattering, in an inhomogeneous or random medium it can improve beyond the diffraction limit. We also show that the back-propagated signal from a spatially localized narrow-band source is self-averaging, or statistically stable, and relate this to the self-averaging properties of functionals of the Wigner distribution in phase space. Time reversal from spatially distributed sources is self-averaging only for broad-band signals. The array of transducers operates in a remote-sensing regime so we analyze time reversal with the parabolic or paraxial wave equation.

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Section: Discussionmentioning

confidence: 99%

Statistical Stability in Time Reversal

Papanicolaou¹,

Sølna²,

Ryzhik³

2004

SIAM J. Appl. Math.

107

146

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“…More generally, white noise limits for random ordinary differential equations are studied in [8] and for partial differential equations in [12]. A recent study of white noise limits for Schrödinger and Wigner equations is given in [15,16].…”

Section: The White Noise Limitmentioning

confidence: 99%

“…When the propagation distance is large compared to the correlation length, then the random potential in the Schrödinger equation tends to white noise in the propagation direction [12,1,15,16]. We begin here with this white noise, Itô-Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

Self-Averaging from Lateral Diversity in the Itô–Schrödinger Equation

Papanicolaou¹,

Ryzhik²,

Sølna³

2007

Multiscale Model. Simul.

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We consider the random Schrödinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Itô-Schrödinger stochastic partial differential equation (SPDE) which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions tends to zero but is large compared to the correlation length.

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“…One can then do the narrow beam approximation as we did in section 7, and this can be found in the literature in many places, in [Fur93] as well as in this appendix. The white noise or δ-correlation limit leading to (38) is also considered in [BP84].…”

Section: A Comments and References For The Transport Approximationmentioning

confidence: 99%

Super-resolution in time-reversal acoustics

Blomgren

Papanicolaou

Zhao

2002

The Journal of the Acoustical Society of America

325

339

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We analyze theoretically and with numerical simulations the phenomenon of super-resolution in time-reversal acoustics. A signal that is recorded and then re-transmitted by an array of transducers, propagates back though the medium and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution.We give a theoretical treatment of this phenomenon and present numerical simulations which confirm the theory.

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Asymptotic analysis of P.D.E.s with wide–band noise disturbances, and expansion of the moments

Cited by 44 publications

References 9 publications

Statistical Stability in Time Reversal

Statistical Stability in Time Reversal

Self-Averaging from Lateral Diversity in the Itô–Schrödinger Equation

Super-resolution in time-reversal acoustics

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