Loss of polarization of elliptically polarized collapsing beams

Patwardhan, Gauri; Gao, Xiaohui; Sagiv, Amir; Dutt, Avik; Ginsberg, Jared S.; Ditkowski, Adi; Fibich, Gadi; Gaeta, Alexander L.

doi:10.1103/physreva.99.033824

Cited by 9 publications

(2 citation statements)

References 57 publications

(67 reference statements)

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“…where 0 < ǫ ≪ 1, t ≥ 0, and x ∈ R, describes the propagation of elliptically polarized, ultrashort pulses in optical fibers [2], of elliptically polarized continuous-wave (CW) beams in a bulk medium [37,46], Stokes and anti-Stokes radiation in Raman amplifiers [40], and rogue water-waves formation at the interaction of crossing seas [1]. We consider (7.1) with an elliptically-polarized Gaussian input pulse with a random amplitude [37,46] (7.2)…”

Section: And Both Approximations Use N = 64 Sample Points (C) L 1 Err...mentioning

confidence: 99%

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

Ditkowski¹,

Fibich²,

Sagiv³

2020

SIAM/ASA J. Uncertainty Quantification

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The effect of uncertainties and noise on a quantity of interest (model output) is often better described by its probability density function (PDF) than by its moments. Although density estimation is a common task, the adequacy of approximation methods (surrogate models) for density estimation has not been analyzed before in the uncertainty-quantification (UQ) literature. In this paper, we first show that standard surrogate models (such as generalized polynomial chaos), which are highly accurate for moment estimation, might completely fail to approximate the PDF, even for one-dimensional noise. This is because density estimation requires that the surrogate model accurately approximates the gradient of the quantity of interest, and not just the quantity of interest itself. Hence, we develop a novel spline-based algorithm for density-estimation whose convergence rate in L q is polynomial in the sampling resolution. This convergence rate is better than that of standard statistical density-estimation methods (such as histograms and kernel density estimators) at dimensions 1 ≤ d ≤ 5 2 m, where m is the spline order. Furthermore, we obtain the convergence rate for density estimation with any surrogate model that approximates the quantity of interest and its gradient in L ∞ . Finally, we demonstrate our algorithm for problems in nonlinear optics and fluid dynamics.1 g in L q , the corresponding density p g might not be a good approximation of p f . Indeed, for p g to approximate p f , then g ′ needs to be close to f ′ , and g ′ (α) should also vanish if and only if f ′ (α) does. These conditions might not be satisfied by some of the above-mentioned standard UQ methods. In contrast, spline interpolation approximates both the function and its gradient [3,25,41,45], and does not tend to produce spurious extremal points. Therefore, we construct a novel algorithm for density estimation in uncertainty-propagation problems using splines as our surrogate model. With cubic splines, our density-estimation algorithm has a guaranteed convergence rate of at least h 3 , where h is the maximal sampling spacing (resolution). More generally, with splines of order m, the convergence rate is at least h m . These rates are superior to those of the standard kernel density estimators [52,59], for noise dimension 1 ≤ d ≤ 5 2 m. Our choice of splines is motivated by the availability and efficiency of one-and multi-dimensional spline toolboxes. Nevertheless, other surrogate models can be used in this algorithm, and indeed this paper lays the theoretical framework for deriving the convergence rate of such methods (Corollaries 4.8 and 5.5). We show, essentially, that density estimation convergence can be performed with any surrogate model for which the L ∞ error of both the function and its gradient converge to zero as the spacing resolution h vanishes. Because we only rely on solving the underlying deterministic model (i.e., our method is non-intrusive), and because interpolation by spline is a standard numerical procedure, our proposed method is...

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Section: And Both Approximations Use N = 64 Sample Points (C) L 1 Err...mentioning

confidence: 99%

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

Ditkowski¹,

Fibich²,

Sagiv³

2020

SIAM/ASA J. Uncertainty Quantification

Self Cite

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“…Indeed, the general settings presented above have spurred numerous papers in a field of computational science known as Uncertainty-Quantification (UQ), see e.g., [14,22,43,53,54,55]. Perhaps surprisingly, the full approximation of µ (rather than its moments alone) in these particular settings received little theoretical attention in the literature, even though it is of practical importance in diverse fields such as ocean waves [1], computational fluid dynamics [8], hydrology [9], aeronautics [17], biochemistry [25], and nonlinear optics [30,36]. Even though f − g q does not control p µ − p ν p in general (see e.g., Fig.…”

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confidence: 99%

The Wasserstein Distances Between Pushed-Forward Measures with Applications to Uncertainty Quantification

Sagiv¹

2019

Preprint

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In the study of dynamical and physical systems, the input parameters are often uncertain or randomly distributed according to a measure ̺. The system's response f pushes forward ̺ to a new measure f * ̺ which we would like to study. However, we might not have access to f , but to its approximation g. This problem is common in the use of surrogate models for numerical uncertainty quantification (UQ). We thus arrive at a fundamental question -if f and g are close in an L q space, does the measure g * ̺ approximate f * ̺ well, and in what sense? Previously, it was demonstrated that the answer to this question might be negative when posed in terms of the L p distance between probability density functions (PDF). Instead, we show in this paper that the Wasserstein metric is the proper framework for this question. For domains in R d , we bound the Wasserstein distance Wp(f * ̺, g * ̺) from above by f − g q . Furthermore, we prove lower bounds for for the cases where p = 1 and p = 2 (for d = 1) in terms of moments approximation. From a numerical analysis standpoint, since the Wasserstein distance is related to the cumulative distribution function (CDF), we show that the latter is well approximated by methods such as spline interpolation and generalized polynomial chaos (gPC).

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Spectral convergence of probability densities for forward problems in uncertainty quantification

Sagiv

2022

Numer. Math.

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Loss of polarization of elliptically polarized collapsing beams

Cited by 9 publications

References 57 publications

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

The Wasserstein Distances Between Pushed-Forward Measures with Applications to Uncertainty Quantification

Spectral convergence of probability densities for forward problems in uncertainty quantification

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