Particle-in-Wavelets scheme for the 1D Vlasov-Poisson equations

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Application to Tore Supra movies

In this section we present preliminary results obtained by applying the WVD-reconstruction method to an experimental movie frame acquired during the Tore Supra shot 42967. The plasma is in a detached phase, and therefore is strongly radiative in the neighborhood of a magnetic surface located at r ≃ 0.4. The camera is oriented so that its line of sight is approximately tangent to this magnetic surface. The geometric parameters are provided in the last column of Table. II.4.1. Note that these parameters were estimated from the movie itself using a key-point detection method based on some visible features of the vessel. One frame (Fig. II.4.5, a) was picked out randomly within the acquired movie to test WVD-inversion. Note that the time average of the whole movie has been subtracted from the frame, so that only the fluctuations are visible. The inverted emissivity map is shown in Fig. II.4.5 (b). The strong radiative activity going on around r = 0.4 is well detected by the algorithm. Except some very intense and localized artifacts, the field is quite smooth. The movie reveals that the structures are preserved in time and propagate counter-clockwise. In Fig. II.4.5 (c), we show the artificial movie frame obtained by applying K to the inverted emissivity map. The main features that were visible by eye in the original movie frame (Fig. II.4.5, a) are strongly enhanced in the artificial one, while the noise has been reduced to a very low level.

Kernel density estimation

Given a sequence of independent and identically distributed measurements, the nonparametric density estimation problem consists in finding the underlying probability density function (PDF), with no a priori assumptions on its functional form. Here we discuss general ideas on this difficult problem for which a variety of statistical methods have been developed. Further details can be found in the statistics literature, e.g. Ref. (Silverman, 1986). Consider a number Np of statistically independent particles with phase space coordinates (Xn)1≤n≤Np distributed in Rd according to a PDF f. This data can come from a PIC or a Monte-Carlo, full f or δf simulation. Formally, the sample PDF can be written as fδ(x) = 1 Np XNp n=1 δ(x − Xn) (III.1.1).
where δ is the Dirac distribution. Because of its lack of smoothness, Eq. (III.1.1) is far from the actual distribution f according to most reasonable definitions of the error. The dependence of fδ on the statistical fluctuations in (Xn) can lead to an artificial increase of the collisionality, which could be problematic in the modeling of near collisionless plasmas of interest to controlled fusion. Beyond introducing dissipation, noise can lead to other problems including self-heating and momentum spread which, for example, is known to be an issue in laser-plasma interaction computations. Also, computations involving derivatives of f, like for example quasilinear fluxes in wave-particle interaction calculations, can be seriously compromised by poor reconstruction techniques.

Bases of orthogonal wavelets

Wavelets are a standard mathematical tool to analyze and compute non stationary signals. Here we recall basic concepts and definitions. Further details can be found in Ref. (Farge, 1992) and references therein. The construction takes place in the Hilbert space L2(R) of square integrable functions. An orthonormal family (ψj,i(x))j∈N,i∈Z is called a wavelet family when its members are dilations and translations of a fixed function ψ called the mother wavelet: ψj,i(x) = 2j/2ψ(2jx − i) (III.1.5) where j indexes the scale of the wavelets and i their positions, and ψ satisfies R ψ = 0. In the following we shall always assume that ψ has compact support of length S. The coefficients hf | ψj,ii = R fψj,i of a function f for this family are denoted by ( ˜ fj,i). These coefficients describe the fluctuations of f at scale 2−j around position i 2j . Large values of j correspond to fine scales, and small values to coarse scales. Some members of the commonly used Daubechies 6 wavelet family are shown in the left panel of Fig. 1.

Wavelet based density estimation

The multiscale nature of wavelets allows them to adapt locally to the smoothness of the analyzed function (Mallat, 1999). This fundamental property has triggered their use in a variety of problems. One of their most fruitful applications has been the denoising of intermittent signals (Donoho and Jonhstone, 1994). The practical success of wavelet thresholding to reduce noise relies on the fact that the expansion of signals in a wavelet basis is typically sparse. Sparsity means that the interesting features of the signal are well summarized by a small fraction of large wavelet coefficients. On the contrary, the variance of the noise is spread over all the coefficients appearing in Eq. (III.1.12). Although the few large coefficients are of course also affected by noise, curing the noise in the small coefficients is already a very good improvement. The original setting of this technique, hereafter referred to as global wavelet shrinkage, requires the noise to be additive, stationary, Gaussian and white. It found a first application in plasma physics in Ref. (Farge et al., 2006), where coherent bursts were extracted out of plasma density signals. Since Ref. (Donoho and Jonhstone, 1994), wavelet denoising has been extended to a number of more general situations, like non-Gaussian or correlated additive noise, or to denoise the spectra of locally stationary time series (von Sachs and Schneider, 1996). In particular, the same ideas were developed in Ref. (Vannucci and Vidakovic, 1998; Donoho et al., 1996) to propose a wavelet-based density estimation (WBDE) method based on independent observations. At this point we would like to stress that WBDE assumes nothing about the Gaussianity of the noise, nor on its stationarity. In fact, under the independence hypothesis – which is admittedly quite strong – the statistical properties of the noise are entirely determined by standard probability theory. We refer to Ref. (Vidakovic, 1999) for a review on the applications of wavelets in statistics.
In Ref. (Gassama et al., 2007), global wavelet shrinkage was applied directly to the charge density of a 2D PIC code, in a case were the statistical fluctuations were quasi Gaussian and stationary. In particular, an iterative algorithm (Azzalini et al., 2004), which crucially relies on the stationnarity hypothesis, was used to determine the level of fluctuations. However,in the next section we will show an example where the noise is clearly non-stationary, and this procedure fails.

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Table of contents :

Remerciements
Introduction
I Context and open questions 
I.1 Physical concepts
I.1.1 Turbulent fluid flows
I.1.2 Filamentary plasma flows
I.1.3 Dissipation .
I.2 Statistical models
I.2.1 (Un-)Predictability
I.2.2 The K41 and KBL67 theories
I.2.3 Turbulence models
I.3 Mathematical view
I.3.1 Weak solutions and well-posedness
I.3.2 Dynamical systems and attractors
I.3.3 Singular limits
I.4 Numerical approach
I.4.1 Foundations .
I.4.2 Technological requirements
I.4.3 Examples of current achievements
II Wavelet tools for flow analysis 
II.1 Mathematical theory
II.1.1 Multiresolution analysis
II.1.2 Fast wavelet transform
II.1.3 Wavelet families
II.1.4 Representation of differential operators
II.1.5 Denoising .
II.2 Implementation
II.2.1 Review of some existing implementations
II.2.2 General structure of our approach
II.2.3 Adaptive wavelet transform
II.2.4 Parallelization
II.3 Verification and benchmarking
II.3.1 Parallel efficiency
II.3.2 Representation of translation operators
II.3.3 Power spectrum estimation
II.3.4 Denoising correlated noises
II.4 Application: edge plasma tomography
II.4.1 Introduction .
II.4.2 Reconstruction method
II.4.3 Validation .
II.4.4 Application to Tore Supra movies
II.4.5 Conclusion .
III Particle-in-Wavelets approach for the Vlasov equation 
III.1 Wavelet-based density estimation
III.1.1 Introduction
III.1.2 Methods .
III.1.3 Applications
III.1.4 Summary and Conclusion
III.2 Particle-in-Wavelets scheme for the 1D Vlasov-Poisson equations
III.2.1 Background .
III.2.2 Description of the PIW scheme
III.2.3 Numerical results
III.2.4 Discussion .
IV Regularization of inviscid equations 
IV.1 1D Burgers equation
IV.1.1 Introduction .
IV.1.2 Numerical method
IV.1.3 Deterministic initial condition
IV.1.4 Random initial condition
IV.1.5 Conclusion .
IV.2 Incompressible 2D Euler equations
IV.2.1 Introduction .
IV.2.2 Numerical method
IV.2.3 Results .
IV.2.4 Conclusion and Perspectives
IV.3 Remarks on Galerkin discretizations
V Dissipation at vanishing viscosity 
V.1 Volume penalization
V.2 Molecular dissipation in the presence of walls
V.2.1 Introduction .
V.2.2 Model and numerical method
V.2.3 Results .
V.2.4 Conclusion .
V.3 Turbulent dissipation in 2D homogeneous turbulence
V.3.1 Introduction .
V.3.2 Conditional statistical modelling
V.3.3 Mathematical framework and numerical method
V.3.4 Statistical analysis
V.3.5 Scale-wise coherent vorticity extraction
V.3.6 Interscale enstrophy transfers and production of incoherent enstrophy .
V.3.7 Dynamical influence of the incoherent part
V.3.8 Conclusion .
V.4 Analysis of 3D turbulent boundary layers
V.4.1 Introduction .
V.4.2 Flow configuration and parameters
V.4.3 Orthogonal wavelet decomposition of the turbulent boundary layer flow .
V.4.4 Numerical results
V.4.5 Conclusions and perspectives
Conclusion

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