Shearlets and Optimally Sparse Approximations

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springe

Comparison of Stochastic Methods for the Variability Assessment of Technology Parameters

This paper provides and compares two alternative solutions for the simulation of cables and interconnects with the inclusion of the effects of parameter uncertainties, namely the Polynomial Chaos (PC) method and the Response Surface Modeling (RSM). The problem formulation applies to the telegraphers equations with stochastic coefficients. According to PC, the solution requires an expansion of the unknown parameters in terms of orthogonal polynomials of random variables. On the contrary, RSM is based on a least-square polynomial fitting of the system response. The proposed methods offer accuracy and improved efficiency in computing the parameter variability effects on system responses with respect to the conventional Monte Carlo approach. These approaches are validated by means of the application to the stochastic analysis of a commercial multiconductor flat cable. This analysis allows us to highlight the respective advantages and disadvantages of the presented method

A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older and BMO spaces.Comment: 24 page

Conditioning bounds for traveltime tomography in layered media

This paper revisits the problem of recovering a smooth, isotropic, layered wave speed profile from surface traveltime information. While it is classic knowledge that the diving (refracted) rays classically determine the wave speed in a weakly well-posed fashion via the Abel transform, we show in this paper that traveltimes of reflected rays do not contain enough information to recover the medium in a well-posed manner, regardless of the discretization. The counterpart of the Abel transform in the case of reflected rays is a Fredholm kernel of the first kind which is shown to have singular values that decay at least root-exponentially. Kinematically equivalent media are characterized in terms of a sequence of matching moments. This severe conditioning issue comes on top of the well-known rearrangement ambiguity due to low velocity zones. Numerical experiments in an ideal scenario show that a waveform-based model inversion code fits data accurately while converging to the wrong wave speed profile

Sparse Deterministic Approximation of Bayesian Inverse Problems

We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. To this end, we estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number $N$ of unknowns appearing in the parameteric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise

A Probabilistic proof of the breakdown of Besov regularity in LL-shaped domains

{We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in $L$-shaped domains. In particular, we obtain (probabilistic) integral representations for the solution. We also recover Grisvard's classic result on the angle-dependent breakdown of the regularity of the solution measured in a Besov scale

Disk and circumsolar radiances in the presence of ice clouds

The impact of ice clouds on solar disk and circumsolar radiances is investigated using a Monte Carlo radiative transfer model. The monochromatic direct and diffuse radiances are simulated at angles of 0 to 8° from the center of the sun. Input data for the model are derived from measurements conducted during the 2010 Small Particles in Cirrus (SPARTICUS) campaign together with state-of-the-art databases of optical properties of ice crystals and aerosols. For selected cases, the simulated radiances are compared with ground-based radiance measurements obtained by the Sun and Aureole Measurements (SAM) instrument. First, the sensitivity of the radiances to the ice cloud properties and aerosol optical thickness is addressed. The angular dependence of the disk and circumsolar radiances is found to be most sensitive to assumptions about ice crystal roughness (or, more generally, non-ideal features of ice crystals) and size distribution, with ice crystal habit playing a somewhat smaller role. Second, in comparisons with SAM data, the ice cloud optical thickness is adjusted for each case so that the simulated radiances agree closely (i.e., within 3 %) with the measured disk radiances. Circumsolar radiances at angles larger than ≈ 3° are systematically underestimated when assuming smooth ice crystals, whereas the agreement with the measurements is better when rough ice crystals are assumed. Our results suggest that it may well be possible to infer the particle roughness directly from ground-based SAM measurements. In addition, the results show the necessity of correcting the ground-based measurements of direct radiation for the presence of diffuse radiation in the instrument's field of view, in particular in the presence of ice clouds.Peer reviewe

Investigating the driving mechanisms of coronal mass ejections

The objective of this investigation was to first examine the kinematics of coronal mass ejections (CMEs) using EUV and coronagraph images, and then to make a comparison with theoretical models in the hope to identify the driving mechanisms of the CMEs. We have studied two CMEs which occurred on 2006 Dec. 17 (CME06) and 2007 Dec. 31 (CME07). The models studied in this work were catastrophe, breakout, and toroidal instability models. We found that after the eruption, the accelerations of both events exhibited a drop before increasing again. Our comparisons with the theories suggested that CME06 can be best described by a hybrid of the catastrophe and breakout models while CME07 is most consistent with the breakout model.Comment: 9 pages 7 figure