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In this paper we treat statistical inference for a wavelet estimator of curves of symmetric positive definite (SPD) using the log-Euclidean distance. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation (AI) and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our AI wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes. • Confidence regions for curves in the space of symmetric positive-definite matrices • Based on second-generation wavelet estimators of the curves of SPD matrices • Asymptotic normality of estimators including derivation of their asymptotic variance • Wild bootstrap confidence regions shown to be valid via derived asymptotic normality [ABSTRACT FROM AUTHOR]

A mesh adaptation method is proposed for solving optimal control problems with non-smooth control. The original optimal control problem (OCP) is transcribed into a nonlinear programming (NLP) problem by using the Runge-Kutta discretization method, in which the NLP can be solved by using standard nonlinear programming codes. The method employs collocations from the dyadic background points, which used for the second-generation wavelet (SGW) translation simultaneously. The SGW is used to approximate the control variables and get the wavelet coefficients once they are obtained. In regions contain discontinuities, the magnitude of the relevant wavelet coefficients is large than other regions. The corresponding dyadic background points are reserved as the collocation points. Furthermore, the approximation error of the control and/or state variables can be predicted by a given threshold. Thus, the accuracy and efficiency can be balanced in a simple way. Finally, the method is demonstrated by three numerical examples from the open literature.

Abstract The advantage of accurate forecasts is that it solves the main problem related to renewable energies: their variability. Indeed, while renewable energies has not yet replaced fossil fuels, in spite of the efforts of many governments, it is because of their intermittent nature, hence the importance of prediction in this field. The new approach for energy prediction that we propose in this paper, is founded on the analysis of the dynamical behavior of the photovoltaic production of the Faculty of Sciences and Technology of Beni Mellal, Morocco. It consists in performing the phase space reconstruction, which allowed us later to build a database for the input of the neural network and thus take into account the dynamics of the system in the forecasting process. Then, in search of more precision, we introduce the wavelet transformation, to simplify the database constructed from phase space reconstruction. Finally, comparing between the predictions and the actual observations confirmed the efficiency of our approach. Highlights • PV production forecast has an major role in its integration into electricity mix and thus in the fight against climate change. • In this work, we propose a new approach for forecasting, based on the exploitation of the phase space reconstruction. • The proposed model is a hybrid method combining the phase space reconstruction, DWT method and recurrent neural network. • The method is tested on real observations to confirm the efficiency of our approach. • Comparison between predictions and actual data confirmed the effectiveness of our approach given the improved results. [ABSTRACT FROM AUTHOR]

Micromechanical analysis of polycrystalline microstructures of metals and alloys, using crystal plasticity finite element (CPFE) models is extensively used for predicting deformation and failure under various conditions of strain-rates, creep and fatigue loading. Many CPFE models involve a large number of degrees of freedom for accurate representation of realistic polycrystalline microstructures. This can lead to prohibitively high computational costs to conduct meaningful analyses of phenomena of interest. To overcome this limitation, the authors have recently developed a wavelet enrichment adapted finite element model in Azdoud and Ghosh (2017) for elastic materials. The method adaptively creates an optimal discretization space conforming to the solution profile by projecting the solution field onto a set of scaling and multi-resolution wavelet basis functions. This paper extends this wavelet adapted FE model to finite deformation, crystal plasticity analysis of polycrystalline microstructures. After presenting the formulations, various validation tests are conducted to examine the convergence rates and computational efficiency of this method. [ABSTRACT FROM AUTHOR]

A second generation wavelet-based adaptive finite-difference Lattice Boltzmann method (FD-LBM) is developed in this paper. In this approach, the adaptive wavelet collocation method (AWCM) is firstly, to the best of our knowledge, incorporated into the FD-LBM. According to the grid refinement criterion based on the wavelet amplitudes of density distribution functions, an adaptive sparse grid is generated by the omission and addition of collocation points. On the sparse grid, the finite differences are used to approximate the derivatives. To eliminate the special treatments in using the FD-based derivative approximation near boundaries, the immersed boundary method (IBM) is also introduced into FD-LBM. By using the adaptive technique, the adaptive code requires much less grid points as compared to the uniform-mesh code. As a consequence, the computational efficiency can be improved. To justify the proposed method, a series of test cases, including fixed boundary cases and moving boundary cases, are invested. A good agreement between the present results and the data in previous literatures is obtained, which demonstrates the accuracy and effectiveness of the present AWCM-IB-LBM. [ABSTRACT FROM AUTHOR]

For a tower structure containing many bars, the possible number of damage patterns in a tower is numerous. To solve this problem, a new two-step method was proposed based on a cross-correlation function and second generation wavelets. First step, by comparing the peaks of cross-correlation function, damage basic units can be found. Second step, we select a suitable second generation wavelet for distinguishing damage patterns with vector angle similarity measure. SVM are used to pinpoint damage bar last. Test result indicates that this method has good locating accuracy. [ABSTRACT FROM AUTHOR]

A parallel adaptive wavelet collocation method for solving a large class of Partial Differential Equations is presented. The parallelization is achieved by developing an asynchronous parallel wavelet transform, which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution. The data are stored using tree-like structure with tree roots starting at a priori defined level of resolution. Both static and dynamic domain partitioning approaches are developed. For the dynamic domain partitioning, trees are considered to be the minimum quanta of data to be migrated between the processes. This allows fully automated and efficient handling of non-simply connected partitioning of a computational domain. Dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process. The parallel efficiency of the approach is discussed based on parallel adaptive wavelet-based Coherent Vortex Simulations of homogeneous turbulence with linear forcing at effective non-adaptive resolutions up to 2048 3 using as many as 2048 CPU cores. [ABSTRACT FROM AUTHOR]

A sequential solution approach based on an adaptive wavelet collocation method is proposed for solving optimal control problems. By expanding the state and control variables with wavelet multi-resolution decomposition, an original optimal control problem can be transcribed into a nonlinear programming problem that can be solved by general methods. In the proposed framework, an iterative algorithm starts from an initial coarse grid with few collocations. Subsequent solutions can thus be obtained on a dynamically refined grid by an adaptive wavelet collocation method, while the previous coarser level solution can be taken as the initial guess for the next iteration. Consequently, the computational grid for direct numerical optimization method is able to automatically adapt to any irregularities or discontinuities in the solution. The efficiency and accuracy of the proposed algorithm are verified by two typical examples. [ABSTRACT FROM AUTHOR]

Abstract: We address the problem of spherical deconvolution in a non-parametric statistical framework, where both the signal and the operator kernel are subject to measurement errors. After a preliminary treatment of the kernel, we apply a thresholding procedure to the signal in a second generation wavelet basis. Under standard assumptions on the kernel, we study the minimax performances of the resulting algorithm in terms of losses ( ) on Besov spaces on the sphere. We hereby extend the application of second generation spherical wavelets to the blind spherical deconvolution framework. It is important to stress that the procedure is adaptive with regard to both the target function sparsity and the kernel blurring effect. We end with the study of a concrete example, putting into evidence the improvement of our procedure on the recent blockwise SVD algorithm of Delattre et al. (2012). [Copyright &y& Elsevier]

Wavelet based near-lossless compression techniques are limited in the literature. In this paper, a new approach for the near-lossless compression of images based on the wavelet transform is presented. Initially, the source image is pre-processed using a logarithmic transformation technique to generate a near-lossless image. The logarithmic transformation stage is followed by a second generation wavelet coding and a Huffman entropy coding. Inverse techniques are applied to get the decompressed image. Objective quality parameters are used to analyze the performance of the proposed method and the results are compared with those of the other near-lossless coders like the JPEG-LS and CALIC. The proposed method outperforms these near-lossless image coders, in terms of the PSNR at all error levels. The bit rates obtained are also comparable. [ABSTRACT FROM AUTHOR]

Abstract: We consider the problem of estimating a density of probability from indirect data in the spherical convolution model. We aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated -penalized criterion. The spherical deconvolution setting has been barely studied so far, and the two main approaches to this problem, namely the SVD and the hard thresholding ones considered only one basis at a time. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. We provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that we put forward gives quite satisfying results in the numerical study when compared with other procedures. [Copyright &y& Elsevier]

In this paper we have developed a second generation wavelet based finite element method for solving elliptic PDEs on two dimensional triangulations using customized operator dependent wavelets. The wavelets derived from a Courant element are tailored in the second generation framework to decouple some elliptic PDE operators. Starting from a primitive hierarchical basis the wavelets are lifted (enhanced) to achieve local scale-orthogonality with respect to the operator of the PDE. The lifted wavelets are used in a Galerkin type discretization of the PDE which result in a block diagonal, sparse multiscale stiffness matrix. The blocks corresponding to different resolutions are completely decoupled, which makes the implementation of new wavelet finite element very simple and efficient. The solution is enriched adaptively and incrementally using finer scale wavelets. The new procedure completely eliminates wastage of resources associated with classical finite element refinement. Finally some numerical experiments are conducted to analyze the performance of this method. [ABSTRACT FROM AUTHOR]

In this paper, we propose novel adaptive wavelet filter bank structures based on the lifting scheme. The filter banks are nonseparable, based on quincunx sampling, with their properties being pixel-wise adapted according to the local image features. Despite being adaptive, the filter banks retain a desirable number of primal and dual vanishing moments. The adaptation is introduced in the predict stage of the filter bank with an adaptation region chosen independently for each pixel, based on the intersection of confidence intervals (ICI) rule. The image denoising results are presented for both synthetic and real-world images. It is shown that the obtained wavelet decompositions perform well, especially for synthetic images that contain periodic patterns, for which the proposed method outperforms the state of the art in image denoising. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new method is presented to identify crack location and size, which is based on stress intensity factor suitable for pipe structure and finite element method of second generation wavelets (SGW-FEM). Pipe structure is dispersed into a series of nested thin-walled pipes. By making use of stress intensity factor of the thin-walled pipe, a new calculation method of crack equivalent stiffness is proposed to solve the stress intensity factor of the pipe structure. On this basis, finite element method of second generation wavelets is used to establish the dynamic model of cracked pipe. Then we combine forward problem with inverse problem in order to establish quantitative identification method of the crack based on frequency change, which provides a non-destructive testing technology with vibration for the pipe structure. The efficiency of the proposed method is verified by experiments. [Copyright &y& Elsevier]

Distributed video coding is a new paradigm for video compression in opposition over the existing video coding standards like MPEG-x and H.26x families. These codecs make use of motion estimation algorithms, the main part of the compression process of the video signal. Hence, the compression process demands a high computational cost and high performance of the encoder. The present work proposes a video compression scheme based on the lossy distributed source coding theory. The proposed compressor encodes the odd frames and the even frames separately using turbo codes and the discrete wavelet transform. The decoding process is performed in a iterative manner and explores the statistical dependency of the video frames of the original sequence. This approach gives a great encoding runtime reduction and allows the implementation at devices with limited computation power and memory. Simulation results show the good performance of the proposed codec in comparison with the state of the art video compression standard H.264/AVC. [ABSTRACT FROM PUBLISHER]

An adaptive multilevel wavelet solver for elliptic equations on an optimal spherical geodesic grid is developed. The method is based on second-generation spherical wavelets on almost uniform optimal spherical geodesic grids. It is an extension of the adaptive multilevel wavelet solver [O. V. Vasilyev and N. K.-R. Kevlahan, J. Comput. Phys., 206 (2005), pp. 412-431] to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. A hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate the Laplace- Beltrami operator. The optimal spherical geodesic grid [Internat. J. Comput. Geom. Appl., 16 (2006), pp. 75-93] is convergent in terms of local mean curvature and has lower truncation error than conventional spherical geodesic grids. The overall computational complexity of the solver is O(N), where N is the number of grid points after adaptivity. The accuracy and efficiency of the method is demonstrated for the spherical Poisson equation. Although the present paper considers the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by choosing an appropriate coarse approximation and using recursive surface subdivision. [ABSTRACT FROM AUTHOR]

Abstract: A dynamic adaptive numerical method for solving partial differential equations on the sphere is developed. The method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids, and is an extension of the adaptive wavelet collocation method to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. An hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate Laplace–Beltrami, Jacobian and flux-divergence operators. The accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows. Although the present paper considers only the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level). [Copyright &y& Elsevier]

Abstract: A distinguishing feature of second generation wavelets is that it can be custom designed depending on applications. Based on second generation wavelets, a multiresolution finite element method is discussed, and its adaptive algorithm is constructed. The hierarchical approximation spaces for finite element analysis are produced. The finite element equation is scale-decoupled via eliminating all coupling in the stiffness matrix of element across scales, then resolved in different spaces independently. The coarse solution can be obtained in the coarse approximation space, and refined by adding details in the detail spaces over several levels till the equation is resolved to the desired accuracy. The scale-decoupling condition of the stiffness matrix of element is proposed by introducing wavelet vanishing moments, and the principle of constructing the scale-decoupling wavelet bases is established. The method establishes an important connection between finite element analysis and multiresolution analysis. The numerical examples have illustrated that the proposed method is powerful to analyze the field problems with changes in gradients and singularities. [Copyright &y& Elsevier]

Abstract: Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time. These methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time. The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space–time intermittency. We propose an adaptive wavelet collocation method for solving highly intermittent problems (e.g. turbulence) on a simultaneous space–time computational domain which naturally adapts both the space and time resolution to match the solution. Besides generating a near optimal grid for the full space–time solution, this approach also allows the global time integration error to be controlled. The efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent (1D + t)-dimensional and (2D + t)-dimensional test problems. In particular, we found that the space–time method uses roughly 18 times fewer space–time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method, while achieving similar global accuracy. [Copyright &y& Elsevier]

Abstract: An adaptive multilevel wavelet collocation method for solving multi-dimensional elliptic problems with localized structures is described. The method is based on multi-dimensional second generation wavelets, and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems [Int. J. Comp. Fluid Dyn. 17 (2003) 151]. Wavelet decomposition is used for grid adaptation and interpolation, while a hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid. In order to accelerate the convergence of the solver, an iterative procedure analogous to the multigrid algorithm is developed. The overall computational complexity of the solver is , where is the number of adapted grid points. The accuracy and computational efficiency of the method are demonstrated for the solution of two- and three-dimensional elliptic test problems. [Copyright &y& Elsevier]

- Today, wavelets are recognized to have a wide range of useful properties that allow them to treat effectively multifacet problems, such as data compression, scale-localization analysis, feature extraction, statistics, numerical simulation, visualization, and communication. Second-generation wavelets represent a recent improvement of the wavelet algorithm, allowing for greater flexibility in the spatial domain and other computational advantages. We will show how these wavelets can be employed to extract large-scale coherent structures from (1) three-dimensional turbulent flows and (2) high Rayleigh number thermal convection. We will discuss the concept of modelling via decomposition into coherent and incoherent fields, taking into account the effect of the incoherent field via statistical modelling. We will discuss wavelet properties and how they can be utilized and integrated in handling large-scale problems in earthquake physics and other nonlinear phenomena in the geosciences. [ABSTRACT FROM AUTHOR]

Over the last 3 years or so, first-generation wavelets have been used to realize superresolution from a captured sequence of low-resolution (LR) degraded frames. Here, it is pointed out that second-generation wavelets (SGWs) are inherently more suited for image superresolution. From preliminary results that exploit subpixel displacements between LR frames to attain superresolution, it is concluded that SGWs show promise and potential to be extremely fast, efficient and versatile for superresolution. [Copyright &y& Elsevier]

A dynamically adaptive numerical method for solving multi-dimensional evolution problems with localized structures is developed. The method is based on the general class of multi-dimensional second-generation wavelets and is an extension of the second-generation wavelet collocation method of Vasilyev and Bowman to two and higher dimensions and irregular sampling intervals. Wavelet decomposition is used for grid adaptation and interpolation, while O ( N ) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The prowess and computational efficiency of the method are demonstrated for the solution of a number of two-dimensional test problems. [ABSTRACT FROM AUTHOR]

The general concept of data compression consists in removing the redundancy existing in data to find a more compact representation. This paper is concerned with a new method of compression using the second generation wavelets based on the lifting scheme, which is a simple but powerful wavelet construction method. It has been proved by its successful application to a real-time monitoring system of large hydraulic machines that it is a promising compression method. [ABSTRACT FROM AUTHOR]