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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]