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Deep learning plays a key role in the recent developments of machine learning. This paper develops a deep residual neural network (ResNet) for the regression of nonlinear functions. Convolutional layers and pooling layers are replaced by fully connected layers in the residual block. To evaluate the new regression model, we train and test neural networks with different depths and widths on simulated data, and we find the optimal parameters. We perform multiple numerical tests of the optimal regression model on multiple simulated data, and the results show that the new regression model behaves well on simulated data. Comparisons are also made between the optimal residual regression and other linear as well as nonlinear approximation techniques, such as lasso regression, decision tree, and support vector machine. The optimal residual regression model has better approximation capacity compared to the other models. Finally, the residual regression is applied into the prediction of a relative humidity series in the real world. Our study indicates that the residual regression model is stable and applicable in practice.

In current days, hybrid models have become more essential in a wide range of systems, including medical treatment, aerosol particle handling, laboratory instrument design, industry and naval academia, and more. The influence of linear, nonlinear, and quadratic Rosseland approximations on 3D flow behavior was explored in the presence of Fourier fluxes and Boussinesq quadratic thermal oscillations. Ternary hybrid nanoparticles of different shapes and densities were also included. Using the necessary transformation, the resulting partial differential system is transformed into a governing ordinary differential system, and the solution is then furnished with two mixed compositions (Case-Ⅰ and Case-Ⅱ). Combination one looked at aluminum oxide (Platelet), graphene (Cylindrical), and carbon nanotubes (Spherical), whereas mixture two looked at copper (Cylindrical), copper oxide (Spherical), and silver oxide (Platelet). Many changes in two mixture compositions, as well as linear, quadratic, and nonlinear thermal radiation situations of the flow, are discovered. Case-1 ternary combinations have a wider temperature distribution than Case-2 ternary mixtures. Carbon nanotubes (Spherical), graphene (Cylindrical), and aluminum oxide (Platelet) exhibit stronger conductivity than copper oxide (Spherical), copper (Cylindrical), and silver oxide (Platelet) in Case 1. (Platelet). In copper oxide (Spherical), copper (Cylindrical), and silver (Platelet) compositions, the friction factor coefficient is much higher. The combination of liquids is of great importance in various systems such as medical treatment, manufacturing, experimental instrument design, aerosol particle handling and naval academies, etc. Roseland's quadratic and linear approximation of three-dimensional flow characteristics with the existence of Boussinesq quadratic buoyancy and thermal variation. In addition, we combine tertiary solid nanoparticles with different shapes and densities. In many practical applications such as the plastics manufacturing and polymer industry, the temperature difference is remarkably large, causing the density of the working fluid to vary non-linearly with temperature. Therefore, the nonlinear Boussinesq (NBA) approximation cannot be ignored, since it greatly affects the flow and heat transport characteristics of the working fluid. Here, the flow of non-Newtonian elastomers is controlled by the tension of an elastic sheet subjected to NBA and the quadratic form of the Rosseland thermal radiation is studied.

In nonlinear greedy approximation theory, bidemocratic bases have traditionally played the role of dualizing democratic, greedy, quasi-greedy, or almost greedy bases. In this article we shift the viewpoint and study them for their own sake, just as we would with any other kind of greedy-type bases. In particular we show that bidemocratic bases need not be quasi-greedy, despite the fact that they retain a strong unconditionality flavor which brings them very close to being quasi-greedy. Our constructive approach gives that for each 1 < p < ∞ the space ℓ p has a bidemocratic basis which is not quasi-greedy. We also present a novel method for constructing conditional quasi-greedy bases which are bidemocratic, and provide a characterization of bidemocratic bases in terms of the new concepts of truncation quasi-greediness and partially democratic bases.

We devise a generalization of tree approximation that generates conforming meshes, i.e., meshes with a particular structure like edge-to-edge triangulations. A key feature of this generalization is that the choices of the cells to be subdivided are affected by that particular structure. As main result, we prove near best approximation with respect to conforming meshes, independent of constants such as the completion constant for newest-vertex bisection. Numerical experiments complement the theoretical results.

We study nonlinear n -term wavelet approximation in BMO on R d . Certain Besov-type spaces are naturally involved in the approximation process. Sharp Jackson and Bernstein estimates are established that allow for a complete characterization of the rates of approximation (approximation spaces).

In this paper, we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in Dilworth et al. (Constr Approx 19:575–597, 2003) for Schauder bases. We also give a characterization of 1-strong partial greediness, following the study started in Albiac and Ansorena (Rev Matem Compl 30(1):13–24, 2017), Albiac and Wojtaszczyk (J Approx Theory 138:65–86, 2006).

This paper considers the black-box approximation problem where the goal is to create a regression model using only empirical data without incorporating knowledge about the character of nonlinearity of the approximated function. This paper reports on ongoing work on a nonlinear regression methodology called IBHM which builds a model being a combination of weighted nonlinear components. The construction process is iterative and is based on correlation analysis. Due to its iterative nature, the methodology does not require a priori assumptions about the final model structure which greatly simplifies its usage. Correlation based learning becomes ineffective when the dynamics of the approximated function is too high. In this paper we introduce weighted correlation coefficients into the learning process. These coefficients work as a kind of a local filter and help overcome the problem. Proof of concept experiments are discussed to show

A basic building block in classical potential theory is the fundamental solution of the Laplace equation in ℝ d (Newtonian kernel). The main goal of this article is to study the rates of nonlinear n-term approximation of harmonic functions on the unit ball Bd from shifts of the Newtonian kernel with poles outside B d ¯ in the harmonic Hardy spaces. Optimal rates of approximation are obtained in terms of harmonic Besov spaces. The main vehicle in establishing these results is the construction of highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.

We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted ℓ 1 minimization has recently been shown to be an effective procedure for computing such approximations. We extend this work to the gradient-augmented case. Our main results show that for the same asymptotic sample complexity, gradient-augmented measurements achieve an approximation error bound in a stronger Sobolev norm, as opposed to the L 2 -norm in the unaugmented case. For Chebyshev and Legendre polynomial approximations, this sample complexity estimate is algebraic in the sparsity s and at most logarithmic in the dimension d , thus mitigating the curse of dimensionality to a substantial extent. We also present several experiments numerically illustrating the benefits of gradient information over an equivalent number of function samples only.

We propose a novel algorithm to solve a general class of linear ill-posed inverse problems. For our numerical tests, we consider ill-posed problems on the sphere as they appear in the geosciences. Based on an iterative greedy algorithm, called the orthogonal matching pursuit, the signal is expanded in terms of trial functions which are picked from a large redundant set of functions, the so-called dictionary. The method is able to combine arbitrary trial functions which is a great advantage to former approximation algorithms. In particular, we combine orthogonal polynomials (such as spherical harmonics in the case of the sphere) of low degrees with localized trial functions such as wavelets and/or scaling functions for the reconstruction of global trends and regional details of the signal, respectively. Since we deal with ill-posed problems, we use a Tikhonov-type regularization with a penalty term based on a (spherical) Sobolev norm. There is no need to solve any system of equations or any integration problem which provides the ability to handle very large amounts of data or extremely scattered data sets. The outcome of the algorithm is a smooth and sparse approximation of the unknown signal which is locally adapted to the detail structure of the signal as well as to the density of the given data. Moreover, in the case that wavelets are contained in the dictionary, we additionally obtain a multiresolution of the signal. Several numerical experiments are presented.