Search

We present a novel algorithm attaining excessively fast, the sought solution of linear systems of equations. The algorithm is short in its basic formulation and, by definition, vectorized, while the memory allocation demands are trivial, because, for each iteration, only one dimension of the given input matrix $\mathbf X$ is utilized. The execution time is very short compared with state-of-the-art methods, exhibiting $> \times 10^2$ speed-up and low memory allocation demands, especially for non-square Systems of Linear Equations, with ratio of equations versus features high (tall systems), or low (wide systems) accordingly. The accuracy is high and straightforwardly controlled, and the numerical results highlight the efficiency of the proposed algorithm, in terms of computation time, solution accuracy and memory demands. The paper also comprises a theoretical proof for the algorithmic convergence, and we extend the implementation of the proposed algorithmic rationale to feature selection tasks.

The zonostrophic instability that leads to the emergence of zonal jets in barotropic beta-plane turbulence was analyzed through a geometric decomposition of the eddy stress tensor. The stress tensor is visualized by an eddy variance ellipse whose characteristics are related to eddy properties. The tilt of the ellipse principal axis is the tilt of the eddies with respect to the shear, the eccentricity of the ellipse is related to the eddy anisotropy, while its size is related to the eddy kinetic energy. Changes of these characteristics are directly related to the vorticity fluxes forcing the mean flow. The statistical state dynamics of the turbulent flow closed at second order was employed as it provides an analytic expression for both the zonostrophic instability and the stress tensor. For the linear phase of the instability, the stress tensor was analytically calculated at the stability boundary. For the non--linear equilibration of the instability the tensor was calculated in the limit of small supercriticality in which the amplitude of the jet velocity follows Ginzburg--Landau dynamics. It is found that dependent on the characteristics of the forcing, the jet is accelerated either because it primarily anisotropizes the eddies so as to produce upgradient fluxes or because it changes the eddy tilt. The instability equilibrates as these changes are partially reversed by the non--linear terms. Parameterizations of the ellipse characteristics are also discussed.

The purpose of this letter is to investigate the time complexity consequences of the truncated Taylor series, known as Taylor Polynomials \cite{bakas2019taylor,Katsoprinakis2011,Nestoridis2011}. In particular, it is demonstrated that the examination of the $\mathbf{P=NP}$ equality, is associated with the determination of whether the $n^{th}$ derivative of a particular solution is bounded or not. Accordingly, in some cases, this is not true, and hence in general.

Optimization algorithms appear in the core calculations of numerous Artificial Intelligence (AI) and Machine Learning methods, as well as Engineering and Business applications. Following recent works on the theoretical deficiencies of AI, a rigor context for the optimization problem of a \textit{black-box} objective function is developed. The algorithm stems directly from the theory of probability, instead of a presumed inspiration, thus the convergence properties of the proposed methodology are inherently stable. In particular, the proposed optimizer utilizes an algorithmic implementation of the $n$-dimensional inverse transform sampling as a search strategy. No control parameters are required to be tuned, and the trade-off among exploration and exploitation is by definition satisfied. A theoretical proof is provided, concluding that only falling into the proposed framework, either directly or incidentally, any optimization algorithm converges in the fastest possible time. The numerical experiments, verify the theoretical results on the efficacy of the algorithm apropos reaching the optimum, as fast as possible.

We present a numerical scheme for computation of Artificial Neural Networks (ANN) weights, which stems from the Universal Approximation Theorem, avoiding laborious iterations. The proposed algorithm adheres to the underlying theory, is highly fast, and results in remarkably low errors when applied for regression and classification of complex data-sets, such as the Griewank function of multiple variables $\mathbf{x} \in \mathbb{R}^{100}$ with random noise addition, and MNIST database for handwritten digits recognition, with $7\times10^4$ images. The same mathematical formulation is found capable of approximating highly nonlinear functions in multiple dimensions, with low errors (e.g. $10^{-10}$) for the test-set of the unknown functions, their higher-order partial derivatives, as well as numerically solving Partial Differential Equations. The method is based on the calculation of the weights of each neuron in small neighborhoods of the data, such that the corresponding local approximation matrix is invertible. Accordingly, optimization of hyperparameters is not necessary, as the number of neurons stems directly from the dimensionality of the data, further improving the algorithmic speed. Under this setting, overfitting is inherently avoided, and the results are interpretable and reproducible. The complexity of the proposed algorithm is of class P with $\mathcal{O}(mn^2)+\mathcal{O}(\frac{m^3}{n^2})-\mathcal{O}(\log(n+1))$ computing time, with respect to the observations $m$ and features $n$, in contrast with the NP-Complete class of standard algorithms for ANN training. The performance of the method is high, irrespective of the size of the dataset, and the test-set errors are similar or smaller than the training errors, indicating the generalization efficiency of the algorithm.

Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is demonstrated, by utilizing partial sums of Taylor series in high arithmetic precision. In particular, the proposed method is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method is based on the utilization of Taylor polynomials, by exploiting some hundreds of computer digits, resulting in highly accurate calculations. Interestingly, some well-known problems were found to reason by calculations accuracy, and not methodological inefficiencies, as supposed. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated and the extrapolation extent may a-priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provide a rigor estimation of the prediction ability. The approximation errors have comprehensively been analyzed, for a variety of calculation digits and test problems.

Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies on barotropic beta-plane turbulence have shown that coherent flows emerge out of a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases and the emergence of large scale flows has been attributed to a new type of collective, symmetry breaking instability of the statistical state dynamics of the turbulent flow. In this work we extend the analysis to stratified flows and investigate turbulent self-organization in a two-layer fluid with no imposed mean north-south thermal gradient and turbulence supported by an external random stirring. We use a second order closure of the statistical state dynamics (S3T) with an appropriate averaging ansatz that allows the identification of statistical turbulent equilibria and their structural stability. The bifurcation of the statistically homogeneous equilibrium state to inhomogeneous equilibrium states comprising of zonal jets and/or large scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. The theory predicts that when the flow transitions to a statistical state with large-scale structures, these states are barotropic if the scale of excitation is larger than the deformation radius. Mixed barotropic-baroclinic states with jets and/or waves arise when the excitation is at scales shorter than the deformation radius with the baroclinic component of the flow being subdominant for low energy input rates and non-significant for higher energy input rates. The results of the S3T theory are compared to nonlinear simulations. The theory is found to accurately predict both the critical transition arameters and the scales of the emergent structures but underestimates their amplitude., Comment: submitted to the Journal of Fluid Mechanics

Zonal jets in a barotropic setup emerge out of homogeneous turbulence through a flow-forming instability of the homogeneous turbulent state (`zonostrophic instability') which occurs as the turbulence intensity increases. This has been demonstrated using the statistical state dynamics (SSD) framework with a closure at second order. Furthermore, it was shown that for small supercriticality the flow-forming instability follows Ginzburg-Landau (G-L) dynamics. Here, the SSD framework is used to study the equilibration of this flow-forming instability for small supercriticality. First, we compare the predictions of the weakly nonlinear G-L dynamics to the fully nonlinear SSD dynamics closed at second order for a wide ranges of parameters. A new branch of jet equilibria is revealed that is not contiguously connected with the G-L branch. This new branch at weak supercriticalities involves jets with larger amplitude compared to the ones of the G-L branch. Furthermore, this new branch continues even for subcritical values with respect to the linear flow-forming instability. Thus, a new nonlinear flow-forming instability out of homogeneous turbulence is revealed. Second, we investigate how both the linear flow-forming instability and the novel nonlinear flow-forming instability are equilibrated. We identify the physical processes underlying the jet equilibration as well as the types of eddies that contribute in each process. Third, we propose a modification of the diffusion coefficient of the G-L dynamics that is able to capture the asymmetric evolution for weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability., Comment: 27 pages, 17 figures

Planetary turbulence is observed to self-organize into large-scale structures such as zonal jets and coherent vortices. One of the simplest models that retains the relevant dynamics of turbulent self-organization is a barotropic flow in a beta-plane channel with turbulence sustained by random stirring. Non-linear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is first broken by the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. The emergence of both non-zonal coherent structures and zonal jets is studied using a statistical theory, Stochastic Structural Stability Theory (S3T). S3T directly models a second-order approximation to the statistical mean turbulent state and allows the identification of statistical turbulent equilibria and study of their stability. Using S3T, the bifurcation properties of the homogeneous state in barotropic beta-plane turbulence are determined. Analytic expressions for the zonal and non-zonal large-scale coherent flows that emerge as a result of structural instability are obtained and the equilibration of the incipient instabilities is studied through numerical integrations of the S3T dynamical system. The dynamics underlying the formation of zonal jets are also investigated. It is shown that zonal jets form from the upgradient momentum fluxes that result from the shearing of the eddies by the emerging infinitesimal large-scale flow. Finally, numerical simulations of the nonlinear equations confirm the characteristics (scale, amplitude and phase speed) of the structures predicted by S3T, even in highly non-linear parameter regimes such as the regime of zonostrophic turbulence., Comment: 17 pages, 14 figures, submitted to "Zonal jets" by Galperin B. and Read, P.L. (eds.). arXiv admin note: substantial text overlap with arXiv:1303.6435

Zonal jets and non-zonal large-scale flows are often present in forced-dissipative barotropic turbulence on a beta-plane. The dynamics underlying the formation of both zonal and non-zonal coherent structures is investigated in this work within the statistical framework of Stochastic Structural Stability Theory (S3T). Previous S3T studies have shown that the homogeneous turbulent state undergoes a bifurcation at a critical parameter and becomes inhomogeneous with the emergence of zonal and/or large-scale non-zonal flows and that these statistical predictions of S3T are reflected in direct numerical simulations. In this paper, we study the dynamics underlying the S3T statistical instability of the homogeneous state as a function of parameters. It is shown that for weak planetary vorticity gradient, $\beta$, both zonal jets and non-zonal large-scale structures form from upgradient momentum fluxes due to shearing of the eddies by the emerging infinitesimal flow. For large $\beta$, the dynamics of the S3T instability differs for zonal and non-zonal flows but in both the destabilizing vorticity fluxes decrease with increasing $\beta$. Shearing of the eddies by the mean flow continues to be the mechanism for the emergence of zonal jets while non-zonal large-scale flows emerge from resonant and near resonant triad interactions between the large-scale flow and the stochastically forced eddies. The relation between the formation of large-scale structure through modulational instability and the S3T instability of the homogeneous state is also investigated and it is shown that the modulational instability results are subsumed by the S3T results., Comment: 22 pages, 15 figures, submitted to the Journal of the Atmospheric Sciences

Planetary turbulent flows are observed to self-organize into large scale structures such as zonal jets and coherent vortices. One of the simplest models of planetary turbulence is obtained by considering a barotropic flow on a beta-plane channel with turbulence sustained by random stirring. Non-linear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is broken with the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. We study the emergence of non-zonal coherent structures using a non-equilibrium statistical theory, Stochastic Structural Stability Theory (S3T, previously referred to as SSST). S3T directly models a second order approximation to the statistical mean turbulent state and allows identification of statistical turbulent equilibria and study of their stability. Using S3T, the bifurcation properties of the homogeneous state in barotropic beta-plane turbulence are determined. Analytic expressions for the zonal and non-zonal large scale coherent flows that emerge as a result of structural instability are obtained. Through numerical integrations of the S3T dynamical system, it is found that the unstable structures equilibrate at finite amplitude. Numerical simulations of the nonlinear equations confirm the characteristics (scale, amplitude and phase speed) of the structures predicted by S3T., Comment: submitted to Journal of Fluid Mechanics

Planetary and magnetohydrodynamic drift-wave turbulence is observed to self-organize into large scale structures such as zonal jets and coherent vortices. In this Letter we present a non-equilibrium statistical theory, the Stochastic Structural Stability theory (SSST), that can make predictions for the formation and finite amplitude equilibration of non-zonal and zonal structures (lattice and stripe patterns) in homogeneous turbulence. This theory reveals that the emergence of large scale structure is the result of an instability of the interaction between the coherent flow and the associated turbulent field. Comparison of the theory with nonlinear simulations of a barotropic flow in a beta-plane channel with turbulence sustained by isotropic random stirring, demonstrates that SSST predicts the threshold parameters at which the coherent structures emerge as well as the characteristics of the emerging structures (scale, amplitude, phase speed). It is shown that non-zonal structures (lattice states or zonons) emerge at lower energy input rates of the stirring compared to zonal flows (stripe states) and their emergence affects the dynamics of jet formation., Comment: 5 pages, 5 figures