Orthogonal Function Expansion Research Articles

Variability of SST through Koopman Modes

Abstract The majority of dynamical systems arising from applications show a chaotic character. This is especially true for climate and weather applications. We present here an application of Koopman operator theory to tropical and global sea surface temperature (SST) that yields an approximation to the continuous spectrum typical of these situations. We also show that the Koopman modes yield a decomposition of the datasets that can be used to categorize the variability. Most relevant modes emerge naturally, and they can be identified easily. A difference with other analysis methods such as empirical orthogonal function (EOF) or Fourier expansion is that the Koopman modes have a dynamical interpretation, thanks to their connection to the Koopman operator, and they are not constrained in their shape by special requirements such as orthogonality (as it is the case for EOF) or pure periodicity (as in the case of Fourier expansions). The pure periodic modes emerge naturally, and they form a subspace that can be interpreted as the limiting subspace for the variability. The stationary states therefore are the scaffolding around which the dynamics takes place. The modes can also be traced to the Niño variability and in the case of the global SST to the Pacific decadal oscillation (PDO). Significance Statement We compute the Koopman modes for the tropical and global SST, demonstrating that significant dynamical modes can be identified also in complex high-dimensional datasets with continuous spectra. The Koopman modes are then used to identify the stationary subspace, namely, the limit subspace for the invariant evolution of the system.

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.

Gyro-kinetic simulations of trapped electron collision effects on low-frequency drift-wave instabilities in tokamak plasmas

Low-frequency drift-wave instabilities play a crucial role in the radial transport of present-day tokamaks, and trapped electron collisions can significantly influence these instabilities. In this paper, the effects of trapped electron collisions on these instabilities are investigated based on linear gyro-kinetic simulations. The basic numerical techniques including dispersion relation integral method and orthogonal basis function expansion are presented in detail with necessary benchmark work. The results demonstrate that in medium gradients, the increase of trapped electron proportions promotes the growth rate and radial heat transport largely for quasi-linear trapped electron modes (TEMs) and ion temperature gradient (ITG) modes. Moreover, trapped electron collisions have strong stabilizing effects, especially for TEMs driven by electron temperature gradients. Two distinctive branches, namely Mode #1 and #2, are investigated in steep gradients. Both behave in a varied instability nature during different ranges of the normalized wave vector k^θ . Mode #1 mainly induces radial heat transport during k^θ<0.5 and is significantly suppressed by the collisions. Mode #2 mainly induces radial heat transport during 0.4<k^θ<0.8 , and is largely enhanced by the collisions. When the collisionality is large enough, Mode #2 has stronger transport capacity than the other. Mode #2 at the medium wave vector, known as dissipative TEM, may provide the mechanism of the edge coherent mode observed in EAST H-mode plasmas, wherein collisionality plays an important role in the mode excitation.

Anti‐plane stress analysis for V‐notch in a piezomagnetic half space under SH wave

AbstractIn this paper, the anti‐plane stress analysis of a V‐notch with complex boundary conditions in a piezomagnetic half space is studied. Firstly, SH wave is considered as an external load acting on piezomagnetic half space, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the boundary conditions on the boundary of the half space. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V‐notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green's function method is applied, the half space is divided into two parts along the vertical interface, a pair of in‐plane magnetic field and out‐plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Revisiting the bound states of a confined delta potential

The stationary states of a particle under the influence of a delta potential confined by impenetrable walls are investigated using the method of expansion in orthogonal functions. The eigenfunctions of the time-independent Schrödinger equation are expressed in closed form by using a pair of closed-form expressions for series available in the literature. The analysis encompasses both attractive and repulsive potentials with arbitrary couplings. Confinement significantly impacts the quantum states and introduces a scenario of double degeneracy including the ground state. Analysis extends to discuss the transition to unconfinement. This research holds particular significance for educators and students engaged in mathematical methods applied to physics and quantum mechanics within undergraduate courses, offering valuable insights into the complex relationships among profiles of potentials, boundary conditions, and the resulting quantum phenomena.

Spatio-temporal characteristics of seismic strain anomalies reveal seismic risk zones along the Longmenshan fault zone and adjacent areas

Introduction: Identifying and quantifying earthquake precursors, and analyzing their physical mechanisms, continues to be a challenge for earthquake forecasting. In this study, orthogonal functions were developed to effectively identify precursor anomalies, thereby improving the forecasting of strong earthquakes.Methods: To study the spatio-temporal contour anomalies in seismic strain fields, we assessed them for seismic activity variables and natural orthogonal function expansion, in six strong earthquakes near the Longmenshan Fault Zone, China, that have occurred since 2008.Results: We observed that, prior to these earthquakes, the temporal factor (the time variation characteristics of the strain field) displayed anomalies with high/low values exceeding the mean square error within a stable context. The anomalies exhibited multi-component characteristics and were primarily concentrated in the first four-strain fields. Short-term and impending-earthquake anomalies were observed in the temporal factor before the 2008 Wenchuan (M8.0) and 2013 Lushan (M7.0) earthquakes, while medium-term and long-term anomalies appeared before the other four strong earthquakes, without notable short-term anomalies. The temporal evolution of strain field contour anomalies, and the strain contours positive and negative intersection, showed that central areas surrounded by multiple strain field contour anomalies were potential locations for strong earthquakes. This suggests a potential approach for earthquake location forecasting. Since 2009, there have been five strong earthquakes, each affected to varying degrees by anomalous strain fields from the 2008 Wenchuan (M8.0) earthquake.Conclusion: The results of this study corroborate the findings of the focal mechanism’s node shear stress, indicating significant physical implications of the anomalies and the reliability of these conclusion.

Analysis on scattering characteristics of SH guided wave due to V-notch in a piezoelectric bi-material strip

In this paper, the problem of a V-notch with complex boundary conditions in a piezoelectric bi-material strip is studied. First, SH guided wave is considered as an external load acting on the piezoelectric bi-material strip; on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress-free and electric insulation conditions on the upper and lower horizontal boundaries of the strip. Then, the analytical expression of standing wave is established, which satisfies the stress-free and electric insulation conditions on the boundaries of V-notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green’s function method is applied, the bi-material strip is divided into two parts along the vertical interface, a pair of in-plane electric field and out-plane forces is applied on the vertical interface, and the first kind of Fredholm integral equations is set up and solved by applying the orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Dynamic analysis for V-notch in a piezomagnetic bi-material strip under SH guided wave

In this paper, the problem of a V-notch with complex boundary conditions in a piezomagnetic bi-material strip is studied. Firstly, SH guided wave is considered as an external load acting on piezomagnetic bi-material strip, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress free and magnetic insulation conditions on the upper and lower horizontal boundaries of the strip. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V-notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green’s function method is applied, the bi-material strip is divided into two parts along the vertical interface, a pair of in-plane magnetic field and out-plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Extracting a function encoded in amplitudes of a quantum state by tensor network and orthogonal function expansion

There are quantum algorithms for finding a function f satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when the number d of the variables of f is large. In general, however, these algorithms output the quantum state which encodes f in the amplitudes, and reading out the values of f as classical data from such a state can be so time-consuming that the quantum speedup is ruined. In this study, we propose a general method for this function readout task. Based on the function approximation by a combination of tensor network and orthogonal function expansion, we present a quantum circuit and its optimization procedure to obtain an approximating function of f that has a polynomial number of degrees of freedom with respect to d and is efficiently evaluable on a classical computer. We also conducted a numerical experiment to approximate a finance-motivated function to demonstrate that our method works.

Strain fields of Ms &gt;6.0 earthquakes in Menyuan, Qinghai, China

In predicting earthquakes, it is a major challenge to capture the time factor and spatial isoline anomalies, and understand their physical processes, of the seismic strain field before a strong earthquake. In this study, the seismic strain field was used as representative of seismic activity. The natural orthogonal function expansion method was used to calculate the seismic strain field before the Menyuan Ms 6.4 earthquakes in 1986 and 2016, and the Ms 6.9 earthquake in 2022. Time factor and spatial isoline anomaly of the strain field before each earthquake was extracted. We also compared the evolution of the strain field with numerical simulation results under the tectonic stress system at the source. The results showed that the time factor before the earthquakes had high or low value anomalies, exceeding the mean square error of the stable background. The anomalies were concentrated in the first four typical fields of the strain field, which has multiple components. The abnormal contribution rate of the first typical field is the largest (accounting for 42%–49% of the total field). The long- and medium-term anomalies appear 3-4, and 1-2 years before the earthquake, respectively. There were no short or immediate-term anomalies within 3 months of the earthquake. In addition, during the evolution of the strain field, the abnormal area of the spatial isoline changed with the change in time. Usually, the intersection area of the two isoseismic lines of strain accumulation and strain release becomes a potential location for strong earthquakes. Finally, we found that the high strain field values of the 1986 and 2016 Ms 6.4 earthquakes were equivalent to the numerical simulation results, while the high strain field values of the 2022 Menyuan Ms 6.9 earthquakes were slightly different, but within the accepted error range. These results indicate that the two methods are consistent. We have shown that the natural orgthagonal method can be used to obtain the spatiotemporal anomaly information of strain field preceding strong earthquakes.

Numerical solution of neutral delay differential equations using orthogonal neural network

In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order 2^{-n}, where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials.

The dynamic behavior of a piezoelectric strip with a salient and an inclusion under guided SH waves

Dynamic stress around the inclusion and salient in a piezoelectric strip subjected to dynamic incident anti-plane shearing (guided SH wave) is calculated. Using the repeated image superposition method, the analytical expression for scattering wave, satisfying stress-free and electric insulation conditions at the upper and lower horizontal boundaries of the strip, is obtained. The integral equations are set up with the boundary conditions and solved by orthogonal function expansion and effective truncation techniques. The obtained results clarify the characteristic features of the influence on the dynamic stress concentration factor and electric field intensity concentration factor. The analytical solutions are compared to the finite element modeling results to verify the accuracy of the conclusions of this paper.

Dynamic behavior of an inhomogeneous piezoelectric/piezomagnetic half space with a circular ring structure under SH wave

In this article, the dynamic characteristics of a circular ring structure in an inhomogeneous piezoelectric/piezomagnetic half spaces subjected to anti-plane shear wave is studied. The material parameters are assumed to have an exponential function distribution along the coordinate axis, and the Helmholtz equation includes variable coefficients due to inhomogeneity. First, by introducing new variables, the Helmholtz equation is transformed into standard form. Next, on the basis of boundary conditions, integral equations are setup and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor, electric field intensity concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Research on Stock Price Prediction Based on Orthogonal Gaussian Basis Function Expansion and Pearson Correlation Coefficient Weighted LSTM Neural Network

For stock price prediction in quantitative finance, deep learning techniques such as LSTM neural network do not need the stationarity assumption of traditional time series models (such as ARIMA and GARCH) and can forecast medium and long-term time series, so they have attracted much attention. This paper proposes an improved LSTM neural network based on orthogonal Gaussian basis function expansion and Pearson correlation coefficient weighting. The proposed method uses the functional features of intra-day prices to fit the residual series predicted by the LSTM neural network. Considering that the underlying model structure between each component of the function eigenvector and the residual series is unknown, we use the Bagging method to capture and trade off the variance and bias of the prediction model. In addition, since the dimension of the predictive variable of the LSTM neural network is a parameter to be estimated, we use the model averaging method based on Pearson correlation coefficient weighting for tuning. The results of actual data analysis show that the proposed method can significantly improve the prediction accuracy of the original LSTM neural network and has certain robustness. Finally, the proposed method can be further applied to consumer price index (CPI) prediction, daily average temperature prediction, and real-time monitoring of environmental trace elements.

Scattering of SH guided wave by a circular cavity and a semicircular salient in an infinite piezoelectric strip

In this article, the problem of a salient with complex boundary conditions in a piezoelectric strip is studied by Fourier series expansion method, and SH guided wave is considered as an external load acting on piezoelectric strip. First, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress free and electric insulation conditions on the upper and lower horizontal boundaries of the strip. Next, with the aid of boundary condition, integral equations are setup and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Dynamic anti-plane analysis for an elliptic inclusion in an infinite piezoelectric strip

Dynamic incident anti-plane shearing (SH guided wave) is considered to calculate dynamic stress and electric intensity in an infinite piezoelectric strip that contains an elliptic inclusion. By the means of the conformal mapping method, the field of an ellipse can be transformed into a unit circle. With the aid of series expansion method, the expression of SH guided wave is established, which satisfies the stress-free and electric insulation conditions on the upper and lower horizontal boundaries of the strip. The expression of scattering waves is conducted based on repeated image superposition. Boundary conditions are solved by applying the orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions.

Scattering of SH-guided wave by an elliptic inclusion in an infinite strip region

Dynamic incident anti-plane shearing (SH-guided wave) is considered to calculate dynamic stress in an infinite strip region with an elliptic inclusion. By the means of the conformal mapping method, the field of an ellipse can be transformed into a unit circle. With the aid of series expansion method, the expression of SH-guided wave is established, which satisfies the stress free condition on the upper and lower horizontal boundaries of the strip. The expression of scattering wave is conducted based on repeated image superpositions. Boundary conditions are solved by applying the orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Dynamic anti-plane analysis for a semi-circular salient in an infinite elastic strip region

Dynamic incident anti-plane shearing (SH guided wave) is considered to calculate dynamic stress in an infinite strip region with a semi-salient. With the aid of series expansion method, the expression of SH guided wave is established, which satisfies the stress free condition on the upper and lower horizontal boundaries of the strip. The expression of standing wave is constructed by the means of Fourier series expansion. The expression of scattering wave is conducted based on repeated image superposition. Boundary conditions are solved by applying orthogonal function expansion technique. Results clarified the influence on the dynamic stress concentration factor under proper conditions.

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.

The spherical multipole resonance probe: kinetic damping in its spectrum

The multipole resonance probe is one of the more recently developed measurement devices to measure plasma parameters like electron density and temperature based on the concept of active plasma resonance spectroscopy. The dynamical interaction between the probe and the plasma in an electrostatic, kinetic description can be modeled in an abstract notation based on functional analytic methods. These methods provide the opportunity to derive a general solution, which is given as the response function of the probe-plasma system. It is defined by the matrix elements of the resolvent of an appropriate dynamical operator. Based on the general solution a residual damping for vanishing pressure can be predicted and can only be explained by kinetic effects. Within this manuscript an explicit response function of the multipole resonance probe is derived. Therefore, the resolvent is determined by its algebraic representation based on an expansion in orthogonal basis functions. This allows us to compute an approximated response function and its corresponding spectra, which show additional damping due to kinetic effects.