Orthogonal Basis Functions Research Articles

Data Reconstruction-Based Two-Step Non-Intrusive Reduced-Order Modeling Using Fourier Transform and Interpolations

This study presents a data reconstruction-based two-step non-intrusive reduced-order modeling (ROM) based on discrete Fourier transformation (DFT) and proper orthogonal decomposition-radial basis function (POD-RBF) interpolation. To efficiently approximate a system for various parametric inputs, two offline and one online stage are proposed. The first offline stage adjusts and reconstructs sampled data using a scaling factor. During the adjusting procedure, the fast Fourier transform operation is used to transform a domain between the time and frequency, and the POD-RBF interpolation method efficiently generates adjusted data. The second offline stage constructs multiple ROMs in the frequency domain for interpolation with respect to the parameter. Finally, in the online stage, the solution field depending on the changes in input parameters, is approximated using the POD-RBF interpolation and the inverse Fourier transformation. The accuracy and efficiency of the proposed method are verified using the 2-D unsteady incompressible Newtonian fluid problems and are compared to the OpenFOAM software program showing remarkable efficiencies in computing approximated solutions.

Analysis on Uncertainty of Field-to-Wire Coupling Model in Time Domain

This paper presents an approach for the time domain statistical simulation of field-to-wire coupling model with uncertain wire position in traverse direction. The parameters of governing equations of field-to-wire coupling model about the wire position are expanded to the augmented governing equations by using the generalized polynomial chaos technique with normalized orthogonal basis functions. And the coefficients of the polynomial chaos expansions are solved by combining with the stochastic Galerkin method. However, the solution of augmented governing equations is a complex problem. Therefore, a new finite-difference time-domain (FDTD) algorithm based on the implicit Wendroff difference format is proposed to solve the governing equations, and the general iterative equations are given in this paper. The proposed methods are verified by a field-to-wire coupling model. The effectiveness and advantages of the new FDTD algorithm and the uncertainty analysis method based on generalized polynomial chaos technique are verified, which are achieved by comparing with the conventional leapfrog fashion FDTD algorithm and the Monte Carlo technique, respectively.

Magnetic anomaly detection via a combination approach of minimum entropy and gradient orthogonal functions

Aiming at the problem of magnetic anomaly detection under low signal-to-noise ratio, a full magnetic gradient detection via minimum entropy and gradient orthonormal basis function is proposed in this paper. Firstly, the mathematical model of magnetic dipole gradient tensor and the experimental method of collecting magnetic gradient tensor signals are introduced. Secondly, a new minimum entropy detector is designed, the basic functions which are processed by new minimum entropy detector can be orthogonalized easily. Thirdly, a group of new standard orthogonal basis functions are constructed according to the characteristics of magnetic gradient anomaly signals. Finally, the detector and orthogonal basis functions are used to detect the magnetic anomaly. Both numerical simulation signals and experiment signals are used to prove the effectiveness of the methods.

Joint digital pre-distortion model based on Chebyshev expansion

<span>In this paper, a new low complexity model is proposed for the joint digital pre-distortion of in-phase/quadrature-phase (I/Q) imbalance, local oscillator (LO) leakage, and power amplifier nonlinearity in direct-conversion transmitters (DCTs). In this structure, we proposed a set of orthogonal basis functions based on Chebyshev expansion to attenuate the problem of numerical instability created during the conventional model identification method. This robust joint digital pre-distortion (DPD) utilized the indirect learning architecture and updated the coefficients vector based on the recursive least square (RLS) algorithm. To verify the operation and efficiency of the proposed model, an extensive simulation in MATLAB was carried out. The results showed a significant reduction in the conditional number and the coefficient dispersion of the observation matrix. Furthermore, the power of the signal in the adjacent channel decreased by more than 16 dB for the orthogonal frequency division multiplexing (OFDM), 16 QAM input signal. In comparison to the previous digital pre-distorter models, the proposed DPD builds strong numerical stability with the least coefficients.</span>

A method to characterize climate, Earth or environmental vector random processes

We propose a general methodology to characterize a non-stationary random process that can be used for simulating random realizations that keep the probabilistic behavior of the original time series. The probability distribution of the process is assumed to be a piecewise function defined by several weighted parametric probability models. The weights are obtained analytically by ensuring that the probability density function is well defined and that it is continuous at the common endpoints. Any number of subintervals and continuous probability models can be chosen. The distribution is assumed to vary periodically in time over a predefined time interval by defining the model parameters and the common endpoints as truncated generalized Fourier series. The coefficients of the expansions are obtained with the maximum likelihood method. Different sets of orthogonal basis functions are tested. The method is applied to three time series with different particularities. Firstly, it is shown its good behavior to capture the high variability of the precipitation projected at a semiarid location of Spain for the present century. Secondly, for the Wolf sunspot number time series, the Schwabe cycle and time variations close to the 7.5 and 17 years are analyzed along a 22-year cycle. Finally, the method is applied to a bivariate time series that contains (1) freshwater discharges at the last regulation point of a dam located in a semiarid zone in Andalucía (Spain) which is influenced not only by the climate variability but also by management decisions and (2) the salinity at the mouth of the river. For this case, the analysis, that was combined with a vectorial autoregressive model, focus on the assessment of the goodness of the methodology to replicate the statistical features of the original series. In particular, it is found that it reproduces the marginal and joint distributions and the duration of sojourns above/below given thresholds.

Comparative study on wavelet functional partial least squares soft sensor for complex batch processes

Conventional data-driven models for batch processes conduct unfolding operations and neglect the continuous property. A novel soft sensor method is proposed based on the wavelet functional partial least squares (WFPLS). The three-dimensional (3D) batch data is transferred into an even two-dimensional function matrix no matter whether it has even-length or uneven-length. The orthogonal wavelet basis functions are selected by the proposed active algorithm to approximate the variables' trajectories. Two versions of WFPLS are developed to predict quality based on different assumptions of the loading function. By comparing one of the proposed methods and the classical multi-way partial least squares (MPLS), the root means square errors are relatively descended by 20.6566% and 63.9925%, R2 are improved by 2.5813% and 83.2518%, and the corresponding computation time of the proposed method is only 5.7% and 41% of the time used by MPLS in the numerical case and in the industrial case, respectively.

Evaluating the Influence of ipRGCs on Color Discrimination.

To investigate the influence of intrinsically photosensitive retinal ganglion cells (ipRGCs) on color discrimination, it is necessary to create two metameric light stimuli (metameric ipRGC stimuli) with the same amount of cone and rod stimulation, but different amounts of ipRGC stimulation. However, since the spectral sensitivity functions of cones and rods overlap with those of ipRGCs in a wavelength band, it has been difficult to independently control the amount of stimulation of ipRGCs only. In this study, we first propose a method for calculating metameric ipRGC stimulation based on the orthogonal basis functions of human photoreceptor cells. Then, we clarify the controllable range of metameric ipRGC stimulation within a color gamut. Finally, to investigate the color discrimination by metameric ipRGC stimuli, we conduct subjective evaluation experiments on 24 chromaticity coordinates using a multispectral projector. The results reveal a correlation between differences in the amount of ipRGC stimulation and differences in color appearance, indicating that ipRGCs may influence color discrimination.

Feed-forward neural networks for fitting of kinetic energy and its functional derivative

We present a feed-forward NN approach for fitting of kinetic energy and its functional derivative for a 1-dimensional system. The density is represented in terms of orthogonal basis functions. The coefficients of basis functions forms the inputs to NN. Using this approach we found no oscillatory behaviour in functional derivative. Using NN based functional derivative we determine the ground-state density by solving the Euler–Lagrange equation. The presented approach can open up new ways for accurate calculations kinetic energy and the functional derivative, which can be considered as an important step towards advancement of machine learning based OF-DFT methods.

Maximum Achievable Gain of Annular-Shaped Beams

The maximum achievable gain and the corresponding source distribution for an annular or “donut”-shaped beam produced by a circular aperture are studied. Expressing the source in terms of orthogonal basis functions and then applying Lagrange’s multiplier method, the optimum source distribution is determined. The residue calculus is then invoked for the closed-form expressions. Numerical results are shown and discussed. In applications such as direction finding and satellite tracking systems on moving vehicles, a high-gain annular beam is suitable for maintaining uninterrupted connectivity.

A hybrid spectral element-infinite element approach for 3D controlled-source electromagnetic modeling

In this study, we developed a novel hybrid approach for 3D controlled-source electromagnetic (CSEM) forward modeling. The hybrid approach combines vector spectral element method (SEM) with scalar infinite element method (IFEM) by using infinite element instead of traditional truncated boundary. Based on Maxwell's equations, the curl-curl control equation satisfied by secondary electric field is derived. The boundary value problem in differential form is transformed into integral weak form by using Galerkin weighted residual method. Then the global problem is discretized by element subdivision and high-order orthogonal basis function interpolation. The computational domain is discretized using hexahedral spectral elements and a single layer of infinite elements is added outside the domain. The SEM with high-order Gauss-Lobatto-Legendre (GLL) vector basis functions is used to represent the interior electric field, and the IFEM with high-order Gauss-Lobatto-Legendre (GLL) scalar basis functions is uesd as a boundary condition. Finally, the global numerical solution is obtained by solving large-scale linear equations. The layered model verifies the accuracy and correctness of the approach, and the 3D model verifies the efficiency and numerical stability of the approach. Numerical experiments show that the hybrid SEM-IFEM approach developed in this study can reach similar accuracy to the conventional spectral element method, but the computational domain required can be much smaller, and the corresponding CPU time and memory requirement are greatly reduced, namely, the high precision three-dimensional electromagnetic response can be obtained with less computational resource consumption.

Online monitoring and fault diagnosis for uneven length batch process based on multi‐way orthogonal enhanced neighborhood preserving embedding

AbstractIn the practical batch process, the duration of each batch is probably different, and the key event happened may also vary from batch to batch. This paper proposes an effective fault diagnosis algorithm to synchronize the batch runs with uneven duration and keep the key features that reflect the running state of batch process. To address the batch out of sync, the relaxed greedy time warping is used to accurate on‐line synchronization of ongoing batch and avoid evaluating the optimal path every time a new sample is available. For online monitoring and fault diagnosis, an important step is how to extract key features that reflect the running state of batch process, on which basis a weight matrix considers the distance between the neighbor data is embedded to construct the enhanced objective function, and then the mutually orthogonal basis functions which represent the local geometrical structure are computed to extract key features of batch process. After a fault is detected, the contribution plot method is used to diagnose the variables that cause this fault. The effectiveness of the proposed algorithm is verified by the penicillin fermentation process.

Ionosphere Tomographic Model Based on Neural Network with Balance Cost and Dynamic Correction Using Multi-Constraints

A Neural network (NN) is a promising tool for the tomographic inversion of the ionosphere. However, existing research has adopted an unbalanced cost function for training purposes and a preset image for constraint purposes, resulting in the output image being dominated by measurements. To address these problems, we proposed an NN-based tomographic model with a balance cost function and a dynamic correction process (BCDC) for ionosphere inversion. The cost function is composed of two balance terms corresponding to the measurements and the selected constraints, respectively. The produced image in the forward process of the NN is corrected dynamically by fitting each vertical profile with orthogonal basis functions (EOFs) and the Chapman function and then by smoothing the voxels of each layer with a moving window approach horizontally. The corrected image is then used to calculate the slant total electron content (STEC) parameter, which is further translated into the term of the cost for the vertical and horizontal constraints. Experiments were carried out to validate the BCDC method and compared with a recently developed tomographic method and the international reference ionosphere (IRI) model. Results show that the parameters derived from the BCDC model demonstrate good consistency with the observations. Comparing with the reference methods, the BCDC method performs better in the validations of vertical profiles, F2 layer peak density (NmF2), STEC parameter and vertical total electron content map. Further analysis also shows that a balance cost function is of benefit to achieve an image of better quality.

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.

Exact Solution of New Magnetic Current Based Surface-Volume-Surface EFIE and Analysis of Its Spectral Properties

A novel magnetic current based Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE-M) is presented for the problem of scattering on homogeneous non-magnetic dielectric objects. The exact Galerkin Method of Moments (MoM) utilizing both the rotational and irrotational vector spherical harmonics as orthogonal basis and test functions according to the Helmholtz decomposition is implemented to solve SVS-EFIE-M analytically for the case of dielectric sphere excited by an electric dipole. The field throughout the sphere is evaluated and compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. This exact solution validates the rigorous nature of the new SVS-EFIE-M formulation. It also reveals the spectral properties of its individual operators, their products and their linear combination. The spectrum of the MoM impedance matrix is also obtained. It is shown that upon choosing basis and test functions in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}(S)$</tex-math></inline-formula> space and evaluating testing inner products in the same space, the MoM impedance matrix features bounded condition number with increasing order of discretization and/or at low frequencies. This makes the proposed SVS-EFIE-M formulation free of oversampling and low-frequency breakdowns giving it advantage both over its SVS-EFIE-J predecessor and classical double-source integral equations such as PMCHWT, Muller, and others suffering from this type of numerical instabilities inherent to their inferior spectral properties.

Slepian Scale-Discretised Wavelets on the Sphere

This work presents the construction of a novel spherical wavelet basis designed for incomplete spherical datasets, i.e. datasets which are missing in a particular region of the sphere. The eigenfunctions of the Slepian spatial-spectral concentration problem (the Slepian functions) are a set of orthogonal basis functions which are more concentrated within a defined region. Slepian functions allow one to compute a convolution on the incomplete sphere by leveraging the recently proposed sifting convolution and extending it to any set of basis functions. Through a tiling of the Slepian harmonic line, one may construct scale-discretised wavelets. An illustration is presented based on an example region on the sphere defined by the topographic map of the Earth. The Slepian wavelets and corresponding wavelet coefficients are constructed from this region and are used in a straightforward denoising example.

Space Discretization of Time-Fractional Telegraph Equation with Mamadu-Njoseh Basis Functions

In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first converting from Caputo’s type to Riemann-Liouville’s type. The proposed method was constrained to precise error analysis to establish the accuracy of the method. Numerical experimentation was implemented with the aid of MAPLE 18 to show convergence of the method as compared with the analytic solution.

On the Existence and Uniqueness of Solution for the Fractional Fredholm Integro-Differential Equation through the Fractional Variational Orthogonal Collocation method with Mamadu-Njoseh orthogonal basis functions

In this article, we present a detailed analysis on the existence and uniqueness of solution for the fractional Fredholm integro-differential equation (FFIDE) via the fractional variational orthogonal collocation method (FVOCM) with Mamadu-Njoseh orthogonal basis functions. Relevant theorems were formulated and proven to show that there exist a solution for the problem FFIDE through the FVOCM, and the solution is shown to be unique.

A fourth-order Lagrangian discontinuous Galerkin method using a hierarchical orthogonal basis on curvilinear grids

The existing high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic methods are restricted to using quadratic meshes with quadratic polynomials (P2), which in turn, yield up to third-order accuracy. These existing DG hydrodynamic schemes, when extended to work with cubic meshes and cubic polynomials (P3), can be unstable on strong-shock problems. Therefore, this paper presents a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate compressible material dynamics ( e.g., gasses, fluids, and solids) with strong-shocks using cubic meshes and cubic polynomials, and delivers up to fourth-order accuracy on smooth flows. The stability on shock problems is achieved using new hierarchical orthogonal basis functions and a new subcell mesh stabilization (SMS) scheme for cubic meshes. The accuracy and robustness of the new high-order accurate Lagrangian DG hydrodynamic method is demonstrated by simulating a diverse suite of challenging test problems covering gas and solid dynamic problems on curvilinear meshes.

A deep neural network based method for magnetic anomaly detection

Magnetic anomaly detection (MAD) is a technique to find ferromagnets hiding in strong and complicated magnetic background. In many practical cases, the targets are very far from the detection sensor, which leads to low signal-to-noise ratio (SNR) and high detection difficulty. Most of the current methods determine the existence of target by some approaches based on signal analysis, such as the orthogonal basis function (OBF) and the minimum entropy (ME). However, although these methods consume low resources, the detection performances are not satisfactory enough. In recent years, due to the increase of computer capability, complex methods become applicable in MAD. In this study, a deep neural network (DNN) is adopted to detect the magnetic anomalies. The DNN has shown its better ability to represent natural data in many applications. A feature automatically learned by a DNN from data in the raw form is more effective for detecting target signals and suppressing irrelevant variations. Herein, a convolutional network with residual structure to implement the feature extraction is designed and an MAD method based on it is proposed. Through the semi-real tests, the proposed method exhibits a strong capability to extract features and shows excellent performances on detection.

Stochastic Fracture Analysis Using Scaled Boundary Finite Element Methods Accelerated by Proper Orthogonal Decomposition and Radial Basis Functions

This paper presents a stochastic analysis method for linear elastic fracture mechanics using the Monte Carlo simulations (MCs) and the scaled boundary finite element method (SBFEM) based on proper orthogonal decomposition (POD) and radial basis functions (RBF). The semianalytical solutions obtained by the SBFEM enable us to capture the stress intensity factors (SIFs) easily and accurately. The adoption of POD and RBF significantly reduces the model order and increases computation efficiency, while maintaining the versatility and accuracy of MCs. Numerical examples of cracks in homogeneous and bimaterial plates are provided to demonstrate the effectiveness and reliability of the proposed method, where the crack inclination angles are set as uncertain variables. It is also found that the larger the scale of the problem, the more advantageous the proposed method is.