Search Results (18)

To overcome the limitations of Orthonormal Basis Function (OBF) methods in magnetic anomaly detection, including high false alarm rates and ambiguous target localization due to background noise, this paper introduces a high-confidence detection algorithm based on hierarchical clustering with an optimal cut height. The core of our approach is a theoretically derived optimal cut height, which is calculated from a physical model of the magnetic dipole’s vertical gradient field. This model establishes the implicit functional relationship between the effective detection range and key parameters, including magnetic moment orientation, geomagnetic inclination, and sensor height. The calculated optimal cut height serves as the critical criterion in a complete-linkage hierarchical clustering algorithm, which processes the alarm point clouds generated by a preliminary Greatest-of Cell-Averaging Constant False Alarm Rate (GOCA-CFAR) detector. This effectively suppresses isolated false alarms caused by background fluctuations while preserving spatially coherent alarm clusters within the target’s effective detection range, thereby significantly enhancing detection confidence. Results from both simulations and field experiments validate the efficacy of the proposed algorithm, demonstrating its superior capability to reliably discriminate genuine targets from false alarms compared to traditional one-dimensional CFAR detection. Full article

We present a time-dependent framework that combines a hybrid basis, consisting of Gaussian-type orbitals (GTOs) and finite-element discrete-variable representation (FEDVR) functions, with a multicenter grid to simulate strong-field and attosecond dynamics in atoms and molecules. The method incorporates the construction of the orthonormal hybrid basis, the evaluation of electronic integrals, a unitary time-propagation scheme, and the extraction of optical and photoelectron observables. Its accuracy and robustness are benchmarked on one-electron systems such as atomic hydrogen and the dihydrogen cation (${\mathrm{H}}_2^{+}$) through comparisons with essentially-exact reference results for bound-state energies, high-harmonic generation spectra, photoionization cross sections, and photoelectron momentum distributions. This work establishes the groundwork for its integration with quantum-chemistry methods, which is already operational but will be detailed in future work, thereby enabling ab initio simulations of correlated polyatomic systems in intense ultrafast laser fields. Full article

This study investigates curves in a 7-dimensional space, represented by spatial generalized octonion-valued functions of a single variable, where the general octonions include real, split, semi, split semi, quasi, split quasi, and para octonions. We begin by constructing a new frame, referred to as the $G_2$-frame, for spatial generalized octonionic curves, and subsequently derive the corresponding derivative formulas. We also present the connection between the $G_2$-frame and the standard orthonormal basis of spatial generalized octonions. Moreover, we verify that Frenet–Serret formulas hold for spatial generalized octonionic curves. We establish the $G_2$-congruence of two spatial generalized octonionic curves and present the correspondence between the Frenet–Serret frame and the $G_2$-frame. A key advantage of the $G_2$-frame is that the associated frame equations involve lower-order derivatives. This method is both time-efficient and computationally efficient. To demonstrate the theory, we present an example of a unit-speed spatial generalized octonionic curve and compute its $G_2$-frame and invariants using MATLAB. Full article

In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate the Fekete points (AFPs) on arbitrary shape domains based on the singular value decomposition (SVD) of the Vandermonde matrix. In addition, a novel method is derived to compute the moments on highly complex domains, which may include discontinuities. Then, AFPs are used to generate compatible basis functions using SVD. Equations are derived and presented to determine orthogonal/orthonormal modal basis functions, as well as the Lagrange basis. Furthermore, theorems are proved to show the convergence and accuracy of the proposed method, together with an explicit form of the Weierstrass theorem for polynomial approximation. The method was implemented and some classical cases were analyzed. The results show the superior performance of the proposed method in terms of convergence and accuracy using many fewer DOFs and, thus, a much lower computational cost. It was shown that the orthogonal modal basis is the best choice to decrease the DOFs while maintaining a small Lebesgue constant when very high degree of polynomial is employed. Full article

The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier–Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs. Full article

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm–Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods. Full article

This paper presents a closed-loop continuous-time subspace identification method using prior information. Based on a rational inner function, a generalized orthonormal basis can be constructed, and the transformed noises have ergodicity features. The continuous-time stochastic system is converted into a discrete-time stochastic system by using generalized orthogonal basis functions. As is known to all, incorporating prior information into identification strategies can increase the precision of the identified model. To enhance the precision of the identification method, prior information is integrated through the use of constrained least squares, and principal component analysis is adopted to achieve the reliable estimate of the system. Moreover, the identification of open-loop models is the primary intent of the continuous-time system identification approaches. For closed-loop systems, the open-loop subspace identification methods may produce biased results. Principal component analysis, which reliably estimates closed-loop systems, provides a solution to this problem. The restricted least-squares method with an equality constraint is used to incorporate prior information into the impulse response following the principal component analysis. The input–output algebraic equation yielded an optimal multi-step-ahead predictor, and the equality constraints describe the prior information. The effectiveness of the proposed method is provided by numerical simulations. Full article

This paper presents a continuous-time subspace identification method utilizing prior information and generalized orthonormal basis functions. A generalized orthonormal basis is constructed by a rational inner function, and the transformed noises have ergodic properties. The lifting approach and the Hambo system transform are used to establish the equivalent nature of continuous and transformed discrete-time stochastic systems. The constrained least squares method is adopted to investigate the incorporation of prior knowledge in order to further increase the subspace identification algorithm’s accuracy. The input–output algebraic equation derives an optimal multistep forward predictor, and prior knowledge is expressed as equality constraints. In order to solve an optimization problem with equality constraints characterizing the prior knowledge, the proposed method reduces the computational burden. The effectiveness of the proposed method is provided by numerical simulations. Full article

The article is devoted to Bayes optimization problems of nonlinear observable stochastic systems (NLOStSs) based on wavelet canonical expansions (WLCEs). Input stochastic processes (StPs) and output StPs of considered nonlinearly StSs depend on random parameters and additive independent Gaussian noises. For stochastic synthesis we use a Bayes approach with the given loss function and minimum risk condition. WLCEs are formed by covariance function expansion coefficients of two-dimensional orthonormal basis of wavelet with a compact carrier. New results: (i) a common Bayes’ criteria synthesis algorithm for NLOStSs by WLCE is presented; (ii) partial synthesis algorithms for three of Bayes’ criteria (minimum mean square error, damage accumulation and probability of error exit outside the limits) are given; (iii) an approximate algorithm based on statistical linearization; (iv) three test examples. Applications: wavelet optimization and parameter calibration in complex measurement and control systems. Some generalizations are formulated. Full article

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments. Full article

Many systems with distributed dynamics are described by partial differential equations (PDEs). Coupled reaction-diffusion equations are a particular type of these systems. The measurement of the state over the entire spatial domain is usually required for their control. However, it is often impossible to obtain full state information with physical sensors only. For this problem, observers are developed to estimate the state based on boundary measurements. The method presented applies the so-called modulating function method, relying on an orthonormal function basis representation. Auxiliary systems are generated from the original system by applying modulating functions and formulating annihilation conditions. It is extended by a decoupling matrix step. The calculated kernels are utilized for modulating the input and output signals over a receding time window to obtain the coefficients for the basis expansion for the desired state estimation. The developed algorithm and its real-time functionality are verified via simulation of an example system related to the dynamics of chemical tubular reactors and compared to the conventional backstepping observer. The method achieves a successful state reconstruction of the system while mitigating white noise induced by the sensor. Ultimately, the modulating function approach represents a solution for the distributed state estimation problem without solving a PDE online. Full article

Quantum coherence is known as an important resource in many quantum information tasks, which is a basis-dependent property of quantum states. In this paper, we discuss quantum incoherence based simultaneously on k bases using Matrix Theory Method. First, by defining a correlation function $m(e,f)$ of two orthonormal bases e and f, we investigate the relationships between sets $\mathcal{I}\left(e\right)$ and $\mathcal{I}\left(f\right)$ of incoherent states with respect to e and f. We prove that $\mathcal{I}\left(e\right)=\mathcal{I}\left(f\right)$ if and only if the rank-one projective measurements generated by e and f are identical. We give a necessary and sufficient condition for the intersection $\mathcal{I}\left(e\right)\bigcap \mathcal{I}\left(f\right)$ to include a state except the maximally mixed state. Especially, if two bases e and f are mutually unbiased, then the intersection has only the maximally mixed state. Secondly, we introduce the concepts of strong incoherence and weak coherence of a quantum state with respect to a set $\mathcal{B}$ of k bases and propose a measure for the weak coherence. In the two-qubit system, we prove that there exists a maximally coherent state with respect to $\mathcal{B}$ when $k=2$ and it is not the case for $k=3$. Full article

Generating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been developed for the discrete Chebyshev and Krawtchouk functions, i.e., the orthonormal basis in ${\ell }_2$ derived from the polynomials. Since these functions are limiting cases of the discrete Hahn functions, it suggests that the strategy could also be successful there. We outline the algorithmic strategies including the specific method of generating the initial values, and show that the orthonormal basis can indeed be generated for large supports and polynomial degrees with controlled numerical error. Special attention is devoted to symmetries, as the symmetric windows are most commonly used in signal processing, allowing for simplification of the algorithm due to this prior knowledge, and leading to savings in the required computational power. Full article

Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples. Full article

Functional data analysis has become a research hotspot in the field of data mining. Traditional data mining methods regard functional data as a discrete and limited observation sequence, ignoring the continuity. In this paper, the functional data classification is addressed, proposing a functional generalized eigenvalue proximal support vector machine (FGEPSVM). Specifically, we find two nonparallel hyperplanes in function space, a positive functional hyperplane, and a functional negative hyperplane. The former is closest to the positive functional data and furthest from the negative functional data, while the latter has the opposite properties. By introducing the orthonormal basis, the problem in function space is transformed into the ones in vector space. It should be pointed out that the higher-order derivative information is applied from two aspects. We apply the derivatives alone or the weighted linear combination of the original function and the derivatives. It can be expected that to improve the classification accuracy by using more data information. Experiments on artificial datasets and benchmark datasets show the effectiveness of our FGEPSVM for functional data classification. Full article