Search Results (18)

To address the computational efficiency challenges in two-dimensional (2D) direction-of-arrival (DOA) estimation, a two-stage framework integrating the Nyström approximation with subspace decomposition techniques is proposed in this paper. The methodology strategically integrates the Nyström approximation with subspace decomposition techniques to bridge the critical performance–complexity trade-off inherent in high-resolution parameter estimation scenarios. In the first stage, the Nyström method is applied to approximate the signal subspace while simultaneously enabling construction of a reduced rank covariance matrix, which effectively reduces the computational complexity compared with eigenvalue decomposition (EVD) or singular value decomposition (SVD). This innovative approach efficiently derives two distinct signal subspaces that closely approximate those obtained from the full-dimensional covariance matrix but at substantially reduced computational cost. The second stage employs a sophisticated subspace-based estimation technique that leverages the principal singular vectors associated with these approximated subspaces. This process incorporates an iterative refinement mechanism to accurately resolve the paired azimuth and elevation angles comprising the 2D DOA solution. With the use of the Nyström approximation and reduced rank framework, the entire DOA estimation process completely circumvents traditional EVD/SVD operations. This elimination constitutes the core mechanism enabling substantial computational savings without compromising estimation accuracy. Comprehensive numerical simulations rigorously demonstrate that the proposed framework maintains performance competitive with conventional high-complexity estimators while achieving significant complexity reduction. The evaluation benchmarks the method against multiple state-of-the-art DOA estimation techniques across diverse operational scenarios, confirming both its efficacy and robustness under varying signal conditions. Full article

Recently we studied a collocation–quadrature method in weighted ${\mathbf{L}}^2$ spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form $u\left(x\right)-{\int }_{\alpha x}^{1}h(x-\alpha y)u\left(y\right)dy=f\left(x\right),0<x<1,$ where $h\left(x\right)$ (with a possible singularity at $x=0$) and $f\left(x\right)$ are given (in general complex-valued) functions, and $\alpha \in (0,1)$ is a fixed parameter. Here, we want to investigate the same method for the case when $\alpha =1.$ More precisely, we consider (in general weakly singular) Volterra integral equations of the form $u\left(x\right)-{\int }_0^{x}h(x,y){(x-y)}^{\kappa }u\left(y\right)dy=f\left(x\right),0<x<1,$ where $\kappa >-1$, and $h:D⟶\mathbb{C}$ is a continuous function, $D=\left\{(x,y)\in {\mathbb{R}}^2:0<y<x<1\right\}.$ The passage from $0<\alpha <1$ to $\alpha =1$ and the consideration of more general kernel functions $h(x,y)$ make the studies more involved. Moreover, we enhance the family of interpolation operators defining the approximating operators, and, finally, we ask if, in comparison to collocation–quadrature methods, the application of the Nyström method together with the theory of collectively compact operator sequences is possible. Full article

We consider a two-step numerical approach for solving parabolic initial boundary value problems in 3D simply connected smooth regions. The method uses the Laplace transform in time, reducing the problem to a set of independent stationary boundary value problems for the Helmholtz equation with complex parameters. The inverse Laplace transform is computed using a sinc quadrature along a suitably chosen contour in the complex plane. We show that due to a symmetry of the quadrature nodes, the number of stationary problems can be decreased by almost a factor of two. The influence of the integration contour parameters on the approximation error is also researched. Stationary problems are numerically solved using a boundary integral equation approach applying the Nyström method, based on the quadratures for smooth surface integrals. Numerical experiments support the expectations. Full article

We propose an efficient Nyström-based unitary subspace method for low-complexity two-dimensional (2D) direction-of-arrival (DOA) estimation in uniform rectangular array (URA) signal processing systems. The conventional high-resolution DOA estimation methods often suffer from excessive computational complexity, particularly when dealing with large-scale antenna arrays. The proposed method addresses this challenge by combining the Nyström approximation with a unitary transformation to reduce the computational burden while maintaining estimation accuracy. The signal subspace is approximated using a partitioned covariance matrix, and a real-valued transformation is applied to further simplify the eigenvalue decomposition (EVD) process. Furthermore, the linear prediction coefficients are estimated via a weighted least squares (WLS) approach, enabling robust extraction of the angular parameters. The 2D DOA estimates are then derived from these coefficients through a closed-form solution, eliminating the need for exhaustive spectral searches. Numerical simulations demonstrate that the proposed method achieves comparable estimation performance to state-of-the-art techniques while significantly reducing computational complexity. For a fixed array size of $M=N=20$, the proposed method demonstrates significant computational efficiency, requiring less than $50\%$ of the running time compared to conventional ESPRIT, and only $6\%$ of the time required by ML methods, while maintaining similar performance. This makes it particularly suitable for real-time applications where computational efficiency is critical. The novelty lies in the integration of Nyström approximation and unitary subspace techniques, which jointly enable efficient and accurate 2D DOA estimation without sacrificing robustness against noise. The method is applicable to a wide range of array processing scenarios, including radar, sonar, and wireless communications. Full article

The cooperative localization of Unmanned Aerial Vehicles (UAVs) has emerged as a pivotal application in Internet of Things (IoT) tasks. However, the frequent exchange of localization data among UAVs leads to significant energy consumption and escalates the computational complexity involved in multi-UAV cooperative localization tasks. To address these challenges, this paper proposes a cooperative localization algorithm that integrates a biogeography optimization-based cluster networking and adaptive sampling-improved Nystrom super-multidimensional scaling (BOCN-ASNSMS). The proposed method leverages biogeography optimization (BO), prioritizing nodes with higher residual energy and density to serve as cluster heads, thereby optimizing energy usage. Subsequently, an improved adaptive sampling Nystrom super-multidimensional scaling algorithm is employed to dynamically select the kernel matrix row vectors. This selection process not only reduces data processing requirements but also enhances the accuracy of the similarity matrix approximation, thus diminishing computational complexity and achieving precise relative positioning of UAVs. Furthermore, Procrustes analysis and least squares methods are utilized to fuse coordinates across UAV clusters, aligning them into a unified coordinate system and converting them into absolute coordinates, which facilitates high-precision global localization. Theoretical analysis and simulation results underscore that the proposed algorithm substantially reduces computational complexity and energy consumption while enhancing localization accuracy, compared to conventional multi-UAV cooperative localization approaches. Full article

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order $\alpha$$\left(D^{\alpha }f\right)\left(y\right)=\frac{1}{\Gamma (m-\alpha )}{\int }_0^y{(y-x)}^{m-\alpha -1}{f}^{\left(m\right)}\left(x\right)dx,y>0$, with $m-1<\alpha \le m,m\in \mathbb{N}.$ The numerical procedure is based on approximating $f^{\left(m\right)}$ by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order $\alpha$. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure. Full article

This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good conditioning of the method are proved in suitable subspaces of continuous functions of Sobolev type. The cubature formula, on which the Nyström method is constructed, has an error that behaves like the best polynomial approximation of the integrand function. Consequently, it is also shown how the Nyström method inherits this property and, hence, the proposed numerical strategy is fast when the involved known functions are smooth. Some numerical examples illustrate the efficiency of the method, also in comparison with other methods known in the literature. Full article

This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of Nyström formulas that are used at the fist subinterval. The optimized technique is obtained after imposing the vanishing of some of the local truncation errors, which results in a set of symmetric hybrid points. By solving an algebraic system of equations, our proposed approach generates simultaneous approximations at all grid points, resulting in a highly effective technique that outperforms several existing numerical methods in the literature. To assess the efficiency and accuracy of our approach, we perform some numerical tests on diverse real-world problems, including singular boundary value problems (SBVPs) from chemical kinetics. Full article

Integral equations play an important role for their applications in practical engineering and applied science, and nonlinear Urysohn integral equations can be applied when solving many problems in physics, potential theory and electrostatics, engineering, and economics. The aim of this paper is the use of spline quasi-interpolating operators in the space of splines of degree d and of class $C^{d-1}$ on uniform partitions of a bounded interval for the numerical solution of Urysohn integral equations, by using a superconvergent Nyström method. Firstly, we generate the approximate solution and we obtain outcomes pertaining to the convergence orders. Additionally, we examine the iterative version of the method. In particular, we prove that the convergence order is $(2d+2)$ if d is odd and $(2d+3)$ if d is even. In case of even degrees, we show that the convergence order of the iterated solution increases to $(2d+4)$. Finally, we conduct numerical tests that validate the theoretical findings. Full article

Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article. Full article

In this paper, we propose an efficient Nyström method with theoretical and empirical guarantees. In parallel computing environments and for sparse input kernel matrices, our algorithm can have computation efficiency comparable to the conventional Nyström method, theoretically. Additionally, we derive an important theoretical result with a compacter sketching matrix and faster speed, at the cost of some accuracy loss compared to the existing state-of-the-art results. Faster randomized SVD and more efficient adaptive sampling methods are also proposed, which have wide application in many machine-learning and data-mining tasks. Full article

In this paper, we propose a computationally efficient algorithm with improved effective aperture for coherent angle estimation in a monostatic multiple-input multiple-output (MIMO) radar. First, the direction matrix of MIMO radar is mapped into a low-dimensional matrix of virtual uniform linear array (ULA). Then, an augmented data expansion matrix with improved effective aperture is obtained by exploiting the Vandermonde-like structure of the low-dimensional direction matrix and radar cross section (RCS) matrix to enlarge the aperture of the array. Next, a unitary transformation is used to transform the augmented matrix into a real value and the approximate signal subspace of the augmented matrix is obtained by the Nyström method, which can reduce the computational complexity. The eigenvectors of the approximate signal subspace are used to reconstruct the matrix for direct decorrelation processing. Finally, direction of arrivals (DOAs) can be estimated faster by utilizing the unitary ESPRIT algorithm since the rotation invariance of the extended reconstruction matrix still exists. The proposed algorithm has a lower total computational complexity, and the estimation accuracy is improved by utilizing real values and enlarging the array aperture for estimation. Several theoretical analyses and simulation results confirm the effectiveness and advantages of the proposed method. Full article

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree $⩽r-1$, we obtain convergence order $2r$ for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are $3r+1$ and $4r$, respectively. Moreover, we show that the optimal convergence order $4r$ is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples. Full article

In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on $[-1,1]$. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given. Full article

The traditional frequency-modulated continuous wave (FMCW) multiple-input multiple-output (MIMO) radar two-dimensional (2D) super-resolution (SR) estimation algorithm for target localization has high computational complexity, which runs counter to the increasing demand for real-time radar imaging. In this paper, a fast joint direction-of-arrival (DOA) and range estimation framework for target localization is proposed; it utilizes a very deep super-resolution (VDSR) neural network (NN) framework to accelerate the imaging process while ensuring estimation accuracy. Firstly, we propose a fast low-resolution imaging algorithm based on the Nystrom method. The approximate signal subspace matrix is obtained from partial data, and low-resolution imaging is performed on a low-density grid. Then, the bicubic interpolation algorithm is used to expand the low-resolution image to the desired dimensions. Next, the deep SR network is used to obtain the high-resolution image, and the final joint DOA and range estimation is achieved based on the reconstructed image. Simulations and experiments were carried out to validate the computational efficiency and effectiveness of the proposed framework. Full article