Explore the vast range of titles available.

In this paper, moving least squares (MLS) and modified moving least squares (MMLS) methods have been employed to estimate the solution of two-dimensional linear and nonlinear Fredholm-Volterra integral equations. The modification means that quadratic base functions can be utilized with the same size of the support domain as linear base functions, resulting in better approximation capability. The proposed methods are meshless because they don't require any background mesh or cell structures and so they are independent of the geometry of the domain. The error estimate of the proposed method is provided. The accuracy and computational efficiency of the methods are illustrated by several numerical tests.

In this paper, methods of kinematic synthesis and analysis of the RoboMech class parallel manipulator (PM) with two grippers (end effectors) are presented. This PM is formed by connecting two output objects (grippers) with a base using two passive and one negative closing kinematic chains (CKCs). A PM with two end effectors can be used for reloading operations of stamped products between two adjacent main technologies in a cold stamping line. Passive CKCs represent two serial manipulators with two degrees of freedom, and negative CKC is a three-joined link with three negative degrees of freedom. A negative CKC imposes three geometric constraints on the movements of the two output objects. Geometric parameters of the negative CKC are determined on the basis of the problems of the Chebyshev and least-square approximations. Problems of positions and analogues of velocities and accelerations of the PM with two end effectors have been solved.

Using the complex variable moving least-squares (CVMLS) approximation, a complex variable element-free Galerkin (CVEFG) method for two-dimensional elastoplastic large deformation problems is presented. This meshless method has higher computational precision and efficiency because in the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. For two-dimensional elastoplastic large deformation problems, the Galerkin weak form is employed to obtain its equation system. The penalty method is used to impose essential boundary conditions. Then the corresponding formulae of the CVEFG method for two-dimensional elastoplastic large deformation problems are derived. In comparison with the conventional EFG method, our study shows that the CVEFG method has higher precision and efficiency. For illustration purpose, a few selected numerical examples are presented to demonstrate the advantages of the CVEFG method.

Based on the moving least-square (MLS) approximation, the complex variable moving least-square approximation (CVMLS) is discussed in this paper. The complex variable moving least-square approximation cannot form ill-conditioned equations, and has greater precision and computational efficiency. Using the analytical solution near the tip of a crack, the trial functions in the complex variable moving least-square approxi- mation are extended, and the corresponding approximation function is obtained. And from the minimum potential energy principle, a complex variable meshless method for fracture problems is presented, and the formulae of the complex variable meshless method are obtained. The complex variable meshless method in this paper has greater precision and computational efficiency than the conventional meshless method. Some examples are given.

We construct a least squares approximation method for the recovery of complex-valued functions from a reproducing kernel Hilbert space on D ⊂ R d . The nodes are drawn at random for the whole class of functions, and the error is measured in L 2 ( D , ϱ D ) . We prove worst-case recovery guarantees by explicitly controlling all the involved constants. This leads to new preasymptotic recovery bounds with high probability for the error of hyperbolic Fourier regression on multivariate data. In addition, we further investigate its counterpart hyperbolic wavelet regression also based on least squares to recover non-periodic functions from random samples. Finally, we reconsider the analysis of a cubature method based on plain random points with optimal weights and reveal near-optimal worst-case error bounds with high probability. It turns out that this simple method can compete with the quasi-Monte Carlo methods in the literature which are based on lattices and digital nets.

The variable projection algorithm of Golub and Pereyra (SIAM J. Numer. Anal. 10:413–432, 1973 ) has proven to be quite valuable in the solution of nonlinear least squares problems in which a substantial number of the parameters are linear. Its advantages are efficiency and, more importantly, a better likelihood of finding a global minimizer rather than a local one. The purpose of our work is to provide a more robust implementation of this algorithm, include constraints on the parameters, more clearly identify key ingredients so that improvements can be made, compute the Jacobian matrix more accurately, and make future implementations in other languages easy.

We propose a method of least squares approximation (LSA) for unified yet simple LASSO estimation. Our general theoretical framework includes ordinary least squares, generalized linear models, quantile regression, and many others as special cases. Specifically, LSA can transfer many different types of LASSO objective functions into their asymptotically equivalent least squares problems. Thereafter, the standard asymptotic theory can be established and the LARS algorithm can be applied. In particular, if the adaptive LASSO penalty and a Bayes information criterion-type tuning parameter selector are used, the resulting LSA estimator can be as efficient as the oracle. Extensive numerical studies confirm our theory.

This paper addresses the challenge of function approximation using Hermite interpolation on equally spaced nodes. In this setting, standard polynomial interpolation suffers from the Runge phenomenon. To mitigate this issue, we propose an extension of the constrained mock-Chebyshev least squares approximation technique to Hermite interpolation. This approach leverages both function and derivative evaluations, resulting in more accurate approximations. Numerical experiments are implemented in order to illustrate the effectiveness of the proposed method.

This letter proposes a least squares‐based method for a single near‐field source localisation by jointly estimating bearing angle and range using a uniform linear array. To provide initial estimates in one step, the proposed method uses least squares formulation developed based on the observation that adjacent sensor pairs exhibit far‐field characteristics while the overall array has near‐field properties. To further enhance the performance of range estimation, we re‐estimate the range parameters by utilising geometric relationships. Consequently, the proposed method can provide improved localisation performance with reduced computational complexity. Simulation results demonstrate the superiority of the proposed method including the scheme of enhanced range estimation. This letter proposes a least squares‐based method for a single near‐field source localisation by jointly estimating bearing angle and range using a uniform linear array. To provide initial estimates in one step, the proposed method uses least squares formulation developed based on the observation that adjacent sensor pairs exhibit far‐field characteristics while the overall array has near‐field properties. To further enhance the performance of range estimation, we re‐estimate the range parameters by utilising geometric relationships.

We consider the problem of reconstructing an unknown function u ∈ L 2 ( D , μ ) from its evaluations at given sampling points x 1 , ⋯ , x m ∈ D , where D ⊂ R d is a general domain and μ a probability measure. The approximation is picked from a linear space V n of interest where n = dim ( V n ) . Recent results (Cohen and Migliorati in SMAI J Comput Math 3:181–203, 2017, Doostan and Hampton in Comput Methods Appl Mech Eng 290:73–97, 2015, Jakeman et al. in Math Comput 86:1913–1947, 2017) have revealed that certain weighted least-squares methods achieve near best (or instance optimal) approximation with a sampling budget m that is proportional to n , up to a logarithmic factor ln ( 2 n / ε ) , where ε > 0 is a probability of failure. The sampling points should be picked at random according to a well-chosen probability measure σ whose density is given by the inverse Christoffel function that depends both on V n and μ . While this approach is greatly facilitated when D and μ have tensor product structure, it becomes problematic for domains D with arbitrary geometry since the optimal measure depends on an orthonormal basis of V n in L 2 ( D , μ ) which is not explicitly given, even for simple polynomial spaces. Therefore, sampling according to this measure is not practically feasible. One computational solution recently proposed in Adcock and Huybrechs (Approximating smooth, multivariate functions on irregular domains, forum of mathematics, sigma, Cambridge University Press, Cambridge, 2020) relies on using the restrictions of an orthonormal basis of V n defined on a simpler bounding domain and sampling according to the original probability measure μ , in turn giving up on the optimal sampling budget m ∼ n . In this paper, we discuss practical sampling strategies, which amounts to using a perturbed measure σ ~ that can be computed in an offline stage, not involving the measurement of u , as recently proposed in Adcock and Cardenas (SIAM J Math Data Sci 2:607–630, 2020) and Migliorati (IMA J Numer Anal, 2020. https://doi.org/10.1093/imanum/draa023 ). We show that near best approximation is attained by the resulting weighted least-squares method at near-optimal sampling budget and we discuss multilevel approaches that preserve optimality of the cumulated sampling budget when the spaces V n are iteratively enriched. These strategies rely on the knowledge of a-priori upper bounds B ( n ) on the inverse Christoffel function for the space V n and the domain D . We establish bounds of the form O ( n r ) for spaces V n of multivariate algebraic polynomials of given total degree, and for general domains D . The exact growth rate r depends on the regularity of the domain, in particular r = 2 for domains with Lipschitz boundaries and r = d + 1 d for smooth domains.