Search

The best polynomial approximation and the Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it is not always superior to the latter, as noticed by Trefethen (Math. Today 47:184–188, 2011) for the function f (x) = | x - 0.25 | . Recently, Wang (arxiv, 2106.03456, 2021) observed a similar phenomenon for a function with an algebraic singularity and proved it in theory. In this paper, we find that for functions with logarithmic endpoint singularities, the pointwise errors of the Chebyshev approximation are smaller than those of the best approximations of the same degree, except only at the narrow boundary layer. The pointwise error for the Chebyshev series truncated at the degree n is O (n - κ) ( κ = min { 2 γ + 1 , 2 δ + 1 } ), but increases as O (n - (κ - 1)) at the endpoint singularities. Theorems are given to explain this effect. [ABSTRACT FROM AUTHOR]

Many physical problems represented as initial and boundary value problems are usually solved by transforming them into integral equations on the half-line. Therefore, this paper discusses the Galerkin and collocation methods along with the multi-Galerkin and multi-collocation methods for non-linear integral equations with compact and Wiener-Hopf kernel operators on the half-line in the space of piecewise polynomial subspaces. As Hammerstein integral equation on half-line has the unbounded domain, the finite section approximation method is applied; and afterward, we find an approximate solution for the finite section integral equation. In order to obtain improved superconvergence rates, we apply Sunil Kumar's method and then compare proposed convergence rates with the existing results of Nahid et al. [22] both theoretically and numerically. These results are further improved by applying multi-projection methods. In addition, it has shown that the proposed theory enhances the results of [22]. Finally, numerical examples are presented to demonstrate the given theoretical framework. [ABSTRACT FROM AUTHOR]

This paper concerns the study of matrix polynomials of arbitrary degree. In terms of L (λ) = λ r I - ∑ j = 1 r λ r - j C j with or without commuting coefficients ( C i C j = C j C i , o r C i C j ≠ C j C i f o r C i ∈ C t × t , i , j = 1 , ... , r ) by determinant of block Toeplitz-Hessenberg matrices. [ABSTRACT FROM AUTHOR]

In this paper, the discrete parametrix method is adopted to investigate the estimation of transition kernel for Euler-Maruyama scheme SDEs driven by α -stable noise, which implies Krylov's estimate and Khasminskii's estimate. As an application, the convergence rate of Euler-Maruyama scheme for a class of multidimensional SDEs with singular drift (by the use of Zvonkin's transformation) is obtained. [ABSTRACT FROM AUTHOR]

This paper studies numerical methods for the two-dimensional fractional Gray-Scott (GS) model with fractional Laplacian. A three-level linearized difference scheme for solving the fractional GS model is established. The boundedness, existence, uniqueness and convergence of the finite difference scheme are strictly proved by discrete energy analysis method. Furthermore, the convergence of the difference scheme in l ∞ -norm is of second order in space and time. Numerical experiments are implemented to verify the above theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the accuracy and stability of quaternion Gaussian elimination (qGE). First, considering the noncommutativity of quaternion multiplications, we establish the rules of quaternion floating-point operations. After that, roundoff error analyses of quaternion LU and qGE with partial pivoting (qGEPP) are presented to show the backward stability of qGEPP is affected by the growth factor. Then, the upper bounds for growth factors of qGE with partial or complete pivoting, and the quaternion matrices with maximal growth factor are investigated. Finally, growth factors of some special matrices are also studied, and the theoretical results are confirmed through several numerical experiments. A conjecture about the growth factor of quaternion Hadamard matrix is given. [ABSTRACT FROM AUTHOR]

This paper presents a new sufficiently descent algorithm for system of nonlinear equations where the underlying operator is pseudomonotone. The conditions imposed on the proposed algorithm to achieve convergence are Lipschitz continuity and pseudomonotonicity which is weaker than monotonicity assumption forced upon many algorithms in this area found in the literature. Numerical experiments on selected test problems taken from the literature validate the efficiency of the new algorithm. Moreover, the new algorithm demonstrates superior performance in comparison with some existing algorithms. Furthermore, the proposed algorithm is applied to reconstruct some disturbed signals. [ABSTRACT FROM AUTHOR]

When measuring the value of a function to be minimized is not only expensive but also with noise, the popular simultaneous perturbation stochastic approximation (SPSA) algorithm requires only two function values in each iteration. In this paper, we present a method requiring only one function measurement value per iteration in the average sense. We prove the strong convergence and asymptotic normality of the new algorithm. Limited experimental results demonstrate the effectiveness and potential of our algorithm for solving low-dimensional problems. [ABSTRACT FROM AUTHOR]

Al-Baali et al. (Comput. Optim. Appl. 60:89–110, 2015) have proposed a three-term conjugate gradient method which satisfies a sufficient descent condition and global convergence. In this paper, we extend this method to a family of three-term conjugate gradient projection methods with a restart procedure and their relaxed-inertial versions for constrained nonlinear pseudo-monotone equations. The accelerated gradient-descent method MSM and the relaxed-inertial strategy are incorporated into the proposed methods to obtain better computational performance. The global convergence of the extended methods are established theoretically without the Lipschitz continuity of the underlying mapping. Numerical results on constrained nonlinear equations show that the extended methods with different settings are efficient. The applicability and efficiency of the extended methods in the regularized decentralized logistic regression and sparse signal restoration problems are also presented and verified. [ABSTRACT FROM AUTHOR]

This-36pt paper considers a discontinuous Galerkin finite element method for the 2D transient and incompressible Navier-Stokes model. Following the analysis of Heywood and Rannacher (SIAM J. Numer. Anal. 19:275–311, 1982), we derive optimal velocity and pressure error estimates in L ∞ (L 2) and L ∞ (L 2) -norms, respectively, for the discontinuous Galerkin case. We use standard L 2 -projection and modified Stokes operator but on appropriate broken Sobolev spaces, and then standard duality arguments to achieve these results. For sufficiently small data, uniform in time estimates are proved. Based on the backward Euler method, time discretization is carried out and fully discrete error estimates are derived. Finally, numerical experiments are conducted to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving the one-dimensional global optimization problems where the objective function f satisfies a Hölder condition over a real interval [a, b]. We suggest two algorithms as an extended version of the Piyavskii's method. The first one is based on the use of the tangent at the midpoint of each subinterval of [a, b] of the parabolic lower bounding functions of f. The second one is a combination of two successive phases. In phase 1, we obtain with a few iterations, a subinterval which often contains the point of the global minimum using a K α -Lipschitz lower bounding functions with a tolerance δ much larger than the given accuracy ε . In phase 2, we apply the first algorithm on the interval obtained in phase 1. A convergence result is proved and compared with the existing methods. The numerical experiments of the two algorithms on some test functions are encouraging. [ABSTRACT FROM AUTHOR]

An algorithm and a MATLAB implementation for computing the Kummer function U(a,b,x) and its derivative is given in this paper. The algorithm is efficient and accurate. Numerical tests show that the MATLAB algorithm allows the computation of the function with ∼ 1 0 − 14 relative accuracy in the parameter region (a,b,x) ∈ (0,500) × (0,500) × (0,1000) in double-precision floating point arithmetic. [ABSTRACT FROM AUTHOR]

This paper proposes an accelerating outer space algorithm for globally solving generalized linear multiplicative problems (GLMP). Utilizing the logarithmic and exponential function properties, we begin with transforming the GLMP into an equivalent problem (EP) by introducing some outer space variables. Then, a two-stage outer space relaxation method is developed to convert the EP into a series of relaxed linear problems. Furthermore, to improve the convergence speed of the algorithm, we develop some accelerating techniques to remove the domains that do not contain a global optimal solution. Then, fusing the linear relaxed method and accelerating techniques into the branch-and-bound framework, we provide the accelerating outer space algorithm for solving the EP. Additionally, we analyze the global convergence of the proposed algorithm. Meanwhile, by investigating the algorithmic complexity, we estimate the maximum iterations required by the proposed algorithm in the worst case. Finally, in contrast to other algorithms in the currently known literature, numerical experimental results indicate that the proposed algorithm is feasible with better efficiency and robustness. [ABSTRACT FROM AUTHOR]

Consider the problem of solving systems of linear algebraic equations A x = b with a real symmetric positive definite matrix A using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the A -norm of the error. This quantity cannot be easily computed; however, it can be estimated. In this paper we discuss and analyze the behavior of the Gauss-Radau upper bound on the A -norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate μ to the smallest eigenvalue of A . We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of μ . We construct a model problem that is used to demonstrate and study the behavior of the upper bound in dependence of μ , and developed formulas that are helpful in understanding this behavior. We show that the above-mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of A . It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate μ . We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations. [ABSTRACT FROM AUTHOR]

Matrix completion is usually formulated as a low-rank matrix approximation problem. Several methods have been proposed to solve this problem, e.g., truncated nuclear norm regularization (TNNR) which performs well in recovery accuracy and convergence speed, and hybrid truncated norm regularization (HTNR) method which has better stability compared to TNNR. In this paper, a modified hybrid truncated norm regularization method, named WHTNR, is proposed to accelerate the convergence of the HTNR method. The proposed WHTNR method can preferentially restore rows with fewer missing elements in the matrix by assigning appropriate weights to the first r singular values. The presented experiments show empirical evidence on significant improvements of the proposed method over the closest four methods, both in convergence speed or in accuracy, it is robust to the parameter truncate singular values r. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the strong convergence order for the truncated Euler-Maruyama (TEM) method to solve stochastic differential delay equations (SDDEs) with multiple time delays and super-linearly growing coefficients. The strong Lp (1 ≤ p < 2) convergence rate of the TEM method under the one-sided polynomial growth condition is first established. Imposing additional conditions on the diffusion coefficient, the p-th moment uniform boundedness of both the exact and approximate solutions is then proved. Next, we show that the strong order of Lq-convergence can be arbitrarily close to 1/2 for 2 ≤ q ≤ p. Several examples and a numerical simulation are provided to illustrate the main results at the end. [ABSTRACT FROM AUTHOR]

Algebraically stabilized finite element discretizations of scalar steady-state convection–diffusion–reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). However, it was observed that a deterioration of the accuracy and convergence rates may occur for some problems if meshes without local symmetries are used. The paper investigates these phenomena both numerically and analytically and the findings are used to design a new algebraic stabilization called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is proved that the SMUAS method is linearity preserving and satisfies the DMP on arbitrary simplicial meshes. Moreover, numerical results indicate that the SMUAS method leads to optimal convergence rates on general simplicial meshes. [ABSTRACT FROM AUTHOR]

We consider Nitsche's method for solving elliptic Dirichlet boundary control problems on curved domains with control constraints. By using Nitsche's method for the treatment of inhomogeneous Dirichlet boundary conditions, the L2 boundary control enters in the variational formulation in a natural sense. The idea was first used in Chang, et al. (Math. Anal. Appl.453, 529–557 2017) where the curved boundary was approximated by a broken line and a locally defined mapping was needed to obtain the numerical control on the curved boundary. In this paper, we develop a method defined on curved domains directly. We derive a priori estimates of quasi-optimal order for the control in the L2 norm, and quasi-optimal order for the state and adjoint state in energy norms. Numerical examples are provided to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Adopting a suitable approximation strategy can both enhance the robustness and improve the efficiency of the numerical algorithms. In this paper, we suggest combining two approximation criteria, the absolute one and the relative one, to the asymmetric forward-backward-adjoint splitting (AFBA) algorithm for a class of convex-concave saddle point problems, resulting in two inexact AFBA variants. These two approximation criteria are low-cost, since verifying them just involves the subgradient of a certain function. For both the absolute error AFBA and the relative error AFBA, we establish the global convergence and the O(1/N) convergence rate measured by the gap function in the ergodic sense, where N is the number of iterations. For the absolute error AFBA, we show that it possesses an O(1/N2) (linear convergence) rate of convergence, under the assumption that a part of (both) the underlying functions are strongly convex. We report some numerical results which demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we develop a second-order backward differentiation formula (BDF) numerical scheme for the Ericksen-Leslie model of nematic liquid crystal. It combines a convex splitting method to discretize the penalty function. In the meanwhile, a pressure-correction strategy is used to decouple the pressure from that of the velocity. We prove that the proposed scheme is uniquely solvable and unconditionally stable in energy. Some numerical simulations are also performed to demonstrate the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In the present paper, two integrable and one non-integrable semi-discrete analogues of a generalized sine-Gordon (sG) equation are constructed. The keys of the construction are the Bäcklund transformation of bilinear equations and appropriate definition of the discrete hodograph transformation. We construct N-soliton solutions for the semi-discrete analogues of the generalized sG equation in the determinant form. In the continuous limit, we show that the semi-discrete generalized sG equations converge to the continuous generalized sG equation. Furthermore, we propose four self-adaptive moving mesh methods for the generalized sG equation, two are integrable and two are non-integrable. Integrable and non-integrable self-adaptive moving mesh methods are proposed and used for simulations of regular, irregular and loop soliton while comparing with the Crank-Nicolson (C-N) scheme. The numerical solutions show that the proposed self-adaptive moving methods perform better than the C-N scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number k, we establish global O (1 / k) pointwise and O (1 / k) ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the quaternion equality constrained weighted least squares problem. First, according to the properties of the Moore-Penrose generalized inverse matrix, the singular value decomposition, the CS decomposition, and the real representation matrices of quaternion matrices, we obtain a real unconstrained equivalent problem of the quaternion equality constrained weighted least squares problem. And then we give the expressions of its general solution and minimal norm solution. Finally, according to the properties of the real representation matrices of quaternion matrices, the special structure of real representation matrices and the real structure-preserving singular value decomposition of quaternion matrix, we propose a real structure-preserving algorithm for the minimal norm solution of the quaternion equality constrained weighted least squares problem, and illustrate the effectiveness of proposed algorithm by a numerical example. [ABSTRACT FROM AUTHOR]