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Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lions et al. in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulam (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the computation of the Lerch transcendent by means of the Gauss-Laguerre formula. An a priori estimate of the quadrature error, that allows to compute the number of quadrature nodes necessary to achieve an arbitrary precision, is derived. Exploiting the properties of the Gauss-Laguerre rule and the error estimate, a truncated approach is also considered. The algorithm used and its Matlab implementation are reported. The numerical examples confirm the reliability of this approach. [ABSTRACT FROM AUTHOR]

Singular value decomposition plays a prominent role in the theoretical study and numerical calculation of a quaternion matrix in applied sciences. This paper, by means of a complex representation of a quaternion matrix, studies the algorithm for the singular value decomposition of a quaternion matrix, and derives a complex structure-preserving algorithm for the singular value decomposition of a quaternion matrix. This paper also gives two examples to demonstrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The BFGS method is a common and effective method for solving unconstrained optimization problems in quasi-Newton algorithm. However, many scholars have proven that the algorithm may fail in some cases for nonconvex problems under Wolfe conditions. In this paper, an adaptive projection BFGS algorithm is proposed naturally which can solve nonconvex problems, and the following properties are shown in this algorithm: ➀ The generation of the step size α j satisfies the popular Wolfe conditions; ➁ a specific condition is proposed which has sufficient descent property, and if the current point satisfies this condition, the ordinary BFGS iteration process proceeds as usual; ➂ otherwise, the next iteration point x j + 1 is generated by the proposed adaptive projection method. This algorithm is globally convergent for nonconvex problems under the weak-Wolfe-Powell (WWP) conditions and has a superlinear convergence rate, which can be regarded as an extension of projection BFGS method proposed by Yuan et al. (J. Comput. Appl. Math. 327:274-294, 2018). Furthermore, the final numerical results and the application of the algorithm to the Muskingum model demonstrate the feasibility and competitiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a projection-based hybrid spectral gradient algorithm for nonlinear equations with convex constraints, which is based on a certain line search strategy. Convex combination technique is used to design a novel spectral parameter that is inspired by some classical spectral gradient methods. The search direction also meets the sufficient descent condition and trust region feature. The global convergence of the proposed algorithm has been established under reasonable assumptions. The results of the experiment demonstrate the proposed algorithm is both more promising and robust than some similar methods, and it is also capable of handling large-scale optimization problems. Furthermore, we apply it to problems involving sparse signal recovery and blurred image restoration. [ABSTRACT FROM AUTHOR]

In this paper, we propose a constraint preserving algorithm for the smallest Z-eigenpair of the compact Laplacian tensor of an even-uniform hypergraph, where Householder transform is employed and a family of modified conjugate directions with sufficient descent is determined. Besides, we prove that there exists a positive step size in the new constraint preserving update scheme such that the Wolfe conditions hold. Based on these properties, we prove the convergence of the new algorithm. Furthermore, we apply our algorithm to the hypergraph partitioning and image segmentation, and numerical results are reported to illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper constructs an interval partition linearization algorithm for solving minimax linear fractional programming (MLFP) problem. In this algorithm, MLFP is converted and decomposed into a series of linear programs by dividing the outer 1-dimensional space of the equivalent problem (EP) into polynomially countable intervals. To improve the computational efficiency of the algorithm, two new acceleration techniques are introduced, in which the regions in outer space where the optimal solution of EP does not exist are largely deleted. In addition, the global convergence of the proposed algorithm is summarized, and its computational complexity is illustrated to reveal that it is a fully polynomial time approximation scheme. Finally, the numerical results demonstrate that the proposed algorithm is feasible and effective. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the application of the fading regularization method with the method of fundamental solutions (MFS) to the ill-posed Cauchy-Stokes problem. For both smooth and piecewise smooth two-dimensional geometries, we present a numerical reconstruction of the missing velocity and normal stress tensor on an inaccessible part of the boundary based on knowledge of over-prescribed noisy data acquired on the remaining accessible boundary part. Three numerical examples demonstrate the proposed numerical algorithm's accuracy, convergence, stability, and efficiency, as well as its ability to deblur the data. [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]

In this paper, we consider the continuous relaxation reformulation of cardinality-constrained optimization problems that has become more popular in recent years and propose a new sequential optimality condition (approximate stationarity) for cardinality-constrained optimization problems, which is proved to be a genuine necessary optimality condition that does not require any constraint qualification to hold. We compare this condition with the rest of the sequential optimality conditions and prove that our condition is stronger and closer to the local minimizer. A problem-tailored regularity condition is proposed, and we show that this regularity condition ensures that the approximate stationary point proposed in this paper is the exact stationary point and is the weakest constraint qualification with this property. Finally, we apply the results of this paper to safeguarded augmented Lagrangian method and prove that the algorithm converges to the approximate stationary point proposed in this paper under mild assumptions, the existing theoretical results of this algorithm are further enhanced. [ABSTRACT FROM AUTHOR]

In this paper, we design a low-cost feasible projection algorithm for variational inequalities by replacing the projection onto the feasible set with the projection onto a ball. In each iteration, it only needs to calculate the value of the mapping once, and the projection onto the ball contained in the feasible set (which has an explicit expression), so the algorithm is easier to implement and feasible. The convergence of the algorithm is proved when the Slater condition holds for the feasible set and the mapping is pseudomonotone, Lipschitz continuous. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. [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]

In this paper, we focus on a descent algorithm for solving nonsmooth nonconvex optimization problems. The proposed method is based on the proximal bundle algorithm and the gradient sampling method and uses the advantages of both. In addition, this algorithm has the ability to handle inexact information, which creates additional challenges. The global convergence is proved with probability one. More precisely, every accumulation point of the sequence of serious iterates is either a stationary point if exact values of gradient are provided or an approximate stationary point if only inexact information of the function and gradient values is available. The performance of the proposed algorithm is demonstrated using some academic test problems. We further compare the new method with a general nonlinear solver and two other methods specifically designed for nonconvex nonsmooth optimization problems. [ABSTRACT FROM AUTHOR]

This paper considers a high-order numerical method for a computed solution of multidimensional convection-diffusion-reaction equation with time-fractional derivative subjected to appropriate initial and boundary conditions. The stability and error estimates of the proposed numerical approach are analyzed using the L ∞ (0 , T ; L 2) -norm. The theoretical study suggests that the new technique is unconditionally stable and temporal accurate with order O(τ2+α), where τ denotes the time step and 0 < α < 1. This result shows that the developed algorithm is faster and more efficient than a broad range of numerical techniques widely studied in the literature for the considered problem. Numerical experiments confirm the theory and they indicate that the proposed numerical scheme converges with accuracy O(τ2+α + h4), where h represents the space step. [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 propose and analyze a second-order partitioned method with multiple-time-step technique for the nonstationary Stokes-Darcy model. This method allows different time steps in different subdomains and improves the accuracy by the time filters. Besides, by designing new error estimate and time step adjustment strategy, we extend this method to variable timestep and develop single and double adaptive algorithms. Constant and variable time step tests are given to confirm the theoretical analysis and illustrate the effectiveness of the proposed method. [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 propose a filter sequential adaptive regularization algorithm using cubics (ARC) for solving nonlinear equality constrained optimization. Similar to sequential quadratic programming methods, an ARC subproblem with linearized constraints is considered to obtain a trial step in each iteration. Composite step methods and reduced Hessian methods are employed to tackle the linearized constraints. As a result, a trial step is decomposed into the sum of a normal step and a tangential step which is computed by a standard ARC subproblem. Then, the new iteration is determined by filter methods and ARC framework. The global convergence of the algorithm is proved under some reasonable assumptions. Preliminary numerical experiments and comparison results are reported. [ABSTRACT FROM AUTHOR]

When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z2 − c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p(z) = zd − c. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods. [ABSTRACT FROM AUTHOR]

In a very interesting paper (SIAM J. Control Optim. 37(3): 765–776, 1999), Solodov and Svaiter introduced an effective projection algorithm with linesearch for finding a solution of a variational inequality problem (VIP) in Euclidean space. They showed that the iterative sequence generated by their algorithm converges to a solution of (VIP) under the main assumption that the cost mapping is pseudomonotone and continuous. In this paper, we propose to modify this algorithm for solving variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. Moreover, we do not use the embedded projection methods as in methods used in literature and the linesearch procedure is not necessary when the cost mapping is Lipschitz. Several numerical examples are also provided to illustrate the efficient of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets, such as the intersection of many closed convex sets, the set of all minimizers of a nonsmooth convex function, and the intersection of sublevel sets of nonsmooth convex functions. We focus on adaptive learning rate optimization algorithms, which adapt step-sizes (referred to as learning rates in the machine learning field) to find optimal solutions quickly. We then propose a Riemannian stochastic fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with the adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm for nonsmooth convex and smooth nonconvex optimization. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem with a guaranteed convergence rate. This paper also provides numerical comparisons that demonstrate the effectiveness of the proposed algorithms with formulas based on the adaptive learning rate optimization algorithms, such as Adam and AMSGrad. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm for finding a zero of a pseudomonotone vector field with a convex constraint on a Hadamard manifold. This new method is the combination of the hyperplane projection method with specially constructed search directions. The global convergence property of this algorithm is established under the assumptions that the constructed halfspace is closed and convex, the tangent vector field is continuous, and the solution set is nonempty. Numerical experiments show the efficiency of this new derivative-free iterative method. [ABSTRACT FROM AUTHOR]

As a generalization of k-tridiagonal matrices, many variations of (p,q)-tridiagonal matrices have attracted much attention over the years. In this paper, we present an efficient algorithm for numerically computing the inverses of n-square (p,q)-tridiagonal matrices under a certain condition. The algorithm is based on the combination of a bidiagonalization approach which preserves the banded structure and sparsity of the original matrix, and a recursive algorithm for inverting general lower bidiagonal matrices. Some numerical results with simulations in MATLAB implementation are provided in order to illustrate the accuracy and efficiency of the proposed algorithms, and its competitiveness with MATLAB built-in function. [ABSTRACT FROM AUTHOR]

The mutual coherence of a matrix, defined as the maximum absolute value of the normalized inner-products between different columns, is an important property that characterizes the similarity between different matrix columns. Redundant matrices with very low mutual coherence are referred to as incoherent redundant matrices which play an important role in mathematical signal processing tasks. The problem of minimizing the mutual coherence in a give matrix space where each matrix has normalized columns is called the coherence optimization problem. In this paper, we transform equivalently the coherence optimization problem as a rank constrained semidefinite ℓ ∞ -minimization problem. It is critical to analyze the projection operator onto the nonconvex set in the new matrix optimization constraints, i.e., the nonconvex set of symmetric positive semidefinite matrices whose rank is not greater than a give positive integer. By exploiting the projection operator, we express the nonconvex set mentioned above as the set of zero roots of a Difference of two Convex (DC) functions. For the convex function related to the projection operator in the DC function, we characterize its properties and obtain the concise form of its subdifferential. With the help of the DC function, a new algorithm based on DCA (DC Algorithms) is proposed to solve the Coherence Optimization problem, and thus the proposed algorithm is called DCACO for short. We also study the convergence analysis of DCACO. An advantage of DCACO is that subproblems in each iteration have closed-form solutions. Experimental results demonstrate that DCACO leads to state-of-art performance on generating highly incoherent redundant matrices, and DCACO can also compete with several other algorithms on designing optimized projection matrices for improving the performance of compressed sensing. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem of nonlinear systems with mixed control variables and probability constraints. To obtain a numerical solution of this optimal control problem, our target is to formulate this problem as a constrained nonlinear parameter optimization problem (CNPOP), which can be solved by using any gradient-based numerical computation method. Firstly, some binary functions are introduced for each value of the discrete-valued control variable (DCV). Following that, we relax these binary functions and impose a penalty term on the relaxation such that the solution of the resulting relaxed problem (RP) can converge to the solution of the original problem as the penalty parameter increases. Secondly, we introduce a simple initial transformation for the probability constraints. Following that, an adaptive sample approximation method (ASAM) and a novel smooth approximation technique (NSAT) are adopted to formulate the probability constraints as some deterministic constraints. Thirdly, a control parameterization approach (CPA) is used to transform the deterministic problem (i.e., an infinite dimensional problem) into a finite dimensional CNPOP. Fourthly, in order to combine the advantages of limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithms and differential evolution (DE) algorithms, a L-BFGS-based DE (L-BFGS-DE) algorithm is proposed for solving the resulting approximation problem based on an improvied L-BFGS (IL-BFGS) method and an improved DE (IDE) algorithm. Following that, we establish the convergence result of the L-BFGS-DE algorithm. The L-BFGS-DE algorithm consists of two stages. The objectives of the first and second stages are to obtain a probable position of the global solution and to accelerate the convergence rate, respectively. In the IL-BFGS method, we propose a novel updating rule (NUR), which uses not only the gradient information of the objective function but also the value of the objective function. This will improved the performance of the IL-BFGS method. In the IDE algorithm, a novel adaptive parameter adjustment (NAPA) method, a novel population size decrease (NPSD) strategy, and an improved mutation (IM) scheme are proposed to improve its performance. Finally, an anti-cancer drug therapy problem (ADTP) is further extended to illustrate the effectiveness of the L-BFGS-DE algorithm by taking into account some probability constraints. Numerical results show that the L-BFGS-DE algorithm has good performance and can obtain a stable and robust performance when considering the small noise perturbations in initial state. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of hybrid inertial and contraction proximal point algorithm for the variational inclusion problem of the sum of two mappings in Hilbert spaces. We prove that the proposed algorithm converges strongly to a solution of the variational inclusion problem whenever its solution set is nonempty and the single-valued mapping f is Lipschitz continuous, monotone, and the set-valued mapping A is maximal monotone in infinite-dimensional real Hilbert spaces. Our work generalize and extend some related existing results in the literature. Finally, we illustrate the numerical performance of our Algorithm 1 and we give an application to the split feasibility problem. [ABSTRACT FROM AUTHOR]

In this paper, a two-sided, parallel Kogbetliantz-type algorithm for the hyperbolic singular value decomposition (HSVD) of real and complex square matrices is developed, with a single assumption that the input matrix, of order n, admits such a decomposition into the product of a unitary, a non-negative diagonal, and a J-unitary matrix, where J is a given diagonal matrix of positive and negative signs. When J = ±I, the proposed algorithm computes the ordinary SVD. The paper's most important contribution—a derivation of formulas for the HSVD of 2 × 2 matrices—is presented first, followed by the details of their implementation in floating-point arithmetic. Next, the effects of the hyperbolic transformations on the columns of the iteration matrix are discussed. These effects then guide a redesign of the dynamic pivot ordering, being already a well-established pivot strategy for the ordinary Kogbetliantz algorithm, for the general, n × n HSVD. A heuristic but sound convergence criterion is then proposed, which contributes to high accuracy demonstrated in the numerical testing results. Such a J-Kogbetliantz algorithm as presented here is intrinsically slow, but is nevertheless usable for matrices of small orders. [ABSTRACT FROM AUTHOR]

It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of f . Future research directions are outlined at the end of the paper. [ABSTRACT FROM AUTHOR]

We formulate the Multi-Constrained Dynamic Traffic Assignment (DTA) problem as an instance of the nonlinear composite problem. To solve the problem, this paper introduces then the penalized nonlinear alternating direction method of multipliers (ADMM), a numerical algorithm that combines the nonlinear ADMM algorithm with the external penalty method. Numerical results are then presented, analyzed and compared against those obtained by applying the Reformulation-Linearization Technique (RLT)-based convex relaxation method together with piecewise linear approximation of the objective function. [ABSTRACT FROM AUTHOR]

This paper proposes a new definition for the term filled function, which weakens the third condition of the filling properties. Based on this new definition, a general form for parameter-free filled functions is created. One of the specific parameter-free filled functions of the proposed general form is then implemented in some comparable global optimization problems. The data generated from the numerical process reveals that the proposed filled functions are dependable and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we study the split feasibility problem with multiple output sets in Hilbert spaces. For solving the aforementioned problem, we propose two new self-adaptive relaxed CQ algorithms which involve computing of projections onto half-spaces instead of computing onto the closed convex sets, and it does not require calculating the operator norm. We establish a weak and a strong convergence theorems for the proposed algorithms. We apply the new results to solve some other problems. Finally, we present some numerical examples to show the efficiency and accuracy of our algorithm compared to some existing results. Our results extend and improve some existing methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized discrete Lotka-Volterra equation and explore its connections with symmetric orthogonal polynomials, Hankel determinants and convergence acceleration algorithms. Firstly, we extend the fully discrete Lotka-Volterra equation to a generalized one with a sequence of given constants { u 0 (n) } and derive its solution in terms of Hankel determinants. Then, it is shown that the discrete equation of motion is transformed into a discrete Riccati system for a discrete Stieltjes function, hence leading to a complete linearization. Besides, we obtain its Lax pair in terms of symmetric orthogonal polynomials by generalizing the Christoffel transformation for the symmetric orthogonal polynomials. Moreover, a generalization of the famous Wynn's ε-algorithm is also derived via a Miura transformation to the generalized discrete Lotka-Volterra equation. Finally, the numerical effects of this generalized ε-algorithm are discussed by applying to some linearly, logarithmically convergent sequences and some divergent series. [ABSTRACT FROM AUTHOR]

Theory of quasi-orthogonal polynomials is significantly related to constructing vector Padé approximations. The present paper introduces an efficient procedure to compute adjacent families of quasi-orthogonal polynomials as defined in Sadaka (Appl. Numer. Math. 21:57–70, 1996) and Sadaka (Appl. Numer. Math. 24:483–499, 1997). The derived algorithm uses short recursive relations whose coefficients are written in terms of lower triangular block determinants. The strategy of computing such determinants is given and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, based on the Gauss transformation of a quaternion matrix, we study the full rank decomposition of a quaternion matrix, and obtain a direct algorithm and complex structure-preserving algorithm for full rank decomposition of a quaternion matrix. In addition, we expand the application of the above two full rank decomposition algorithms and give a fast algorithm to calculate the quaternion linear equations. The numerical examples show that the complex structure-preserving algorithm is more efficient. Finally, we apply the structure-preserving algorithm of the full rank decomposition to the sparse representation classification of color images, and the classification effect is well. [ABSTRACT FROM AUTHOR]

In this paper, we consider the computational problem of the quaternion equality constrained least squares problem. First, applying the quaternion QR decomposition and an equivalence problem of the quaternion equality constrained least squares problem, we obtain the expressions of the general solution and the minimal norm solution of the quaternion equality constrained least squares problem. And then by using the real representation matrices of quaternion matrices, the special structure of the real representation matrices and the real structure-preserving algorithm of the quaternion QR decomposition, we purpose a real structure-preserving algorithm for the minimal norm solution of the quaternion equality constrained least squares problem. The purposed algorithm only involves real algebraic operations, and the number of real floating-point operations and assignments has been minimized. Finally, we give two examples, which show that our purposed algorithm is more efficient and time-saving than direct computation in the quaternion Toolbox for Matlab. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new column selection strategy, named here "Deviation Maximization", and apply it to compute rank-revealing QR factorizations as an alternative to the well-known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK's xgeqp3 routine. We show that the resulting algorithm, named QRDM, has similar rank-revealing properties of QP3 and better execution times. We present experimental results on a wide data set of numerically singular matrices, which has become a reference in the recent literature. [ABSTRACT FROM AUTHOR]

This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular M -tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative componentwise nonincreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method. [ABSTRACT FROM AUTHOR]

This paper proposes an arc-search interior-point algorithm for convex quadratic programming with box constraints. The problem has many applications, such as optimal control with actuator saturation. It is shown that an explicit feasible starting point exists for this problem. Therefore, an efficient feasible interior-point algorithm is proposed to tackle the problem. It is proved that the proposed algorithm is polynomial and has the best known complexity bound O (n log (1 / 휖)) . Moreover, the search neighborhood for this special problem is wider than an algorithm for general convex quadratic programming problems, which implies that longer steps and faster convergence are expected. Finally, an engineering design problem is considered and the algorithm is applied to solve the engineering problem. [ABSTRACT FROM AUTHOR]

At present, many conjugate gradient methods with global convergence have been proposed in unconstrained optimization, such as MPRP algorithm proposed by Zhang et al. (IMA J. Numer. Anal.26(4):629–640, 2006). Unfortunately, almost all of these methods require gradient Lipschitz continuity condition. As far as we know, how do the current conjugate gradient methods deal with gradient non-Lipschitz continuity problems is basically blank. For gradient non-Lipschitz continuity problems, Algorithm 1 and Algorithm 2 are proposed in this paper based on MPRP algorithm. The proposed algorithms have the following characteristics: (i) Algorithm 1 retains sufficient descent property independent of line search technology in MPRP algorithm; (ii) for nonconvex and gradient non-Lipschitz continuous functions, the global convergence of Algorithm 1 is obtained in combination with the trust region property and the weak Wolfe-Powell line search technique; (iii) based on Algorithm 1, Algorithm 2 is further improved which global convergence can be obtained independently of line search technique; (iv) according to numerical experiments, the proposed algorithms perform competitively with other similar algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such a novel strategy is based on the newly developed exponential scalar auxiliary variable (ESAV) approach that can remove the bounded-from-below restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products. So it is more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize symplectic Runge-Kutta methods for both solution variables and the auxiliary variable, where the values of the internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms have constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive modified weak Galerkin (AMWG) method is considered to solve second-order elliptic problem. Under the assumption of a penalty parameter, by showing reliability of error estimator, comparison of solutions and reduction of error estimator, the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops, is proved to be a contraction, namely, the adaptive algorithm is convergent. Numerical experiments are implemented to support the theoretical results. [ABSTRACT FROM AUTHOR]

We propose a new numerical method for semi-infinite optimization problems whose objective function is a nonsmooth function. Existing numerical methods for solving semi-infinite programming (SIP) problems make strong assumptions on the structure of the objective function, e.g., differentiability, or are not guaranteed to furnish a feasible point on finite termination. In this paper, we propose a feasible proximal bundle method with convexification for solving this class of problems. The main idea is to derive upper bounding problems for the SIP by replacing the infinite number of constraints with a finite number of convex relaxation constraints, introduce improvement functions for the upper bounding problems, construct cutting plane models of the improvement functions, and reformulate the cutting plane models as quadratic programming problems and solve them. The convex relaxation constraints are constructed with ideas from the α BB method of global optimization. Under mild conditions, we showed that every accumulation point of the iterative sequence is an 𝜖-stationary point of the original SIP problem. Under slightly stronger assumptions, every accumulation point of the iterative sequence is a local solution of the original SIP problem. Preliminary computational results on solving nonconvex, nonsmooth constrained optimization problems and semi-infinite optimization problems are reported to demonstrate the good performance of the new algorithms. [ABSTRACT FROM AUTHOR]

Recently, a number of efficient iteration methods for the solution of a special class of block 3 × 3 saddle point systems have been proposed by some authors. In order to easily evaluate the strong stability of these numerical algorithms and provide a practical and reliable termination criterion, in this paper, we perform the structured backward error analysis for this type of block 3 × 3 saddle point system and present an explicit and computable formula of the normwise structured backward error. Some numerical experiments are performed to demonstrate that our results can be used to easily test the stability of running algorithms, and the new stopping criterion based on the derived structured backward error is more suitable and efficient than the commonly used residue-based one. [ABSTRACT FROM AUTHOR]

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders. [ABSTRACT FROM AUTHOR]

It is well known that the monotonicity of the underlying mapping of variational inequalities plays a central role in the convergence analysis. In this paper, we propose an infeasible projection algorithm (IPA for short) for nonmonotone variational inequalities. The next iteration point of IPA is generated by projecting a vector onto a half-space. Hence, the computational cost of computing the next iteration point of IPA is much less than the algorithm of Ye and He (Comput. Optim. Appl. 60, 141–150, 2015) (YH for short). Moreover, if the underlying mapping is Lipschitz continuous with its modulus is known, by taking suitable parameters, IPA requires only one projection onto the feasible set per iteration. The global convergence of IPA is obtained when the solution set of its dual variational inequalities is nonempty. Moreover, if in addition error bound holds, the convergence rate of IPA is Q-linear. IPA can be used for a class of quasimonotone variational inequality problems and a class of quasiconvex minimization problems. Comparing with YH and Algorithm 2 in Deng, Hu and Fang (Numer. Algor. 86, 191–221, 2021) (DHF for short) by solving high-dimensional nonmonotone variational inequalities, numerical experiments show that IPA is much more efficient than YH and DHF from CPU time point of view. Moreover, IPA is less dependent on the initial value than YH and DHF. [ABSTRACT FROM AUTHOR]

In our previous paper (Camargo, Numer. Algor., 85:591–606, 2020), we proved that the algorithms in a certain class of divided differences schemes are backward stable and, in particular, we proved that Neville's algorithm for Lagrange interpolation is backward stable for extrapolation for monotonically ordered nodes. That proof was based on a very particular pattern of the signs of the components of the divided differences which, in the case of Neville's algorithm for monotonically ordered nodes, is not satisfied when interpolation is considered instead of extrapolation. In this note we present a different argument that shows that Neville's algorithm is backward stable on the whole real line for monotonically ordered nodes. Our reasoning is based on a pyramid algorithm for the computation of Lebesgue functions. We also explain that obtaining sharp upper bounds for the numerical error in the computation of Neville's algorithm for generic sets of nodes is difficult. [ABSTRACT FROM AUTHOR]

In this paper, a proximal bundle-based method for solving nonsmooth nonconvex constrained multiobjective optimization problems with inexact information is proposed and analyzed. In this method, each objective function is treated individually without employing any scalarization. Using the improvement function, we transform the problem into an unconstrained one. At each iteration, by the proximal bundle method, a piecewise linear model is built and by solving a convex subproblem, a new candidate iterate is obtained. For locally Lipschitz objective and constraint functions, we study the problem of computing an approximate substationary point (a substationary point), when only inexact (exact) information about the functions and subgradient values are accessible. At the end, some numerical experiments are provided to illustrate the effectiveness of the method and certify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an explicit subgradient extragradient algorithm for solving equilibrium problem with a bifunction satisfying pseudomonotone and Lipschitz-like condition in a 2-uniformly convex and uniformly smooth Banach space. We also defined a new self-adaptive stepsize rule and prove a convergence result for solving the equilibrium problem without any prior estimate of the Lipschitz-like constants of the bifunction. Furthermore, we provide some numerical examples to illustrate the efficiency and accuracy of the proposed algorithm. This result improves and extends many recent results in this direction in the literature. [ABSTRACT FROM AUTHOR]

It is well-known that conjugate gradient algorithms are widely applied in many practical fields, for instance, engineering problems and finance models, as they are straightforward and characterized by a simple structure and low storage. However, challenging problems remain, such as the convergence of the PRP algorithms for nonconvexity under an inexact line search, obtaining a sufficient descent for all conjugate gradient methods, and other theory properties regarding global convergence and the trust region feature for nonconvex functions. This paper studies family conjugate gradient formulas based on the six classic formulas, PRP, HS, CD, FR, LS, and DY, where the family conjugate gradient algorithms have better theory properties than those of the formulas by themselves. Furthermore, this technique of the presented conjugate gradient formulas can be extended to any two-term conjugate gradient formula. This paper designs family conjugate gradient algorithms for nonconvex functions, which have the following features without other conditions: (i) the sufficient descent property holds, (ii) the trust region feature is true, and (iii) the global convergence holds under normal assumptions. Numerical results show that the given conjugate gradient algorithms are competitive with those of normal methods. [ABSTRACT FROM AUTHOR]

In this paper, we study an inverse source problem for the Helmholtz equation from measurements. The purpose of this paper is to reconstruct the salient features of the hidden sources within a body. We propose three stable reconstruction algorithms to detect the number, the location, the size, and the shape of the hidden sources along with compact support from a single measurement of near-field Cauchy data on the external boundary. This problem is nonlinear and ill-posed; thus, we should consider regularization techniques in reconstruction algorithms. We give several numerical experiments to demonstrate the viability of our proposed reconstruction algorithms. [ABSTRACT FROM AUTHOR]