Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method

In this paper, a Kansa’s method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa’s method is a meshfree method which applying the combination of some radial basis functions (such as Hardy’s MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.     

Approximate solutions of partial differential equations by some Meshfree Greedy Algorithms

In this article, we use some greedy algorithms to avoid the ill‐conditioning of the final linear system in unsymmetric Kansa collocation method. The greedy schemes have the same background, but we use them in different settings. In the first algorithm, the optimal trial points for interpolation obtained among a huge set of initial points are used for numerical solution of partial differential equations (PDEs). In the second algorithm, based on the Kansa's method, the PDE is discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. The third greedy algorithm is used to generate the test points. This paper shows that the greedily selection of nodes yields a better conditioning in contrast with usual full meshless methods. Some well‐known PDE examples are solved and compared with the full unsymmetric Kansa's technique. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1884–1899, 2017     

A modified adaptive Lasso for identifying interactions in the Cox model with the heredity constraint

In many biomedical studies, identifying effects of covariate interactions on survival is a major goal. Important examples are treatment–subgroup interactions in clinical trials, and gene–gene or gene–environment interactions in genomic studies. A common problem when implementing a variable selection algorithm in such settings is the requirement that the model must satisfy the strong heredity constraint, wherein an interaction may be included in the model only if the interaction’s component variables are included as main effects. We propose a modified Lasso method for the Cox regression model that adaptively selects important single covariates and pairwise interactions while enforcing the strong heredity constraint. The proposed method is based on a modified log partial likelihood including two adaptively weighted penalties, one for main effects and one for interactions. A two-dimensional tuning parameter for the penalties is determined by generalized cross-validation. Asymptotic properties are established, including consistency and rate of convergence, and it is shown that the proposed selection procedure has oracle properties, given proper choice of regularization parameters. Simulations illustrate that the proposed method performs reliably across a range of different scenarios.     

An improved working set selection for SMO-type decomposition method

The sequential minimization optimization (SMO) is a simple and efficient decomposition algorithm for solving support vector machines (SVMs). In this paper, an improved working set selection and a simplified minimization step are proposed for the SMO-type decomposition method that reduces the learning time for SVM and increases the efficiency of SMO. Since the working set is selected directly according to the Karush–Kuhn–Tucker (KKT) conditions, the minimization step of subproblem is simplified, accordingly the learning time for SVM is reduced and the convergence is accelerated. Following Keerthi’s method, the convergence of the proposed algorithm is analyzed. It is proven that within a finite number of iterations, solution that is based on satisfaction of the KKT conditions will be obtained by using the improved algorithm.     

Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations

For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton’s method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton’s method. In those cases, Newton’s method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton’s method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton–Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton–Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.     

Strong convergence properties of a modified nonmonotone smoothing algorithm for the SCCP

The smoothing algorithms have been successfully applied to solve the symmetric cone complementarity problem (denoted by SCCP), which in general have the global and local superlinear/quadratic convergence if the solution set of the SCCP is nonempty and bounded. Huang, Hu and Han [Science in China Series A: Mathematics, 52: 833–848, 2009] presented a nonmonotone smoothing algorithm for solving the SCCP, whose global convergence is established by just requiring that the solution set of the SCCP is nonempty. In this paper, we propose a new nonmonotone smoothing algorithm for solving the SCCP by modifying the version of Huang-Hu-Han’s algorithm. We prove that the modified nonmonotone smoothing algorithm not only is globally convergent but also has local superlinear/quadratical convergence if the solution set of the SCCP is nonempty. This convergence result is stronger than those obtained by most smoothing-type algorithms. Finally, some numerical results are reported.     

Convergence analysis of the corrected Uzawa algorithmfor symmetric saddle point problems简

For the large sparse saddle point problems, Pan and Li recently proposed in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31（3）： 231-242] a corrected Uzawa algorithm based on a nonlinear Uzawa algorithm with two nonlinear approximate inverses, and gave the detailed convergence analysis. In this paper, we focus on the convergence analysis of this corrected Uzawa algorithm, some inaccuracies in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31（3）： 231-242] are pointed out, and a corrected convergence theorem is presented. A special case of this modified Uzawa algorithm is also discussed.     

A Dwindling Filter Algorithm with a Modified Subproblem for Nonlinear Inequality Constrained Optimization

The authors propose a dwindling filter algorithm with Zhou＇s modified subprob- lem for nonlinear inequality constrained optimization. The feasibility restoration phase, which is always used in the traditional filter method, is not needed. Under mild conditions, global convergence and local superlinear convergence rates are obtained. Numerical results demonstrate that the new algorithm is effective.     

Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

In this paper we present the relaxed inertial proximal algorithm for Ky Fan minimax inequalities. Based on Opial lemma, we propose a weak convergence result to a solution of the problem by eliminating in the algorithm (RIPAFAN) the Browder–Halpern’s factor of contraction. We present after, a first result of strong convergence by adding a strong monotonicity condition. Secondly, we eliminate the strong monotonicity and add a Browder–Halpern’s contraction factor in the algorithm (RIPAFAN) and then ensure the strong convergence to a selected solution with respect to the contraction factor. Some examples are proposed. The first one concerns the convex minimization where the objective function is only controlled with a provided well conditioning. In the second one, we propose monotone set-valued variational inequalities. The last example deals with the problem of fixed point for a nonexpansive set-valued operator.     

Automatic Response Category Combination in Multinomial Logistic Regression

We propose a penalized likelihood method that simultaneously fits the multinomial logistic regression model and combines subsets of the response categories. The penalty is nondifferentiable when pairs of columns in the optimization variable are equal. This encourages pairwise equality of these columns in the estimator, which corresponds to response category combination. We use an alternating direction method of multipliers algorithm to compute the estimator and we discuss the algorithm’s convergence. Prediction and model selection are also addressed. Supplemental materials for this article are available online.     

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection–diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c* ) and weight parameter (ϵ) that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black–Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L_∞, L₂, L_rms , and L_rel error norms as well as number of nodes N over space domain and time-step δt. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill-conditioning problem greatly, a major issue in the Kansa method.     

Variable Selection Via Subtle Uprooting

This article proposes a variable selection method termed “subtle uprooting” for linear regression. In this proposal, variable selection is formulated into a single optimization problem by approximating cardinality involved in the information criterion with a smooth function. A technical maneuver is then employed to enforce sparsity of parameter estimates while maintaining smoothness of the objective function. To solve the resulting smooth nonconvex optimization problem, a modified Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm with established global and super-linear convergence is adopted. Both simulated experiments and an empirical example are provided for assessment and illustration. Supplementary materials for this article are available online.     

Modification of Karmarkar’s projective scaling algorithm

This paper presents a new conversion technique of the standard linear programming problem into a homogenous form desired for the Karmarkar’s algorithm, where we employed the primal–dual method. The new converted linear programming problem provides initial basic feasible solution, simplex structure, and homogenous matrix. Apart from the transformation, Hooker’s method of projected direction is employed in the Karmarkar’s algorithm and the modified algorithm is presented. The modified algorithm has a faster convergence with a suitable choice of step size.     

A class of nonlinear Lagrangians for nonconvex second order cone programming

This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.     

The generalized residual cutting method and its convergence characteristics

Iterative methods and especially Krylov subspace methods (KSM) are a very useful numerical tool in solving for large and sparse linear systems problems arising in science and engineering modeling. More recently, the nested loop KSM have been proposed that improve the convergence of the traditional KSM. In this article, we review the residual cutting (RC) and the generalized residual cutting (GRC) that are nested loop methods for large and sparse linear systems problems. We also show that GRC is a KSM that is equivalent to Orthomin with a variable preconditioning. We use the modified Gram–Schmidt method to derive a stable GRC algorithm. We show that GRC presents a general framework for constructing a class of “hybrid” (nested) KSM based on inner loop method selection. We conduct numerical experiments using nonsymmetric indefinite matrices from a widely used library of sparse matrices that validate the efficiency and the robustness of the proposed methods.     

The general analytical and numerical solution for the modified KdV equation with convergence analysis

We investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge–Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge–Kutta discontinuous Galerkin method by some numerical illustrations. Copyright © 2012 John Wiley & Sons, Ltd.     

Iterative solutions of the split common fixed point problem for strictly pseudo-contractive mappings

In this paper, we study the the split common fixed point problem in Hilbert spaces. We establish a weak convergence theorem for the method recently introduced by Wang, which extends a existing result from firmly nonexpansive mappings to strictly pseudo-contractive mappings. Moreover, our condition that guarantees the weak convergence is much weaker than that of Wang’s. A strong convergence theorem is also obtained under some additional conditions. As an application, we obtain several new methods for solving various split inverse problems and split equality problems. Numerical examples are included to illustrate the applications in signal processing of the proposed algorithm.     

An Alternating Direction Method of Multipliers for MCP-penalized Regression with High-dimensional Data

The minimax concave penalty (MCP) has been demonstrated theoretically and practically to be effective in nonconvex penalization for variable selection and parameter estimation. In this paper, we develop an efficient alternating direction method of multipliers (ADMM) with continuation algorithm for solving the MCP-penalized least squares problem in high dimensions. Under some mild conditions, we study the convergence properties and the Karush–Kuhn–Tucker (KKT) optimality conditions of the proposed method. A high-dimensional BIC is developed to select the optimal tuning parameters. Simulations and a real data example are presented to illustrate the efficiency and accuracy of the proposed method.     

Convergence of a parameter switching algorithm for a class of nonlinear continuous systems and a generalization of Parrondo’s paradox

In this paper, we prove the convergence of a numerical algorithm that switches in some deterministic or random manner, the control parameter of a class of continuous-time nonlinear systems while the underlying initial value problem is numerically integrated. The numerically obtained attractor is a good approximation of the attractor obtained when the control parameter is replaced with the average of the switched values. In this way, a generalization of Parrondo’s paradox can be obtained. As an application, the Lorenz and Rabinovich–Fabrikant systems are used for illustration.     

RBFs approximation method for time fractional partial differential equations

In this paper, radial basis functions (RBFs) approximation method is implemented for time fractional advection–diffusion equation on a bounded domain. In this method the first order time derivative is replaced by the Caputo fractional derivative of order α ∈ (0, 1], and spatial derivatives are approximated by the derivative of interpolation in the Kansa method. Stability and convergence of the method is discussed. Several numerical examples are include to demonstrate effectiveness and accuracy of the method.