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A numerical scheme is presented to solve the one source near field refractor problem to arbitrary precision and it is proved that the scheme terminates in a finite number of iterations. The convergence of the algorithm depends upon proving appropriate Lipschitz estimates for the refractor measure. The algorithm is presented in general terms and has independent interest., Comment: 23 pages, 1 figure, a reference was corrected

We propose a variational principle combining a phase-field functional for structural topology optimization with a mixed (three-field) Hu-Washizu functional, then including directly in the formulation equilibrium, constitutive, and compatibility equations. The resulting mixed variational functional is then specialized to derive a classical topology optimization formulation (where the amount of material to be distributed is an \emph{a priori} assigned quantity acting as a global constraint for the problem) as well as a novel topology optimization formulation (where the amount of material to be distributed is minimized, hence with no pre imposed constraint for the problem). Both formulations are numerically solved by implementing a mixed finite element scheme, with the second approach avoiding the introduction of a global constraint, hence respecting the convenient local nature of the finite element discretization. Furthermore, within the proposed approach it is possible to obtain guidelines for settings proper values of phase-field-related simulation parameters and, thanks to the combined phase-field and Hu-Washizu rationale, a monolithic algorithm solution scheme can be easily adopted. An insightful and extensive numerical investigation results in a detailed convergence study and a discussion on the obtained final designs. The numerical results clearly highlight differences between the two formulations as well as advantages related to the monolithic solution strategy; numerical investigations address both two-dimensional and three-dimensional applications., Comment: 35 pages, 14 figures

In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and $$\theta -$$ θ - weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), $$L_2$$ L 2 and $$L_\infty $$ L ∞ error norms. The obtained results are then compared with those available in the literature.

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as $$1/\epsilon $$ 1 / ϵ , and we prove the convergence in the fast-reaction limit $$\epsilon \rightarrow 0$$ ϵ → 0 . We establish a $$\Gamma $$ Γ -convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the $$\Gamma $$ Γ -convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

We are concerned with finite linear constraint systems in a parametric framework where the right-hand side is an affine function of the perturbation parameter. Such structured perturbations provide a unified framework for different parametric models in the literature, as block, directional and/or partial perturbations of both inequalities and equalities. We extend some recent results about calmness of the feasible set mapping and provide an application to the convergence of a certain path-following algorithmic scheme. We underline the fact that our formula for the calmness modulus depends only on the nominal data, which makes it computable in practice.

A Monte Carlo technique for the solution of the Wigner transport equation has been developed, based on the generation and annihilation of signed particles (Nedjalkov et al. in Phys Rev B 70:115319, 2004). A stochastic algorithm without time discretization error has been recently introduced (Muscato and Wagner in Kinet Relat Models 12(1):59–77, 2019). Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in the case of a GaAs resonant tunneling diode. Convergence of the time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.

The surface area of a set which is only observed as a binary pixel image is often estimated by a weighted sum of pixel configurations counts. In this paper we examine these estimators in a design based setting—we assume that the observed set is shifted uniformly randomly. Bounds for the difference between the essential supremum and the essential infimum of such an estimator are derived, which imply that the variance is in $$O(t^2)$$ O ( t 2 ) as the lattice distance t tends to zero. In particular, it is asymptotically neglectable compared to the bias. A simulation study shows that the theoretically derived convergence order is optimal in general, but further improvements are possible in special cases.

International audience; In this paper we present a global-in-time non-overlapping Schwarz method applied to the Ekman boundary layer problem. Such a coupled problem is representative of large-scale atmospheric and oceanic flows in the vicinity of the air-sea interface. Schwarz waveform relaxation (SWR) algorithms provide attractive methods for ensuring a "tight coupling" between the ocean and the atmosphere. However the convergence study of such algorithms in this context raises a number of challenges. Numerous convergence studies of Schwarz methods have been carried out in idealized settings, but the underlying assumptions to make these studies tractable may prohibit them to be directly extended to the complexity of climate models. We illustrate this aspect on the coupled Ekman problem, which includes several essential features inherent to climate modeling while being simple enough for analytical results to be derived. We investigate its well-posedness and derive an appropriate SWR algorithm. Sufficient conditions for ensuring its convergence for different viscosity profiles are then established. Finally, we illustrate the relevance of our theoretical analysis with numerical results and suggest ways to improve the computational cost of the coupling. Our study emphasizes the fact that the convergence properties can be highly sensitive to some model characteristics such as the geometry of the problem and the use of continuously variable viscosity coefficients.

The paper is devoted to Krylov type modifications of the Uzawa method on the operator level for the Stokes problem in order to accelerate convergence. First block preconditioners and their effect on convergence are studied. Then it is shown that a Krylov–Uzawa iteration produces superlinear convergence on smooth domains, and estimation is given on its speed.

The study of convergence of time integrators, applied to linear discretized PDEs, relies on the power boundedness of the stability matrix R. The present work investigates power boundedness in the maximum norm for ADI-type integrators in arbitrary space dimension m. Examples are the Douglas scheme, the Craig–Sneyd scheme, and W-methods with a low stage number. It is shown that for some important integrators $$\Vert R^n\Vert _\infty $$ is bounded in the maximum norm by a constant times $$\min \bigl ( (\ln (1+n))^m, (\ln N)^m \bigr )$$ , where m is the space dimension of the PDE, and $$N\ge 2$$ is the space discretization parameter. For $$m\le 2$$ sharper bounds are obtained that are independent of n and N.

The worst-case linear optimization (WCLO) with uncertainties in the right-hand-side of the constraints often arises from numerous applications such as systemic risk estimate in finance and stochastic optimization, which is known to be NP-hard. In this paper, we investigate the efficient global algorithm for WCLO based on its nonlinear semidefinite relaxation (SDR). We first derive an enhanced nonlinear SDR for WCLO via secant cuts and RLT approaches. A secant search algorithm is then proposed to solve the nonlinear SDR and its global convergence is established. Second, we propose a new global algorithm for WCLO, which integrates the nonlinear SDR with successive convex optimization method, initialization and branch-and-bound, to find a globally optimal solution to the underlying WCLO within a pre-specified $$\epsilon$$ -tolerance. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to the WCLO instances.

This paper proposes and analyzes a globalized version of the Newton method for finding a singularity of the nonsmooth vector fields. Basically, the new method combines a version of nonsmooth Newton method with a nonmonotone line search strategy. The global convergence analysis of the proposed method as well as results on its rate are established under mild assumptions. Finally, numerical experiments illustrating the practical advantages of the proposed scheme are reported.

We suggest the local analysis of a class of iterative schemes based on decomposition technique for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the fourth derivative to establish convergence. But we only apply the first derivative in our convergence theorem. We also provide computable radius of convergence ball, error estimates and uniqueness of the solution results not studied in earlier works. Hence, we enhance the applicability of these schemes. Furthermore, we explore, using basin of attraction tool, the dynamics of the schemes when they are applied on various complex polynomials. This article is concluded with numerical experiments.

In this paper, we study the global convergence of successive approximations as well as the uniqueness of the random solution of a coupled random Hilfer fractional differential system. We prove a theorem on the global convergence of successive approximations to the unique solution of our problem. In the last section, we present an illustrative example.

The paper presents two inertial viscosity-type extragradient algorithms for finding a common solution of the variational inequality problem involving a monotone and Lipschitz continuous operator and of the fixed point problem with a demicontractive mapping in real Hilbert spaces. Our algorithms use a simple step size rule which is generated by some calculations at each iteration. Two strong convergence theorems are obtained without the prior knowledge of the Lipschitz constant of the operator. The numerical behaviors of the proposed algorithms in some numerical experiments are reported and compared with previously known ones.

Sparse multinomial logistic regression has recently received widespread attention. It provides a useful tool for solving multi-classification problems in various fields, such as signal and image processing, machine learning and disease diagnosis. In this paper, we first study the group sparse multinomial logistic regression model and establish its optimality conditions. Based on the theoretical results of this model, we hence propose an efficient algorithm called the subspace quadratic regularization algorithm to compute a stationary point of a given problem. This algorithm enjoys excellent convergence properties, including the global convergence and locally quadratic convergence. Finally, our numerical results on standard benchmark data clearly demonstrate the superior performance of our proposed algorithm in terms of logistic loss value, sparsity recovery and computational time.

This article gives a numerical solution for solving the two-dimensional nonlinear Fredholm–Volterra partial integro-differential equations with boundary conditions with weakly singular kernels. The collocation method has been used for these operational matrices of the Taylor wavelet along with the Newton method to reduce the given partial integro-differential equation to the system of algebraic equations. Error analysis is considered to indicate the convergence of the approximation used in this method. Attaining this purpose, first, two-dimensional Taylor wavelet and then operational matrices should be defined. Regarding the characteristics of the Taylor wavelet, we were obtaining high accuracy of the method. Finally, examples are provided to demonstrate that the proposed method is effective.

This paper is concerned with the nonlocal dispersal problem in inhomogeneous media. Our goal is to show the limiting behavior of perturbation equation with parameters. By analyzing the asymptotic behavior of solutions when the parameter is small, we find that convection appears in inhomogeneous media. Moreover, if the effect of inhomogeneous media changes, then we prove a convergence result that convection disappears in nonlocal dispersal problems.

The fractional differential equation has been used to describe many phenomenons in almost all applied sciences, such as fluid flow in porous materials, anomalous diffusion transport, acoustic wave propagation in viscoelastic materials, and others. In some cases, it is complicated to find the analytical solution of a fractional differential equation, so there are some special techniques to approximate the solution. In this paper, a new collocation method for a class of semilinear fractional differential equations has been constructed. The convergence properties of the proposed methods and the computational complexity are treated. Some numerical experiments are presented to validate the theoretical results. The proposed numerical method provides an alternative way to study some physical phenomena by fractional differential equations.

We explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.

We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra’s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas–Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas–Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas–Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants.

Ant colony optimization has insufficient convergence and tends to fall into the local optima when solving the traveling salesman problem. This paper proposes a co-evolutionary multi-colony ant colony optimization (MCGACO) to overcome this deficiency and applies it to the Robot Path Planning. First, a dynamic grouping cooperation algorithm, combined with Ant Colony System and Max-Min Ant System, is introduced to form a heterogeneous multi-population structure. Each population co-evolves and complements each other to improve the overall optimization performance. Second, an adaptive guidance mechanism is proposed to accelerate convergence speed. The mechanism includes two parts: One is a dynamic evaluation network, which is used to evaluate and divide all solutions by the evaluation function. The other is a positive-negative incentive strategy, which can enhance the guiding role of solutions with higher evaluation value. Besides, to jump out of the local optima, an inter-specific co-evolution mechanism based on the game model is proposed. By dynamically determining the optimal communication combination, the diversity among populations can be well balanced. Finally, the experimental results demonstrate that MCGACO outperforms in terms of solution accuracy and convergence. Meanwhile, the proposed algorithm is also feasible for application in Robot Path Planning.

Motivated by the modulus-based Newton method for solving linear complementarity problems (Zheng and Li, J. Comput. Appl. Math. 288, 116–126 2015; Wu and Li, Calcolo 54, 43–56 2017), we propose a generalized modulus-based Newton method to solve a class of non-linear complementarity problems with P-matrices. A sufficient condition to guarantee the convergence of the proposed method is obtained. Numerical experiments further demonstrate that the proposed method is efficient and has better performance than the existing modulus-based iteration method and the Newton method.

This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient method and prove that these methods satisfy the sufficient descent condition on Riemannian manifolds. One is a hybrid method combining a Fletcher–Reeves-type method with a Polak–Ribiere–Polyak-type method, and the other is a Hager–Zhang-type method, both of which are generalizations of those used in Euclidean space. Moreover, we prove that the hybrid method has a global convergence property under the strong Wolfe conditions and the Hager–Zhang-type method has the sufficient descent property regardless of whether a line search is used or not. Further, we review two kinds of line search algorithm on Riemannian manifolds and numerically compare our generalized methods by solving several Riemannian optimization problems. The results show that the performance of the proposed hybrid methods greatly depends on the type of line search used. Meanwhile, the Hager–Zhang-type method has the fast convergence property regardless of the type of line search used.

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph (SRG). The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence., Published in Mathematical Programming

In this paper, we propose a new linesearch iterative scheme for finding a common solution of split equilibrium and fixed point problems without pseudomonotonicity of the bifunction f in a real Hilbert space. When setting the solution of dual equilibrium problem is nonempty, we obtain a strong convergence theorem which is generated by the iterative scheme. Moreover, we also receive a new linesearch iterative scheme for finding a solution of the split equilibrium problem in suitable assumptions, and report some numerical results to illustrate the convergence of the proposed scheme.

International audience; This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on integral representation formulas. For the parallel and sequential forms of the method appearing in the literature, we propose a general abstract formulation in a given Hilbert setting and interpret the procedure in terms of subspace corrections. We then prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. An alternative proof of convergence is provided in a case which does not fit into the previous framework. We finally present some numerical tests.

In this paper, we propose a projected subgradient method for solving constrained nondifferentiable quasiconvex multiobjective optimization problems. The algorithm is based on the Plastria subdifferential to overcome potential shortcomings known from algorithms based on the classical gradient. Under suitable, yet rather general assumptions, we establish the convergence of the full sequence generated by the algorithm to a Pareto efficient solution of the problem. Numerical results are presented to illustrate our findings.

DC (Difference of Convex functions) programming and DCA (DC Algorithm) play a key role in nonconvex programming framework. These tools have a rich and successful history of thirty five years of development, and the research in recent years is being increasingly explored to new trends in the development of DCA: design novel DCA variants to improve standard DCA, to deal with the scalability and with broader classes than DC programs. Following these trends, we address in this paper the two wide classes of nonconvex problems, called partial DC programs and generalized partial DC programs, and investigate an alternating approach based on DCA for them. A partial DC program in two variables $$(x,y)\in \mathbb {R}^{n}\times {\mathbb {R}}^{m}$$ takes the form of a standard DC program in each variable while fixing other variable. A so-named alternating DCA and its inexact/generalized versions are developed. The convergence properties of these algorithms are established: both exact and inexact alternating DCA converge to a weak critical point of the considered problem, in particular, when the Kurdyka–Łojasiewicz inequality property is satisfied, the algorithms furnish a Frechet/Clarke critical point. The proposed algorithms are implemented on the problem of finding an intersection point of two nonconvex sets. Numerical experiments are performed on an important application that is robust principal component analysis. Numerical results show the efficiency and the superiority of the alternating DCA comparing with the standard DCA as well as a well known alternating projection algorithm.

This paper describes the formulation and experimental testing of an estimation of submanifold models of animal motion. It is assumed that the animal motion is supported on a configuration manifold, Q, and that the manifold is homeomorphic to a known smooth, Riemannian manifold, S. Estimation of the configuration submanifold is achieved by finding an unknown mapping, $$\gamma $$ , from S to Q. The overall problem is cast as a distribution-free learning problem over the manifold of measurements. This paper defines sufficient conditions that show that the rates of convergence in $$L^2_\mu (S)$$ of approximations of $$\gamma $$ correspond to those known for classical distribution-free learning theory over Euclidean space. This paper concludes with a study and discussion of the performance of the proposed method using samples from recent reptile motion studies.

We consider a non-convex quadratic program (QP) with linear and convex quadratic constraints that arises from a broad range of applications and is known to be NP-hard. In this paper, we first prove that the alternative direction method converges to a local solution of the underlying QP problem. We then propose a new branch-and-cut algorithm that finds a globally optimal solution to the underlying QP problem within a pre-specified 𝜖-tolerance by integrating the alternative direction method with semidefinite relaxation and disjunctive cut techniques. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to medium-scale QP instances in which the number of negative eigenvalues of the Hessian matrix in the objective function is less than or equals 20.

Using iterative regularizations, we introduce a new proximal-like algorithm for solving split variational inclusion problems in Hilbert space and establish a strong convergence theorem for it. As applications, we also investigate the split feasibility and split optimization problems. Several numerical experiments are presented in support of our theoretical findings.

We propose two algorithms for solving global optimization problems on a hyperrectangle with an objective function satisfying the Vanderbei condition (this function is also called an $\varepsilon$-Lipschitz continuous function). The algorithms belong to the class of non-uniform cover-ings methods. For the algorithms we prove propositions about convergence to an $\varepsilon$-solution in terms of the objective function. We illustrate the performance of the algorithms using several test numerical examples with non-Lipschitz continuous objective functions., 21 pages, 4 tables, 29 figure, in Russian. (v4) corrected the third numerical example

In this paper, we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems (cMOP). Specifically, we reviewed two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex matrix optimization problems regularized by nonsmooth spectral functions. Under a mild quadratic growth condition on the dual of cMOP, we further discussed the R-superlinear convergence of the Karush–Kuhn–Tucker (KKT) residuals of the sequence generated by the augmented Lagrangian methods (ALM) for solving convex matrix optimization problems. Implementation details of the ALM for solving core convex matrix optimization problems are also provided.

In this paper, a modification of Szasz-Kantorovich type operators based on Brenke-type polynomials is introduced, and the convergence properties of the proposed operators with the help of Korovkin’s theorem are discussed. The order of convergence of these operators with the aid of classical and second-order modulus of continuity is studied. A Voronovkaja-type theorem is also established. Lastly, the rate of convergence and error estimation of these operators compared with the existing operators with the help of some graphs and tables using Mathematica.

This work is focused on developing a finite volume scheme for approximating a fragmentation equation. The mathematical analysis is discussed in detail by examining thoroughly the consistency and convergence of the numerical scheme. The idea of the proposed scheme is based on conserving the total mass and preserving the total number of particles in the system. The proposed scheme is free from the trait that the particles are concentrated at the representative of the cells. The verification of the scheme is done against the analytical solutions for several combinations of standard fragmentation kernel and selection functions. The numerical testing shows that the proposed scheme is highly accurate in predicting the number distribution function and various moments. The scheme has the tendency to capture the higher order moments even though no measure has been taken for their accuracy. It is also shown that the scheme is second-order convergent on both uniform and nonuniform grids. Experimental order of convergence is used to validate the theoretical observations of convergence.

In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search since they compute a next iterate on a straight line determined by a search direction which approximates the central path. The proposed arc-search interior-point algorithm uses an arc for the approximation. We discuss convergence properties of the proposed algorithm. We also conduct numerical experiments on the CUTEst benchmark problems and compare the performance of the proposed arc-search algorithm with that of a line-search algorithm. Numerical results indicate that the proposed arc-search algorithm reaches the optimal solution using fewer iterations but longer times than a line-search algorithm. A modification that leads to a faster arc-search algorithm is also discussed.

In this paper, Modified Crank–Nicolson method is combined with Richardson extrapolation to solve the 1D heat equation. The method is found to be unconditionally stable, consistent and hence the convergence of the method is guaranteed. The method is also found to be second-order convergent both in space and time variables. When combined with the Richardson extrapolation, the order of the method is improved from second-to fourth-order. To validate the proposed method, two model examples are considered and solved for different values of spatial and temporal step lengths.

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.

Jordan Algebras are an important tool for dealing with semidefinite programming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate complementarity measure in this context is typically defined in terms of the eigenvalues of the constraint matrix and the eigenvalues of an approximate Lagrange multiplier. By exploiting the Jordan Algebra structure of the problem, we show that a simpler complementarity measure, defined in terms of the Jordan product, is stronger than the one defined in terms of eigenvalues. Thus, besides avoiding a tricky analysis of eigenvalues, a stronger necessary optimality condition is presented. We then prove the global convergence of an Augmented Lagrangian algorithm to this improved necessary optimality condition. The results are also extended to an interior point method. The optimality conditions we present are sequential ones, and no constraint qualification is employed; in particular, a global convergence result is available even when Lagrange multipliers are unbounded.

We present sufficient conditions for the uniform convergence of interpolation processes constructed on the basis of solutions to Cauchy problems in terms of the one-sided modulus of continuity and modulus of variation of the approximated function on a compact connected subset of the domain.

In this paper, we propose a preconditioned modified positive/negative-stable splitting (PMPNS) iteration method to solve complex symmetric indefinite linear system more efficiently. By analyzing the convergence of the PMPNS iteration method and discussing the spectral properties of the PMPNS iteration method, we construct a new preconditioner to make the eigenvalues of the coefficient matrix more aggregated, which leads to fast convergence of Krylov subspace iteration methods such as GMRES. Numerical example is given to illustrate the efficiency of the PMPNS preconditioner used in GMRES method. In particular, the GMRES method with the PMPNS preconditioner demonstrates meshsize-independent convergence behavior.

In this paper, the authors consider a class of generalized curve flow for convex curves in the plane. They show that either the maximal existence time of the flow is finite and the evolving curve collapses to a round point with the enclosed area of the evolving curve tending to zero, i.e., $$\mathop {\lim}\limits_{t \to T} A(t) = 0$$ , or the maximal time is infinite, that is, the flow is a global one. In the case that the maximal existence time of the flow is finite, they also obtain a convergence theorem for rescaled curves at the maximal time.

The authors propose an analog of the Galerkin method for the initial–boundary-value problem that describes drug delivery in the artery wall using a drug-coated stent. The method of numerical solution of the initial–boundary-value problem is constructed and the theorems on its convergence to the solution are proved.

Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.

Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss order continuity and reflexivity of Banach lattices. Some related results are obtained as well., Comment: 9 pages

This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints on input values. This paper analyzes the characteristics of optimal solutions of neural network inverse problems with rectified activation units and proposes an outer-approximation algorithm by exploiting their characteristics. The proposed outer-approximation guided optimization comprises primal and dual phases. The primal phase incorporates neighbor curvatures with neighbor outer-approximations to expedite the process. The dual phase identifies and utilizes the structure of local convex regions to improve the convergence to a local optimal solution. At last, computation experiments demonstrate the superiority of the proposed algorithm compared to a projected gradient method.

We put forward and analyze the high-order (up to fourth) strong stability-preserving implicit-explicit Runge-Kutta schemes for the time integration of the space-fractional Allen-Cahn equation, which inherits the maximum principle preserving and energy stability. The space-fractional Allen-Cahn equation with homogeneous Dirichlet boundary condition is first discretized in the spatial direction by using a second-order fractional centered difference scheme that preserves the semi-discrete maximum principle. It is subsequently integrated in the temporal direction by a class of strong stability-preserving implicit-explicit Runge-Kutta schemes that are specifically designed to preserve the maximum principle to the optimal time step size. The convergence order in the discrete $L^{\infty }$ norm and energy boundedness are provided by using the established maximum principle. Finally, a series of numerical experiments are carried out to demonstrate the high-order convergence, maximum principle preserving, and energy stability of the proposed schemes.

This work bridges two research fields i.e. metaheuristics and wind farm layout design. Comparative performance of twelve metaheuristics (MHs) on wind farm layout optimisation (WFLO) was conducted. Four WFLO problems are proposed for benchmarking the various metaheuristics while the design problem is an attempt to simultaneously minimise wind farm cost and maximise wind farm totally produced power. Design variables are wind turbine placement with fixed and varied number of wind turbines. The Jansen’s wake model is used while two types of energy estimation with and without considering partially overshadowed wake areas are studied. The results obtained from using various MHs are statistically compared in terms of convergence and consistency while the best performer is obtained. Comparison results indicated that moth-flame optimisation (MFO) algorithm is the most efficient algorithms. The results obtained in this work are said to be the baseline for future study on WFLO using metahueristics.

We establish the local existence and uniqueness of solutions to the free-boundary ideal compressible magnetohydrodynamic equations with surface tension in three spatial dimensions by a suitable modification of the Nash--Moser iteration scheme. The main ingredients in proving the convergence of the scheme are the tame estimates and unique solvability of the linearized problem in the anisotropic Sobolev spaces $H_*^m$ for $m$ large enough. In order to derive the tame estimates, we make full use of the boundary regularity enhanced from the surface tension. The unique solution of the linearized problem is constructed by designing some suitable $\varepsilon$--regularization and passing to the limit $\varepsilon\to 0$., Comment: To appear in: Mathematische Annalen