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Abstract In this paper, we present a scheme for solving two-dimensional miscible displacement problems using Eulerian–Lagrangian localized adjoint methods and mixed finite element methods. Since only the velocity and not the pressure appears explicitly in the concentration equation, an Eulerian–Lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation. To linearize and decouple the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this fully discrete problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid using Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O (h 1 / 2) in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study a maximization and a minimization problem associated with a Poisson boundary value problem. Optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two dimensions. The main contribution of this paper is to determine the optimal solutions. At first, we determine a nearly optimal solution which is an approximation of the optimal solution when the problems are in low contrast regime. Secondly, for the high contrast regime, two optimization algorithms are developed. For the minimization problem, we prove that our algorithm converges to the global minimizer regardless of the initializer. The maximization algorithm is capable of deriving all local maximizers including the global one. Numerical experiments lead us to a conjecture about the location of the maximizers in the set of rearrangements of a function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff type equations (0.1) { − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + λ V ( x ) u = q ( x ) f ( u ) in R 3 , u ∈ H 1 ( R 3 ) , where a , b , λ > 0 , V ∈ C ( R 3 , R ) is a potential well, q ( x ) is a positive bounded function, f ( s ) is either asymptotically linear or asymptotically 3-linear in s at infinity. Under some other suitable conditions on V , q and f , the existence, nonexistence and concentrating behavior of solutions to problem (0.1) are obtained by using variational methods. We mainly extend the results in J. Sun and T. Wu (2014) [26] , which dealt with Kirchhoff type equations with positive potential well, to Kirchhoff type equations with sign-changing potential well. [ABSTRACT FROM AUTHOR]

This paper deals with improvements of the Trudinger–Moser inequality related to the operator Q V ( u ) : = − Δ n u + V ( x ) | u | n − 2 u , where n ≥ 2 and the potential V : R n → R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E : = { u ∈ W 1 , n ( R n ) : ∫ R n V ( x ) | u | n d x < ∞ } endowed with the norm ‖ u ‖ : = [ ∫ R n ( | ∇ u | n + V ( x ) | u | n ) d x ] 1 / n and we prove that if ( u k ) is a sequence in E such that ‖ u k ‖ = 1 , u k ⇀ u ≢ 0 in E and 0 < p < p n ( u ) : = β n ( 1 − ‖ u ‖ n ) − 1 / ( n − 1 ) , then ( ⁎ ) sup k ⁡ ∫ R n Ψ ( p | u k | n / ( n − 1 ) ) d x < ∞ , where Ψ ( t ) : = e t − ∑ i = 0 n − 2 t i i ! , β n : = n ω n − 1 1 / ( n − 1 ) and ω n − 1 is the measure of the unit sphere in R n . Furthermore, p n ( u ) is sharp in the sense that there exists a sequence ( u k ) ⊂ E satisfying ‖ u k ‖ = 1 and u k ⇀ u ≢ 0 in E such that the supremum (⁎) is infinite for p ≥ p n ( u ) . As an application of the previous result we prove the following sharp form of the Trudinger–Moser inequality for the subspace E . Considering ℓ ( α ) : = sup { u ∈ E : ‖ u ‖ = 1 } ⁡ ∫ R n Ψ ∘ ν α ( u ) d x , where ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) , assuming some conditions of symmetry on V it is established (1) for 0 ≤ α < λ 1 ( V ) we have ℓ ( α ) < ∞ , (2) for α ≥ λ 1 ( V ) , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < λ 1 ( V ) , an extremal function for ℓ ( α ) exists. Here λ 1 ( V ) denotes the first eigenvalue of Q V ( u ) . [ABSTRACT FROM AUTHOR]

In this paper, we study the backward–forward algorithm as a splitting method to solve structured monotone inclusions, and convex minimization problems in Hilbert spaces. It has a natural link with the forward–backward algorithm and has the same computational complexity, since it involves the same basic blocks, but organized differently. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators. First, we show that these two methods enjoy remarkable involutive relations, which go far beyond the evident inversion of the order in which the forward and backward steps are applied. Next, we establish several convergence properties for both methods, some of which were unknown even for the forward–backward algorithm. This brings further insight into this well-known scheme. Finally, we specialize our results to structured convex minimization problems, the gradient-projection algorithms, and give a numerical illustration of theoretical interest. [ABSTRACT FROM AUTHOR]

In this paper, we study Noether type symmetry theorem to fractional Birkhoffian systems with Riemann–Liouville derivatives. This theorem provides an explicit algorithmic way to compute a constant for any Birkhoffian systems admitting a symmetry. Finally, we extend our Noether's theorem to fractional Birkhoffian systems base on Caputo or Riesz derivatives. [ABSTRACT FROM AUTHOR]

In this paper we consider continued fraction (CF) expansions on intervals different from [ 0 , 1 ] . For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss–Kuzmin–Lévy based approximation method is used. Convergence of this method follows from [32] but the speed of convergence remains unknown. For a lot of known densities the method gives a very good approximation in a low number of iterations. Finally, a subfamily of the N -expansions is studied. In particular, the entropy as a function of a parameter α is estimated for N = 2 and N = 36 . Interesting behavior can be observed from numerical results. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the existence of strong Nash equilibria (SNE) in continuous and concave games. It is shown that the coalition consistency property introduced in the paper, together with concavity and continuity of payoffs, permits the existence of SNE in games with compact and convex strategy spaces. We also characterize the existence of SNE by providing necessary and sufficient conditions. We suggest an algorithm for computing SNE. The results are illustrated with applications to economies with multilateral environmental externalities and to the static oligopoly model. [Copyright &y& Elsevier]

In this paper, we study the abelian complexity of the Rudin–Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function ρ ( n ) , which satisfies certain recurrence relations. As a consequence, the abelian complexity function is 2-regular. Further, we prove that the box dimension of the graph of the asymptotic function λ ( x ) is 3/2, where λ ( x ) = lim k → ∞ ⁡ ρ ( 4 k x ) / 4 k x and ρ ( x ) = ρ ( ⌊ x ⌋ ) for every x > 0 . [ABSTRACT FROM AUTHOR]

In this paper we propose a new over-relaxed variant of the hybrid proximal extragradient (HPE) algorithm, for the monotone inclusion problem, which uses a projection-free extragradient step with explicit over relaxed stepsize. Its global convergence as well as ergodic and nonergodic complexity rates are established. Moreover, local linear convergence rates are derived under some mild regularity condition. One benefit of the new over relaxed variant of the HPE is that it covers a large class of popular operator splitting methods and their over relaxed versions, thus providing a comprehensive insight on these operators splitting methods. In particular, forward Douglas Rachford splitting method, forward Spingarn's Partial Inverse method, forward Spingarn's partial inverse forward method and Davis–Yin's three operator splitting method are all included as special cases of the over relaxed HPE algorithm. Another benefit is that the interval of stepsize relaxation is easily estimated for these operator splitting methods under the presented framework. Additionally, the over relaxed Korpelevich's method and over relaxed forward–backward–forward method are formulated directly with convergence guarantee based on the proposed framework. The third benefit is that the local linear convergence for a large class of operator splitting methods is established effortlessly under metric subregularity condition. Moreover, this linear convergence condition is shown weaker than some existing ones that almost all require the strong monotonicity of the composite operators. [ABSTRACT FROM AUTHOR]

In this paper, a numerical algorithm for computing the connecting orbits in piecewise smooth dynamical systems is derived and is analyzed. A nondegenerate condition for the connecting orbit with respect to its bifurcation parameter is presented to ensure the defining equation being well posed, which is a generalization of the Melnikov condition for smooth systems. The error caused by the truncation of time interval is also analyzed. Some numerical calculations are carried out to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Within the framework of numerical simulation and optimal control of partial differential equations, in this work we deal with the mathematical modelling and control of the processes related to the urban heat island effect. In particular, we are interested in finding the optimal locations of green zones inside metropolitan areas in order to mitigate the consequences of this harmful phenomenon. So, we consider a three-dimensional climate model and formulate a constrained optimal control problem, that is extensively analyzed in the first part of the paper. Then, we propose a complete numerical algorithm for its resolution, interfacing the interior point algorithm IPOPT with the FreeFem++ software package. Finally, we present several numerical tests for a simple realistic case, where the advantages of our approach are shown. [ABSTRACT FROM AUTHOR]

Abstract: Trigonometric non-Fourier moment problems arise as a result of various control problem study. In current paper, the extremal solution, i.e. the one with the least -norm is searched for. Proposed is an algorithm that allows to change an infinite system of equations into the linear one with only a finite number of equations. The mentioned algorithm is based on the fact, that in the case of a Fourier moment problem, the extremal solution is periodic and easy to construct. The extremal solution of a non-Fourier moment problem close to a Fourier one is approximated by a sequence of solutions with periodicity disturbed in a finite number of equations. It is proved that this sequence of approximations converges to the desired extremal solution. The paper is concluded with the particular example whose consideration leads to a moment problem elaborated in the first part of the article. [Copyright &y& Elsevier]

Abstract: In this paper, a boundary perturbation interior point homotopy method is proposed to give a constructive proof of the general Brouwer fixed point theorem and thus solve fixed point problems in a class of nonconvex sets. Compared with the previous results, by using the newly proposed method, initial points can be chosen in the whole space of , which may improve greatly the computational efficiency of reduced predictor–corrector algorithms resulted from that method. Some numerical examples are given to illustrate the results of this paper. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of . Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa–Holm equation, Kolmogorov–Petrovskii–Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the -dimensional asymmetric version of the Nizhnik–Novikov–Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems. [Copyright &y& Elsevier]

This paper is concerned with nonhomogeneous refinement equations of the form ϕ(x)=∑lower limit α∈Zs a(α)ϕ(Mx−α)+g(x), x∈Rs, where the vector of functions ϕ=(ϕ1,…,ϕr)T is unknown, g is a given vector of compactly supported functions on Rs, a is a finitely supported sequence of r×r matrices called the refinement mask, and M is an s×s integer matrix such that limn→∞M−n=0. Our approach will be to consider the convergence rates of the cascade algorithms associated with nonhomogeneous refinement equations mentioned above. The cascade algorithms associated with mask a, nonhomogeneous term g, and dilation matrix M generates a sequence ϕn, n=1,2,…, by the iterative process ϕn(x)=∑lower limit α∈Zs a(α)ϕn−1(Mx−α)+g(x), x∈Rs, from a starting vector of function ϕ0 in (Lp(Rs))r (0The aim of this paper is to give a characterization of the convergence rates of the cascade algorithms associated with a,g,ϕ0 and dilation matrix M in (Lp(Rs))r (0 in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set E restricted to a certain invariant subspace, where the set E is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0. Some examples are provided to illustrate the method. [Copyright &y& Elsevier]

This paper presents a numerical algorithm for computing ISS Lyapunov functions for discrete-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on solving a linear optimisation problem and delivers a continuous and piecewise affine ISS Lyapunov function on a suitable triangulation covering the given compact set excluding a small neighbourhood of the origin. The objective of the linear optimisation problem is to minimise the ISS gain. It is shown that for every ISS system there exists a suitable triangulation such that the proposed algorithm terminates successfully. [ABSTRACT FROM AUTHOR]

In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements. [ABSTRACT FROM AUTHOR]

In this paper, we consider the least squares regression algorithm with a generalized coefficient regularization term. A novel error decomposition involving a constructive stepping-stone function is introduced. By choosing appropriate parameters for the constructive function we finally derive a satisfactory learning rate under some condition for the goal function and capacity of the hypothesis space. [ABSTRACT FROM AUTHOR]

In this paper we identify weak foci and centers in the Maxwell–Bloch system, a three dimensional quadratic system whose three equilibria are all possible to be of center-focus type. Applying irreducible decomposition and the isolation of real roots in computation of algebraic varieties of Lyapunov quantities on an approximated center manifold, we prove that at most 6 limit cycles arise from Hopf bifurcations and give conditions for exact number of limit cycles near each weak focus. Further, applying algorithms of computational commutative algebra we find Darboux polynomials and give some center manifolds in closed form globally, on which we identify equilibria to be centers or singular centers by integrability and time-reversibility on a center manifold. We prove that those centers are of at most second order. [ABSTRACT FROM AUTHOR]

This paper proposes an identification algorithm for identifying multipolar sources F in the elliptic equation Δ u + μ u = F from boundary measurements. The reconstruction question of this class of sources appears naturally in Helmholtz equation ( μ > 0 ) and in biomedical phenomena particularly in EEG/MEG problems ( μ = 0 ) and bioluminescence tomography (BLT) applications ( μ < 0 ) . Previous works have treated the inverse multipolar source problems, only for equations with μ = 0 , using algebraic approaches depending on the complex calculation of determinants. Knowing that the novelty in our method concerns several points, the principal one is its simplicity where its proof is not founded on the determinants calculation and its ease in implementation. Moreover, this work involves the general form of equations considering μ ∈ R and at the same time considers a more general type of sources than former related works including sources having small compact support within a finite number of subdomains. Finally, some numerical results are shown to prove the robustness of our identification algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we show several formulas for the lemniscate function which include an infinite product formula for the lemniscate sine. Furthermore, we show the relation between the product formula and Carlson's algorithm which is known as the variant of the arithmetic geometric mean of Gauss. [ABSTRACT FROM AUTHOR]

Abstract: Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang (2008) [32]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, based on differential characteristic set theory and the associated algorithm (also called Wuʼs method), an algorithmic method is presented to decide on the existence of a nontrivial non-classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system. The theory and algorithm give a partial answer for the open problem posed by P.A. Clarkson and E.L. Mansfield in [21] on non-classical symmetries of partial differential equations. As applications of our algorithm, non-classical symmetries and corresponding invariant solutions are found for several evolution equations. [Copyright &y& Elsevier]

Abstract: In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising. [Copyright &y& Elsevier]

Abstract: It is known that a consensus problem on any connected complete Riemannian manifold can be transformed into the one on its strongly convex balls via the compression–decompression along geodesics. From the viewpoint of interior metrics, this paper mainly provides a consensus protocol for strongly convex geodesic balls, in which the communication can be switching and directed. With the aid of nonsmooth analysis tools on Riemannian manifolds, our analysis shows that all dynamical points involved can achieve consensus in finite time. Meanwhile, the corresponding global algorithm is given, with its application to the consensus problem of rotation attitudes, as well as a case simulation, to demonstrate and verify our proposed techniques. [Copyright &y& Elsevier]

Abstract: In this paper, we present an algebraic perspective of the de Rham transform of a binary subdivision scheme and propose an elegant strategy for constructing dual -ary approximating subdivision schemes of de Rham-type, starting from two primal schemes of arity and 2, respectively. On the one hand, this new strategy allows us to show that several existing dual corner-cutting subdivision schemes fit into a unified framework. On the other hand, the proposed strategy provides a straightforward algorithm for constructing new dual subdivision schemes having higher smoothness and higher polynomial reproduction capabilities with respect to the two given primal schemes. [Copyright &y& Elsevier]

CG_DESCENT is a state-of-the-art algorithm to solve large-scale unconstrained minimization problems. However, research activities on CG_DESCENT in some other scenarios are relatively fewer. In this paper, by combining with the projection method of Solodov and Svaiter, we extend CG_DESCENT to solve large-scale nonlinear convex constrained monotone equations. The proposed method does not require the Jacobian information, even though it does not store any matrix at each iteration. It thus has the potential to solve large-scale non-smooth problems. Under some mild conditions, we show that the proposed method converges globally. Primary numerical results illustrate that the proposed method works quite well. Moreover, we also extend this method to solve the -norm regularized problems to decode a sparse signal in compressive sensing. Performance comparisons show that the proposed method is practical, efficient and competitive with the compared ones. [ABSTRACT FROM AUTHOR]

Abstract: Previous works describing the generalization performance of bipartite ranking algorithms are usually based on the assumption of (0–1) loss or the area under the receiver operating characteristic (ROC) curve. In this paper we go far beyond this classical framework by investigating the generalization performance of bipartite ranking algorithms with convex losses over reproducing kernel Hilbert spaces. Based on the McDiarmid inequality and Rademacher complexity, we establish the upper bound on the generalization error for a bipartite ranking algorithm. The theoretical analysis is different from the previous results on error analysis and shows the attractive uniform convergence property of regularized bipartite ranking algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a non-iterative imaging algorithm based on the topological derivative in order to retrieve the shape of penetrable electromagnetic inclusions when their dielectric permittivity and/or magnetic permeability differ from those in the embedding (homogeneous) space. The main objective is the imaging of crack-like thin inclusions, but the algorithm can be applied to arbitrarily shaped inclusions. For this purpose, we apply multiple time-harmonic frequencies and normalize the topological derivative imaging function by its maximum value. In order to verify its validity, we apply it for the imaging of two-dimensional crack-like thin electromagnetic inhomogeneities completely hidden in a homogeneous material. Corresponding numerical simulations with noisy data are performed for showing the efficacy of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In the reference paper Wu (2007) [1], a decentralized iterative learning control (ILC) algorithm is proposed aiming at achieving the asymptotic convergence of output errors for a class of linear time-varying large scale interconnected dynamic systems. The key points in the analysis of this decentralized ILC are the adoption of a specific time-weighted norm, and the use of a property of nondecreasing real functions that leads to the cancellation of interactions between subsystems. Here, on the one hand, we show that there exists a derivation problem in the proof, thus the asymptotic convergence property cannot be obtained. On the other hand, we provide an alternative analysis method, the classical contraction-mapping based ILC analysis with the lambda-norm, to demonstrate that the decentralized ILC algorithm is still valid and able to achieve the asymptotic convergence. [Copyright &y& Elsevier]

Abstract: This paper studies the risk probability criteria for finite horizon semi-Markov decision processes. The goal is to find an optimal policy with the minimum risk probability that the total reward produced by a system during a finite horizon does not exceed a reward level, where the optimality is over the class of all randomized historic policies which include states, planning horizons and also reward levels. Under mild conditions, the optimality equation and the existence of optimal policies are established, and in addition, an iteration algorithm for solving optimal policies is developed. Our main results are applied to a manufacturing system. [Copyright &y& Elsevier]

Abstract: In this paper we present a technique for shape reconstruction based on the topological and shape gradients. The shape in consideration is a solution of an inverse conductivity problem. To solve such a problem numerically, we compute the topological gradient of a Kohn–Vogelius-type cost function when the domain under consideration is perturbed by the introduction of a small inclusion instead of a hole. The reconstruction is done by considering the shape as a superposition of very thin elliptic inclusions to get a first approximation. Then, we use a gradient-type algorithm to perform a good reconstruction. Various numerical experiments of single and multiple inclusions demonstrate the viability of the designed algorithm. [Copyright &y& Elsevier]

Abstract: In a Hilbert space, we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors. Most results known in the literature establish the convergence of algorithms, when computational errors are summable. In the present paper, the convergence of the subgradient method is established for nonsummable computational errors. We show that the subgradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. [Copyright &y& Elsevier]

Abstract: In view of the fact that prototypes of Riemannian manifolds range from Euclidean surfaces to Lie groups, operator spaces, etc., the paper explores common consensus schemes for connected complete Riemannian manifolds. After studying the restriction exponential map, we introduce a local consensus protocol with switching communication topologies for the manifolds via the isometric mapping, based on one of the Euclidean consensus protocols. Equipped with the compression–decompression functions, we extend the local consensus protocol, and put forth a global consensus algorithm for any such manifolds. These consensus schemes can provide common solutions to the consensus problems arising from various nonlinear spaces or abstract spaces belonging to connected complete Riemannian manifolds. In particular, they are simple and concise, and can converge in finite time. Simulation on the unit sphere is provided to verify our consensus schemes, and to demonstrate the usage of our proposed techniques. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the extremal behaviour of aggregated risk for a specific parametrised multivariate dependence framework. Furthermore we discuss conditional limit results and extremal behaviour of both maximum and aggregated log-elliptical risk. Our application establishes the logarithmic efficiency of the Rojas-Nandaypa algorithm for rare-event simulation of log-elliptical risks. [Copyright &y& Elsevier]

Abstract: We present a new version of the wave-front tracking algorithm for 2×2 hyperbolic systems of conservation laws. The analysis in this paper is simpler than in the previous algorithms. [Copyright &y& Elsevier]

Abstract: The main goals of this paper are to consider a problem of -convergence () of greedy algorithm with respect to generalized Walsh system. [Copyright &y& Elsevier]

Abstract: The previously known works describing the generalization of least-square regularized regression algorithm are usually based on the assumption of independent and identically distributed (i.i.d.) samples. In this paper we go far beyond this classical framework by studying the generalization of least-square regularized regression algorithm with Markov chain samples. We first establish a novel concentration inequality for uniformly ergodic Markov chains, then we establish the bounds on the generalization of least-square regularized regression algorithm with uniformly ergodic Markov chain samples, and show that least-square regularized regression algorithm with uniformly ergodic Markov chains is consistent. [Copyright &y& Elsevier]

Abstract: In this paper, we show that the convex optimization problem can be solved by the proximal point algorithm in a finite number of steps under the assumption that the solution set is a set of weak sharp minima. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the convergence behavior of the Schwarz waveform relaxation (SWR) algorithms for solving PDEs with time delay. We choose the reaction diffusion equations with a constant time delay as the underlying model problem and try to derive optimized transmission conditions of Robin type. To this end, we propose a new method to get quasi-optimized parameter involved in the transmission conditions and it is shown that this method is essentially different from the existing ones. Moreover, when the situation is reduced into the heat equations with a constant delay, we show that this method results in a more efficient quasi-optimized parameter. Numerical results are provided to validate our theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we present iteration schemes to weakly and strongly approximate common fixed points of a finite family of a class of strict pseudocontractions. [Copyright &y& Elsevier]

Abstract: The principal component analysis is to recursively estimate the eigenvectors and the corresponding eigenvalues of a symmetric matrix A based on its noisy observations , where A is allowed to have arbitrary eigenvalues with multiplicity possibly bigger than one. In the paper the recursive algorithms are proposed and their ordered convergence is established: It is shown that the first algorithm a.s. converges to a unit eigenvector corresponding to the largest eigenvalue, the second algorithm a.s. converges to a unit eigenvector corresponding to either the second largest eigenvalue in the case the largest eigenvalue is of single multiplicity or the largest eigenvalue if the multiplicity of the largest eigenvalue is bigger than one, and so on. The convergence rate is also derived. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the notion of a weak sharp set of solutions to a variational inequality problem (VIP) in a reflexive, strictly convex and smooth Banach space, and present its several equivalent conditions. We also prove, under some continuity and monotonicity assumptions, that if any sequence generated by an algorithm for solving (VIP) converges to a weak sharp solution, then we can obtain solutions for (VIP) by solving a finite number of convex optimization subproblems with linear objective. Moreover, in order to characterize finite convergence of an iterative algorithm, we introduce the notion of a weak subsharp set of solutions to a variational inequality problem (VIP), which is more general than that of weak sharp solutions in Hilbert spaces. We establish a sufficient and necessary condition for the finite convergence of an algorithm for solving (VIP) which satisfies that the sequence generated by which converges to a weak subsharp solution of (VIP), and show that the proximal point algorithm satisfies this condition. As a consequence, we prove that the proximal point algorithm possesses finite convergence whenever the sequence generated by which converges to a weak subsharp solution of (VIP). [ABSTRACT FROM AUTHOR]

Abstract: Many existing algorithms taking the seminorm in for regularization have achieved great success in image processing. However, this paper considers the total bounded variation regularization based approach to perform image deblurring. Based on this novel model, we introduce an extended split Bregman iteration to obtain the optimum solution quickly. We also provide the rigorous convergence analysis of the iterative algorithm here. Compared with the results of the ROF method, numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: Real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient. [Copyright &y& Elsevier]

Abstract: Finding an efficient or weakly efficient solution in a multiobjective linear programming (MOLP) problem is not a difficult task. The difficulty lies in finding all these solutions and representing their structures. Since there are many convenient approaches that obtain all of the (weakly) efficient extreme points and (weakly) efficient extreme rays in an MOLP, this paper develops an algorithm which effectively finds all of the (weakly) efficient maximal faces in an MOLP using all of the (weakly) efficient extreme points and extreme rays. The proposed algorithm avoids the degeneration problem, which is the major problem of the most of previous algorithms and gives an explicit structure for maximal efficient (weak efficient) faces. Consequently, it gives a convenient representation of efficient (weak efficient) set using maximal efficient (weak efficient) faces. The proposed algorithm is based on two facts. Firstly, the efficiency and weak efficiency property of a face is determined using a relative interior point of it. Secondly, the relative interior point is achieved using some affine independent points. Indeed, the affine independent property enable us to obtain an efficient relative interior point rapidly. [Copyright &y& Elsevier]

Abstract: In this paper we present a new primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full Nesterov–Todd step. Moreover, we obtain the currently best known iteration bound for the algorithm with small-update method, namely, , which is as good as the linear analogue. [Copyright &y& Elsevier]

Abstract: Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors. [Copyright &y& Elsevier]