Based on the work of paprer [1], we propose a modified Levenberg-Marquardt algoithm for solving singular system of nonlinear equations $F(x)=0$, where $F(x):R^n\rightarrow R^n$ is continuously differentiable and $F'(x)$ is Lipschitz continuous. The algorithm is equivalent to a trust region algorithm in some sense , and the global convergence result is given. The sequence generated by the algorithm converges to the solution quadratically, if $\|F(x)\|_2$provides a local error bound for the system of nonlinear equations. Numerical results show that the algorithm performs well.

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

Tensor canonical decomposition (shorted as CANDECOMP/PARAFAC or CP) decomposes a tensor as a sum of rank-one tensors, which finds numerous applications in signal processing, hypergraph analysis, data analysis, etc. Alternating least-squares (ALS) is one of the most popular numerical algorithms for solving it. While there have been lots of efforts for enhancing its efficiency, in general its convergence can not been guaranteed.
In this paper, we cooperate the ALS and the trust-region technique from optimization field to generate a trust-region-based alternating least-squares (TRALS) method for CP. Under mild assumptions, we prove that the whole iterative sequence generated by TRALS converges to a stationary point of CP. This thus provides a reasonable way to alleviate the swamps, the notorious phenomena of ALS that slow down the speed of the algorithm. Moreover, the trust region itself, in contrast to the regularization alternating least-squares (RALS) method, provides a self-adaptive way in choosing the parameter, which is essential for the efficiency of the algorithm. Our theoretical result is thus stronger than that of RALS in[26], which only proved the cluster point of the iterative sequence generated by RALS is a stationary point. In order to accelerate the new algorithm, we adopt an extrapolation scheme. We apply our algorithm to the amino acid fluorescence data decomposition from chemometrics, BCM decomposition and rank-(L_r, L_r, 1) decomposition arising from signal processing, and compare it with ALS and RALS. The numerical results show that TRALS is superior to ALS and RALS, both from the number of iterations and CPU time perspectives.

In this paper, two fourth-order accurate compact difference schemes
are presented for solving the Helmholtz equation in two space
dimensions when the corresponding wave numbers are large. The main
idea is to derive and to study a fourth-order accurate compact
difference scheme whose leading truncation term, namely, the
$\mathcal O(h^4)$ term, is independent of the wave number and the
solution of the Helmholtz equation. The convergence property of the
compact schemes are analyzed and the implementation of solving the
resulting linear algebraic system based on a FFT approach is
considered. Numerical results are presented, which support our
theoretical predictions.

The main aim of this paper is to study the error estimates of a nonconforming finite element with some superconvergence results under anisotropic meshes. The anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches and techniques, respectively. Furthermore, the superclose and a superconvergence estimate on the central points of elements are also obtained without the regularity assumption and quasi-uniform assumption requirement on the meshes. Finally, a numerical test is carried out, which coincides with our theoretical analysis.

A shift splitting concept is introduced and, correspondingly,
a shift-splitting iteration scheme and a shift-splitting
preconditioner are presented,
for solving the large sparse system of linear equations of which the
coefficient matrix is an ill-conditioned non-Hermitian
positive definite matrix.
The convergence property of the shift-splitting iteration method
and the eigenvalue distribution of the shift-splitting
preconditioned matrix are discussed in depth,
and the best possible choice of the shift is investigated
in detail. Numerical computations show that
the shift-splitting preconditioner can induce accurate, robust
and effective preconditioned Krylov subspace iteration methods
for solving the large sparse non-Hermitian positive definite
systems of linear equations.

In this paper, we propose an extended Levenberg-Marquardt (ELM) framework that generalizes the classic Levenberg-Marquardt (LM) method to solve the unconstrained minimization problem min ρ (r(x)), where r:Rⁿ → Rm and ρ:R^m → R. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jacobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.

The fast sweeping method is an efficient iterative method for
hyperbolic
problems.
It combines Gauss-Seidel iterations with alternating sweeping orderings.
In this paper several parallel implementations of the fast sweeping method
are presented. These parallel algorithms are simple and efficient due
to the causality of the underlying partial
different equations. Numerical examples are used to verify our algorithms.

We present a program for computing symmetric quadrature rules on
triangles and tetrahedra. A set of rules are obtained by using this
program. Quadrature rules up to order 21 on triangles and up to
order 14 on tetrahedra have been obtained which are useful for use
in finite element computations. All rules presented here have
positive weights with points lying within the integration domain.

In this paper we consider the numerical solution of the
one-dimensional heat equation on unbounded domains. First an exact
semi-discrete artificial boundary condition is derived by
discretizing the time variable with the Crank-Nicolson method. The
semi-discretized heat equation equipped with this boundary condition
is then proved to be unconditionally stable, and its solution is
shown to have second-order accuracy. In order to reduce the
computational cost, we develop a new fast evaluation method for the
convolution operation involved in the exact semi-discrete artificial
boundary condition. A great advantage of this method is that the
unconditional stability held by the semi-discretized heat equation
is preserved. An error estimate is also given to show the dependence
of numerical errors on the time step and the approximation accuracy
of the convolution kernel. Finally, a simple numerical example is
presented to validate the theoretical results.

Higher-order singular value decomposition (HOSVD) is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition.
We also present one algorithm for solving the problem based on block coordinate update (BCU). Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration (HOOI) method, and thus we, for the first time, give global convergence of HOOI.
In addition, we compare the proposed method to state-of-the-art ones for solving incomplete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.

In this paper, a novel class of exponential Fourier collocation methods (EFCMs) is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation methods. We discuss in detail the connections of EFCMs with trigonometric Fourier collocation methods (TFCMs), the well-known Hamiltonian Boundary Value Methods (HBVMs), Gauss methods and Radau ⅡA methods. It turns out that the novel EFCMs are an essential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of firstorder ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM(2,2) as an illustrative example. The numerical experiments using EFCM(2,2) are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM(2,2).

This paper is devoted to give a new proof of Korn's inequality in LT - norm (1 < γ < ∞).

In this paper,we propose two new perturbation simplex variants.Solving linear programming problems without introducing artificial variables,each of the two uses the dual pivot rule to achieve primal feasibility,and then the primal pivot rule two achieve optimality.The second algorithm,a modification of the first,is designed to handle highly degenerate problems more efficiently.Some interesting results concerning merit of the perturbation are established.Numerical results from preliminary tests are also reported.

In this paper, the multisymplectic Fourier pseudospectral scheme
for initial-boundary value problems of nonlinear Schr\"{o}dinger
equations with wave operator is considered. We investigate the
local and global conservation properties of the multisymplectic
discretization based on Fourier pseudospectral approximations. The
local and global spatial conservation of energy is proved. The
error estimates of local energy conservation law are also derived.
Numerical experiments are presented to verify the theoretical
predications.

Let $S_E$ denote the least-squares symmetric solution set of the
matrix equation $AXB=C$, where A, B and C are given matrices of
suitable size. To find the optimal approximate solution in the set
$S_E$ to a given matrix, we give a new feasible method based on the
projection theorem, the generalized SVD and the canonical correction
decomposition.

In this paper, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation with a variable interfacial parameter, is solved numerically by using a convex splitting scheme which is second-order in time for the non-stochastic part in combination with the CrankNicolson and the Adams-Bashforth methods. For the non-stochastic case, the unconditional energy stability is obtained in the sense that a modified energy is non-increasing. The scheme in the stochastic version is then obtained by adding the discretized stochastic term. Numerical experiments are carried out to verify the second-order convergence rate for the non-stochastic case, and to show the long-time stochastic evolutions using larger time steps.

Consider the diffraction of a time-harmonic wave incident upon a
periodic chiral structure. The diffraction problem may be simplified
to a two-dimensional one. In this paper, the diffraction problem is
solved by a finite element method with perfectly matched absorbing
layers (PMLs). We use the PML technique to truncate the unbounded
domain to a bounded one which attenuates the outgoing waves in the
PML region. Our computational experiments indicate that the proposed
method is efficient, which is capable of dealing with complicated
chiral grating structures.

Symplectic integration of separable Hamiltonian ordinary and partial differential equations is discussed. A von Neumann analysis is performed to achieve general linear stability criteria for symplectic methods applied to a restricted class of Hamiltonian PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied.In this way stability criteria are achieved by considering the spectra of linearised Hamiltonian PDEs rather than spatisl step size.

In this paper, we introduce an efficient Chebyshev-Gauss spectral collocation method for initial value problems of ordinary differential equations. We first propose a single interval method and analyze its convergence. We then develop a multi-interval method. The suggested algorithms enjoy spectral accuracy and can be implemented in stable and efficient manners. Some numerical comparisons with some popular methods are given to demonstrate the effectiveness of this approach.

We construct a modified Bernoulli iteration method for solving the
quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and
$C$ are square matrices. This method is motivated from the
Gauss-Seidel iteration for solving linear systems and the
Sherman-Morrison-Woodbury formula for updating matrices. Under
suitable conditions, we prove the local linear convergence of the
new method. An algorithm is presented to find the solution of the
quadratic matrix equation and some numerical results are given to
show the feasibility and the effectiveness of the algorithm. In
addition, we also describe and analyze the block version of the
modified Bernoulli iteration method.

In this paper, we present a weak Galerkin (WG) mixed finite element method for solving the second-order elliptic equations with Robin boundary conditions. Stability and a priori error estimates for the coupled method are derived. We present the optimal order error estimate for the WG-MFEM approximations in a norm that is related to the L² for the flux and H¹ for the scalar function. Also an optimal order error estimate in L² is derived for the scalar approximation by using a duality argument. A series of numerical experiments is presented that verify our theoretical results.

An augmented Lagrangian trust region method with a bi-object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented Lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.

Four new trigonometric Bernstein-like basis functions with two exponential shape parameters are constructed, based on which a class of trigonometric Bézier-like curves, analogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cubic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape parameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C² ∩ FC³ continuous for a non-uniform knot vector, and C³ or C⁵ continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for G1 continuous joining two trigonometric Bézier-like patches over triangular domain are deduced.

The spectral gradient method has proved to be effective for solving large-scale unconstrained optimization problems. It has been recently extended and combined with the projected gradient method for solving optimization problems on convex sets. This combination includes the use of nonmonotone line search techniques to preserve the fast local convergence. In this work we further extend the spectral choice of steplength to accept preconditioned directions when a good preconditioner is available. We present an algorithm that combines the spectral projected gradient method with preconditioning strategies to increase the local speed of convergence while keeping the global properties. We discuss implementation details for solving large-scale problems.

A class of asynchronous matrix multi-splitting multi-parameter
relaxation methods, including the asynchronous matrix multisplitting
SAOR, SSOR and SGS methods as well as the known asynchronous matrix
multisplitting AOR, SOR and GS methods, etc., is proposed for solving
the large sparse systems of linear equations by making use of the
principle of sufficiently using the delayed information. These new
methods can greatly execute the parallel computational efficiency of the
MIMD-systems, and are shown to be convergent when the coefficient
matrices are $H$-matrices. Moreover, necessary and sufficient conditions
ensuring the convergence of these methods are concluded for the case
that the coefficient matrices are $L$-matrices.

This paper first presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDEs). Then the numerical analogous results, of the natural Runge-Kutta (NRK) methods for the same class of nonliner NDDEs,are given.In particular,it is shown thar the (k,l)-algebraic stability of a RK method for ODEs implies the generalized asymptotic stability and the global stability of the induced NRK method.of

In this paper,a kind of rational cubic interpolation function with linear denominator is constructed.The constrained interpolation with constraint on shape of the interpolating curves and on the second-order derivative of the interpolating function is studied by using this interpolation,and as the consequent result,the convex interpolation conditions have been derived.

An upwind difference scheme was given by the author in [5] for the numerical solution of steady-state problems. The present work studies this upwind scheme and its corresponding boundary scheme for the numerical solution of unsteady problems. For interior points the difference equations are approximations of the characteristic relations corresponding to the outgoing characteristics and the "non-reflecting" boundary conditions. Calculation of a Riemann problem in a finite computational region yields promising numerical results.

The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, arti cial boundaries are necessary. A well established arti cial boundary condition for the Navier-Stokes equations discretized by finite elements is the "do-nothing" condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established out ow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a "directional do-nothing" condition. In contrast to the usual "do-nothing" condition this boundary condition has enhanced stability properties. In the case of pure out ow, the condition is equivalent to the original one, whereas in the case of in ow a dissipative e ect appears. We show existence of weak solutions and illustrate the e ect of this boundary condition by computation of steady and non-steady flows.

We consider solving linear ill-posed operator equations. Based on a
multi-scale decomposition for the solution space, we propose a
multi-parameter regularization for solving the equations. We
establish weak and strong convergence theorems for the
multi-parameter regularization solution. In particular, based on the
eigenfunction decomposition, we develop a posteriori choice strategy
for multi-parameters which gives a regularization solution with the
optimal error bound. Several practical choices of multi-parameters
are proposed. We also present numerical experiments to demonstrate
the outperformance of the multi-parameter regularization over the
single parameter regularization.

In this paper, we propose an algorithm for solving nonlinear
monotone equations by combining the limited memory BFGS method
(L-BFGS) with a projection method. We show that the method is
globally convergent if the equation involves a
Lipschitz continuous monotone function. We also present some
preliminary numerical
results.

The monotone variational inequalities VI$(\Omega,F)$ have vast applications,
including optimal controls and convex programming. In this paper we focus
on the VI problems that have a particular splitting structure and in which
the mapping $F$ does not have an explicit form, therefore only its function
values can be employed in the numerical methods for solving such problems.
We study a set of numerical methods that are easily implementable. Each
iteration of the proposed methods consists of two procedures. The first
(prediction) procedure utilizes alternating projections to produce a predictor.
The second (correction) procedure generates the new iterate via some minor
computations. Convergence of the proposed methods is proved under mild
conditions.
Preliminary numerical experiments for some traffic equilibrium problems
illustrate the effectiveness of the proposed methods.

A revised conjugate gradient projection method for nonlinear inequality constrained optimization problems is proposed in the paper, since the search direction is the combination of the conjugate projection gradient and the quasi-Newton direction. It has two merits. The one is that the amount of computation is lower because the gradient matrix only needs to be computed one time at each iteration. The other is that the algorithm is of global convergence and locally superlinear convergence without strict complementary condition under some mild assumptions. In addition the search direction is explicit.

In this paper, we consider the finite element method and
discontinuous Galerkin method for the stochastic Helmholtz equation
in $\mathbb{R}^d$ $(d=2,3)$. Convergence analysis and error
estimates are presented for the numerical solutions. The effects of
the noises on the accuracy of the approximations are illustrated.
Numerical experiments are carried out to verify our theoretical
results.

We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on:
1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities;
2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil.
We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities. Numerical examples are presented.

Characterizations of symmetric and symplectic Runge-Kutta methods, which are based on the W-transformation of Harier and Wanner, are presented. Using these characterizations we construct two classes of high order symplectic (symmetric and algebraically stable or algebraically stable) Runge-Kutta methods. They include and extend known classes of high order implicit Runge-Kutta methods.

We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired $M$-matrix solution of the quadratic matrix equation $X^2-EX-F=0$ by connecting it with the nonsymmetric algebraic Riccati equation, where $E$ is a diagonal matrix and $F$ is an $M$-matrix.

In this paper, three $n$-rectangle
nonconforming elements are proposed with $n\ge3$. They are the extensions of
well-known Morley element, Adini element and Bogner-Fox-Schmit element in two
spatial dimensions to any higher dimensions respectively. These elements are
all proved to be convergent for a model biharmonic equation in $n$ dimensions.

The Laguerre Gauss-Radau interpolation is investigated. Some approximation results are obtained. As an example, the Laguerre pseudospectral scheme is constructed for the BBM equation. The stability and the convergence of proposed scheme are proved. The numerical results show the high accuracy of this approch.