It is known that every Toeplitz matrix $T$ enjoys a circulant and skew circulant splitting (denoted $\text{CSCS}$) i.e., $T=C-S$ a circulant matrix and $S$ a skew circulant matrix. Based on the variant of such a splitting (also referred to as $\text{CSCS}$), we first develop classical $\text{CSCS}$ iterative methods and then introduce shifted $\text{CSCS}$ iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical $\text{CSCS}$ iterative methods work slightly better than the Gauss–Seidel $(\text{GS})$ iterative methods if the $\text{CSCS}$ is convergent, and that there is always a constant $\alpha $ such that the shifted $\text{CSCS}$ iteration converges much faster than the Gauss–Seidel iteration, no matter whether the $\text{CSCS}$ itself is convergent or not.

We construct modulus-based synchronous multisplitting iteration methods to solve a large implicit complementarity problem on parallel multiprocessor systems, and prove their convergence. Numerical results confirm our theoretical analysis and show that these new methods are efficient.

Most current prevalent iterative methods can be classified into the socalled extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with a fixed matrix, the newly introduced methods (the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly an accelerative approach (called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behavior. Comparison between benchmark extended Krylov subspace methods (Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

In this paper, we study to use nonlocal bounded variation (NLBV) techniques to decompose an image intensity into the illumination and reflectance components. By considering spatial smoothness of the illumination component and nonlocal total variation (NLTV) of the reflectance component in the decomposition framework, an energy functional is constructed. We establish the theoretical results of the space of NLBV functions such as lower semicontinuity, approximation and compactness. These essential properties of NLBV functions are important tools to show the existence of solution of the proposed energy functional. Experimental results on both grey-level and color images are shown to illustrate the usefulness of the nonlocal total variation image decomposition model, and demonstrate the performance of the proposed method is better than the other testing methods.

In this paper the problem − div(a(x,y)∇u) = f with Dirichlet boundary conditions on a square is solved iteratively with high accuracy for u and ∇u using a new scheme called “hermitian box-scheme”. The design of the scheme is based on a “hermitian box”, combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2h. The iterative technique is based on the repeated solution by a fast direct method of a discrete Poisson equation on a uniform rectangular mesh. The problem is suitably scaled before iteration. The numerical results obtained show the efficiency of the numerical scheme. This work is the extension to strongly elliptic problems of the hermitian box-scheme presented by Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239-267).

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

An acceleration scheme based on stationary iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method which requires accurate estimation of the bounds for iterative matrix eigenvalues, we use a wide range of Chebyshev-like polynomials for the accelerating process without estimating the bounds of the iterative matrix. A detailed error analysis is presented and convergence rates are obtained. Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods are provided.

We report an efficient phase field formalism to compute the stress distribution in polycrystalline materials with arbitrary elastic inhomogeneity and anisotropy The dependence of elastic stiffness tensor on grain orientation is taken into account, and the elastic equilibrium equation is solved using a spectral iterative perturbation method. We discuss its applications to computing residual stress distribution in systems containing arbitrarily shaped cavities and cracks (with zero elastic modulus) and to determining the effective elastic properties of polycrystals and multilayered composites.

Thermochronometer data offer a powerful tool for quantifying a wide range of geologic processes, such as the deformation and erosion of mountain ranges, topographic evolution, and hydrocarbon maturation. With increasing interest to quantify a wider range of complicated geologic processes, more sophisticated techniques are needed. This paper is concerned with an inverse problem method for interpreting the thermochronometer data quantitatively. Two novel models are proposed to simulate the crustal thermal fields and paleo mountain topography as a function of tectonic and surface processes. One is a heat transport model that describes the change of temperature of rocks; while the other is surface process model which explains the change of mountain topography. New computational algorithms are presented for solving the inverse problem of the coupled system of these two models. The model successfully provides a new tool for reconstructing the kinematic and the topographic history of mountains.

This paper introduces a scheme for the numerical approximation of a model for two turbulent flows with coupling at an interface. We consider the variational formulation of the coupled model, where the turbulent kinetic energy equation is formulated by transposition. We prove the convergence of the approximation to this formulation for 3D flows for large turbulent viscosities and smooth enough flows, whenever bounded in W 1,p Sobolev norms for p large enough. Under the same assumptions, we show that the limit is a solution of the initial problem. Finally, we give some numerical experiments to enlighten the theoretical work.

This paper proposes a method for finding an optimal geometric robot trajectory to perform a specified point-to-point motion without violating joint displacement limits or interference constraints. The problem is discretised, and a quantitative measure of interference is proposed. Constraint violations are represented by exterior penalty functions, and the problem is solved by iteratively improving an initial estimate of the trajectory. This is accomplished by numerically minimizing a cost functional using a modified Newton–Raphson method.

An extension of the inverse kinematics algorithm by Gupta and Kazerounian is presented. The robot kinematics is formulated by using the Zero Reference Position Method. Euler parameters and the related vector forms of the spatial rotation concatenation have been used to improve the efficiency of the velocity Jacobian computation. The joint rates are formally integrated by using a modified predictor-corrector method particularized to robot inverse kinematics – it is a strict descending, p(1)c(0 – n), variable step algorithm. The definitions of the rotational error and overall error measure have been revised. Depending upon the convergence criteria used, these modifications reduce the overall computational time by 20%.