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In this paper, we are interested in the generalized variational inequality problem in real Hilbert spaces. We propose an explicit proximal method which requires only one proximal step and one mapping evaluation per iteration and also uses an adaptive step-size rule that enables to avoid the prior knowledge of the Lipschitz constant of the involved mapping. Weak convergence of the proposed scheme is established under standard assumptions. Under strong monotonicity, we present the R-linear convergence rate of our new method. Intensive numerical experiments illustrate the advantages and the applicability of our scheme. Moreover, our work generalizes theoretically several recent results in this field.

In this paper, we first study a relaxed inertial projection and contraction method for variational inequalities to obtain weak convergence results under standard assumptions in Hilbert spaces. Next, we propose another inertial projection and contraction method for which the inertial factor θ is chosen in [ 0 , 1 ] with θ = 1 possible. Weak and linear convergence are obtained for this second proposed method. As far as we know, the choice θ = 1 has not been considered before in the literature for inertial projection and contraction method. Finally, we give some numerical illustrations of our methods.

In this paper, a reduced spline (RS) method based on a proper orthogonal decomposition (POD) technique for numerical solution of time fractional sub-diffusion equations is investigated. Combining POD with polynomial and non-polynomial spline methods yield a new model with lower dimensions and sufficiently high accuracy, so that the amount of computations and the calculation time decrease in comparison with usual spline methods. Although, the classical L1 scheme on uniform meshes is one of the most successful methods for approximation of the Caputo fractional derivative in time, due to the fact that the solution of time fractional sub-diffusion equations typically has a singularity at the origin, a weak convergence will conclude for such scheme. To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution. Also, the error estimates between the POD approximate solution of the RS scheme and the exact solution of fractional sub-diffusion equation are established for two types of uniform and graded meshes in time. Numerical examples are given to illustrate the feasibility and efficiency of the proposed method.

We present a semilocal and local convergence analysis of Newton's method on a Banach space with a convergence structure to locate zeros of operators. P. Meyer introduced the concept of a Banach space with a convergence structure. Using this setting, he presented a finer semilocal convergence analysis for Newton's method than in related studies using the real norm theory. In all these studies the operator involved as well as its Frechet derivative is bounded above by the same bound-operator. In the present study, we introduce a second bound operator which is a special case of the bound-operator leading to tighter majorizing sequences for Newton's method. Using this more flexible combination of bound-operators, we improve the results in the earlier studies. In the semilocal case, we obtain under the same or weaker sufficient convergence conditions more precise error bounds on the distances involved and in the local case not considered in the earlier studies, we obtain a larger radius of convergence. This way we expand the applicability of Newton's method. Some numerical examples are also provided to show the superiority of the new results over the old results.

We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and also provide the full weak convergence rate for non-self-adjoint linear operator with additive noise. Key part of the proof does not rely on Malliavin calculus. For non-self-adjoint operators, we analyse the optimal strong error for spatially semi-discrete approximations for both multiplicative and additive noise with truncated and non-truncated noise. Depending on the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.

In this paper, we give conditions for stochastic Runge–Kutta (SRK) methods to preserve quadratic invariants. It is shown that SRK methods preserving quadratic invariants are symplectic. Based on both convergence order conditions and quadratic invariant-preserving conditions, we construct some SRK schemes preserving quadratic invariants with strong and weak convergence order with the help of computer algebra, respectively. Numerical experiments are executed to verify our theoretical analysis and show the superiority of these schemes.

This work focuses on the strong solutions of a class of hybrid diffusion processes with state-dependent regime-switching. This important class of processes originates from the purpose of modeling the interest rate in finance. They have no any explicit solutions. Moreover, state-dependent regime-switching and non-Lipschitz diffusion coefficient pose a challenge to our analysis. To overcome all of these, we consider the Euler numerical schemes rather than the traditional Picard iterations in the existing results of solutions of stochastic differential equations. The weak convergence of numerical algorithms is first established by a martingale problem formulation. Using this result, we can also obtain the strong convergence of the algorithms. The existence of strong solutions is then confirmed. In addition, decreasing stepsize iterative algorithms and their weak convergence are presented. Several numerical experiments are also provided for illustration.

We discuss weak-convergence of the splitting methods for solving singular systems and introduce a new concept, that of the k-order quasi-nonnegative splitting. We provide equivalent formulations of this notion. Moreover, we investigate asymptotic and monotone convergence and comparison theorems for the k-order quasinonnegative splittings.