Narrow Results

This paper focuses on developing and analyzing a new AR-iterative scheme designed for estimating fixed points of generalized non-expansive mappings of the Suzuki kind. The scheme’s stability and convergence properties are rigorously proven, highlighting its superior convergence rate for non-expansive mappings when compared to existing schemes in the literature. Additionally, an application of the AR-iterative scheme to differential equations is presented. These findings underscore the effectiveness and versatility of the proposed scheme in numerical analysis and computational mathematics.

We propose the $\delta$-Rescaled Pure Super Greedy Algorithm ($\delta$-RPSGA) for approximation with respect to a dictionary ${𝒟}$ in a Hilbert space. We estimate the upper bound of the error for the $\delta$-RPSGA when ${𝒟}$ is a Riesz dictionary. Our results show that the convergence rate of the $\delta$-RPSGA on the closure of the convex hull of ${𝒟}$ is optimal. Then, we design the $\delta$-Rescaled Pure Super Greedy Learning Algorithm ($\delta$-RPSGLA) for the kernel-based regression in supervised learning. We obtain the convergence rates of the $\delta$-RPSGLA for continuous kernels. When the kernel is infinitely smooth, we derive a convergence rate that can be arbitrarily close to the best rate $O(m^{-1})$ under a mild assumption of the regression function.

We study the greedy algorithms for $m$-term approximation. We propose a modification of the Weak Rescaled Pure Greedy Algorithm (WRPGA) — Approximate Weak Rescaled Pure Greedy Algorithm (AWRPGA) — with respect to a dictionary of a Banach space $X$. By using a geometric property of the unit sphere of $X$, we obtain a general error estimate in terms of some $K$-functional. This estimate implies the convergence condition and convergence rate of the AWRPGA. Furthermore, we obtain the corresponding error estimate for the Vector Approximate Weak Rescaled Pure Greedy Algorithm (VAWRPGA). We show that the AWRPGA (VAWRPGA) performs as well as the WRPGA (VWRPGA) when the noise amplitude changes relatively little. Finally, by using the wavelet bases and trigonometric system of Lebesgue spaces, we show that the convergence rate of the AWRPGA is optimal.

This paper develops a second-order explicit predictor–corrector numerical approach for solving a mathematical model on the dynamics of cytokine expressions and human immune cell activation in response to the bacterium staphylococcus aureus (S. aureus). The proposed algorithm is at least zero-stable and second-order accurate. Mathematical modeling works that analyze the human body in response to some antigens have predicted concentrations of a broad range of cells and cytokines. This study deals with a coupled cellular-cytokine model which predicts cytokine expressions in response to gram-positive bacteria S. aureus. Tumor necrosis factor alpha, interleukin 6, interleukin 8 and interleukin 10 are included to assess the relationship between cytokine release from macrophages and the concentration of the S. aureus antigen. Ordinary differential equations are used to model cytokine levels while the cellular responses are modeled by partial differential equations. Interactions between both components provide a more robust and complete systems of immune activation. In the numerical simulations, a low concentration of S. aureus is used to measure cellular activation and cytokine expressions. Numerical experiments indicate how the human immune system responds to infections from different pathogens. Furthermore, numerical examples suggest that the new technique is faster and more efficient than a large class of statistical and numerical schemes discussed in the literature for systems of nonlinear equations and can serve as a robust tool for the integration of general systems of initial-boundary value problems.

We consider the linearly constrained separable convex programming problem whose objective function is separable into m individual convex functions with non-overlapping variables. The alternating direction method of multipliers (ADMM) has been well studied in the literature for the special case m = 2, but the direct extension of ADMM for the general case m ≥ 2 is not necessarily convergent. In this paper, we propose a new linearized ADMM-based contraction type algorithms for the general case m ≥ 2. For the proposed algorithm, we prove its convergence via the analytic framework of contractive type methods and we derive a worst-case O(1/t) convergence rate in ergodic sense. Finally, numerical results are reported to demonstrate the effectiveness of the proposed algorithm.

In this paper, we conduct a convergence rate analysis of the augmented Lagrangian method with a practical relative error criterion designed in Eckstein and Silva [Mathematical Programming, 141, 319–348 (2013)] for convex nonlinear programming problems. We show that under a mild local error bound condition, this method admits locally a Q-linear rate of convergence. More importantly, we show that the modulus of the convergence rate is inversely proportional to the penalty parameter. That is, an asymptotically superlinear convergence is obtained if the penalty parameter used in the algorithm is increasing to infinity, or an arbitrarily Q-linear rate of convergence can be guaranteed if the penalty parameter is fixed but it is sufficiently large. Besides, as a byproduct, the convergence, as well as the convergence rate, of the distance from the primal sequence to the solution set of the problem is obtained.

This work analyzes the alternating minimization (AM) method for solving double sparsity constrained minimization problem, where the decision variable vector is split into two blocks. The objective function is a separable smooth function in terms of the two blocks. We analyze the convergence of the method for the non-convex objective function and prove a rate of convergence of the norms of the partial gradient mappings. Then, we establish a non-asymptotic sub-linear rate of convergence under the assumption of convexity and the Lipschitz continuity of the gradient of the objective function. To solve the sub-problems of the AM method, we adopt the so-called iterative thresholding method and study their analytical properties. Finally, some future works are discussed.

In this paper, we focus on the primal-dual hybrid gradient (PDHG) method, which is being widely used to solve a broad spectrum of saddle-point problems. Despite of its wide applications in different areas, the study of inexact versions of PDHG still seems to be in its infancy. We investigate how to design implementable inexactness criteria for solving the subproblems in PDHG scheme so that the convergence of an inexact PDHG can be guaranteed. We propose two specific inexactness criteria and accordingly some inexact PDHG methods for saddle-point problems. The convergence of both inexact PDHG methods is rigorously proved, and their convergence rates are estimated under different scenarios. Moreover, some numerical results on image restoration problems are reported to illustrate the efficiency of the proposed methods.

Modern optimization techniques, such as the steepest descent method, Newton's method, Rosen's gradient projection method, genetic algorithms, etc., have been developed and quickly improved with the progress of digital computers. The steepest descent method and Newton's method are applied efficiently to unconstrained problems. For many engineering problems involving constraints, the genetic algorithm and SUMT¹are applied with relative ease.

Genetic algorithms²have global search characteristics and relatively good convergence rates. Recently, a Successive Zooming Genetic Algorithm (SZGA)^3,4 was introduced that can search the precise optimal solution at any level of desired accuracy. In the case of engineering problems involving an equality constraint, even if good optimization techniques are applied to the constraint problems, a proper constraint range can lead to a more rapid convergence and precise solution.

This study investigated the proper band-width of an equality constraint using the Successive Zooming Genetic Algorithm (SZGA) technique both theoretically and numerically. We were able to find a certain band-width range of the rapid convergence for each problem, and a broad but more general one too.

In view of the premature convergence of particle swarm optimization (PSO) that is often caused by the loss of diversity, an improved cooperative PSO (ICPSO) is proposed. The method can dynamically combine the optimum values of the particles themselves, the global particles and the optimum values in groups, use the current optimization stage to dynamically adjust the shared proportion of information and effectively fuse various reference information, which can obtain superior global and local optimization performance. Additionally, to improve the diversity of the algorithm, a dynamic adjustment method using the grouping coefficient $r$ for the convergence rate is put forward. This method makes the algorithm have a more appropriate convergence rate while improving the convergence precision and enhancing the performance of the algorithm. Finally, the algorithm is used to optimize a neural network. The convergence condition and convergence rate of the algorithm are assessed by theoretical analysis and simulation experiments. The results show that ICPSO has more advantages in its diversity and the adjustment of the convergence rate compared to other related algorithms. Regarding neural network optimization, the training speed and optimization precision of the ICPSO-BP neural network are the highest, which has reached the best and average level of classification accuracy 98.5%, 96.3% for 20 iterations in Iris, and 98.7%, 95.1% in Wine. Its average iteration times score the best in five problems out of six.

The high field limit for the semiconductor Boltzmann equation with Pauli exclusion terms is investigated. The limit problem is shown to have a unique solution for every given density. The proof relies on a linearization procedure together with a continuation argument. The density is finally proven to converge in the high field limit towards the solution of a nonlinear hyperbolic equation.

We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

A MUSCL-like cell-centered finite volume method is proposed to approximate the solution of multi-dimensional steady advection–diffusion equations. The second-order accuracy is provided by an appropriate definition of the diffusive and advective numerical fluxes. The method is based on a least squares reconstruction of the vertex values from cell averages. The slope limiter, which is required to prevent the formation and growth of spurious numerical oscillations, is designed to guarantee that the discrete solution of the nonlinear scheme exists. Several theoretical issues regarding the solvability of the resulting discrete problems are thoroughly discussed. Finally, numerical experiments that validate the effectiveness of the approach are presented.

For systems of hyperbolic conservation laws, a new Glimm functional was recently constructed when the linearly degenerate manifold in each characteristic field is either the whole space or it consists of a finitely many smooth and transversal manifolds of codimension one. This new functional leads to the neat consistency and convergence rate estimation of the Glimm scheme. In this paper, by the motivation of the result in Ref. 2, we show that the corresponding new Glimm functional can be constructed for general systems only under the strict hyperbolicity assumption.

We study the global spatial regularity of solutions of generalized elasto-plastic models with linear hardening on smooth domains. Under natural smoothness assumptions on the data and the boundary we obtain u ∈ L^∞((0, T); H^3/2-δ(Ω)) for the displacements and z ∈ L^∞((0, T); H^1/2-δ(Ω)) for the internal variables. The proof relies on a reflection argument which gives the regularity result in directions normal to the boundary on the basis of tangential regularity results. Based on the regularity results we derive convergence rates for a finite element approximation of the models.

We are concerned with the nonlinear stability of contact waves subject to general perturbations of initial datum in the Cauchy problem for a radiating gas model. The model is represented mathematically as a one-dimensional hyperbolic–elliptic system. It is known that general perturbations of contact discontinuities may generate diffusion waves which evolve and interact with the contact wave. In order to quantify the decay to the contact waves exactly, we need to construct the corresponding diffusion waves explicitly depending on the perturbation of the initial datum. Then, the constructed diffusion waves can be added to the viscous contact wave to form a new combined wave pattern. In this paper, we will show that the new combined wave pattern is nonlinear stable provided that the general perturbation of the initial datum and the strength of the contact wave are suitably small. In particular we give detailed convergence rates using anti-derivative methods and elaborated energy estimates. The work extends the results of Huang, Xin and Yang in [Contact discontinuity with general perturbations for gas motions, Adv. Math. 219 (2008) 1246–1297] for compressible Navier–Stokes equations to a system with much weaker dissipation mechanism.

The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications. Due to the singularity of the logarithmic function, it introduces tremendous difficulties in establishing mathematical theories, as well as in designing and analyzing numerical methods for PDEs with such nonlinearity. Here, we take the logarithmic Schrödinger equation (LogSE) as a prototype model. Instead of regularizing $f(\rho )=ln\rho$ in the LogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE by first regularizing $F(\rho )=\rho ln\rho -\rho$ locally near $\rho =0^{+}$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrödinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0<𝜀\ll 1$. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improves the linear convergence rate of the regularization method in the literature. Error estimates are also presented for solving the ERLogSE by using Lie–Trotter splitting integrators. Numerical results are reported to confirm our error estimates of the LER and of the time-splitting integrators for the ERLogSE. Finally, our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.

We obtain quantitative stochastic homogenization results for Hamilton–Jacobi equations arising in front propagation problems which move in the normal direction with a possible unbounded velocity. More precisely, we establish error estimates and rates of convergence for homogenization and effective Hamiltonian. The main idea is to perturb our unbounded problem by a bounded one, and to establish stability results in this context. Then, we combine the estimates that we find with the ones from the bounded case.

It is well known that as the time interval between two consecutive observations shrinks to zero, a properly constructed GARCH model will weakly converge to a bivariate diffusion. Naturally the European option price under the GARCH model will also converge to its bivariate diffusion counterpart. This paper investigates the convergence speed of the GARCH option price. We show that the European option prices under the two corresponding models are equal up to an order near the square root of the length of discrete time interval.

The Longstaff–Schwartz (LS) algorithm is a popular least square Monte Carlo method for American option pricing. We prove that the mean squared sample error of the LS algorithm with quasi-regression is equal to $c_1/N$ asymptotically,^a where $c_1>0$ is a constant, $N$ is the number of simulated paths. We suggest that the quasi-regression based LS algorithm should be preferred whenever applicable. Juneja & Kalra (2009) and Bolia & Juneja (2005) added control variates to the LS algorithm. We prove that the mean squared sample error of their algorithm with quasi-regression is equal to $c_2/N$ asymptotically, where $c_2>0$ is a constant and show that $c_2<c_1$ under mild conditions. We revisit the method of proof contained in Clément et al. [E. Clément, D. Lamberton & P. Protter (2002) An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6 449–471], but had to complete it, because of a small gap in their proof, which we also document in this paper.