The method of simple iterations with correction of convergence and the minimal discrepancy method for plasmonic problems

The problem of searching for complex roots of the dispersion equations of plasmon-polaritons along the boundaries of the layered structure-vacuum interface is considered. Such problems arise when determining proper waves along the interface of structures supporting surface and leakage waves, including plasmons and polaritons along metal, dielectric and other surfaces. For the numerical solution of the problem, we consider a modification of the method of simple iterations with a variable iteration parameter leading to a zero derivative of the right side of the equation at each step, i.e. convergent iterations, as well as a modification of the minimum residuals method. It is shown that the method of minimal residuals with linearization coincides with the method of simple iterations with the specified correction. Convergent methods of higher orders are considered. The results are demonstrated by examples, including complex solutions of dispersion equations for plasmon-polaritons. The advantage of the method over other methods of searching for complex roots in electrodynamics problems is the possibility of ordering the roots and constructing dispersion branches without discontinuities. This allows you to classify modes.

On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

There are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

ParaX

Despite the fact that GPUs and accelerators are more efficient in deep learning (DL), commercial clouds like Facebook and Amazon now heavily use CPUs in DL computation because there are large numbers of CPUs which would otherwise sit idle during off-peak periods. Following the trend, CPU vendors have not only released high-performance many-core CPUs but also developed efficient math kernel libraries. However, current DL platforms cannot scale well to a large number of CPU cores, making many-core CPUs inefficient in DL computation. We analyze the memory access patterns of various layers and identify the root cause of the low scalability, i.e., the per-layer barriers that are implicitly imposed by current platforms which assign one single instance (i.e., one batch of input data) to a CPU. The barriers cause severe memory bandwidth contention and CPU starvation in the access-intensive layers (like activation and BN). This paper presents a novel approach called ParaX, which boosts the performance of DL on many-core CPUs by effectively alleviating bandwidth contention and CPU starvation. Our key idea is to assign one instance to each CPU core instead of to the entire CPU, so as to remove the per-layer barriers on the executions of the many cores. ParaX designs an ultralight scheduling policy which sufficiently overlaps the access-intensive layers with the compute-intensive ones to avoid contention, and proposes a NUMA-aware gradient server mechanism for training which leverages shared memory to substantially reduce the overhead of per-iteration parameter synchronization. We have implemented ParaX on MXNet. Extensive evaluation on a two-NUMA Intel 8280 CPU shows that ParaX significantly improves the training/inference throughput for all tested models (for image recognition and natural language processing) by 1.73X ~ 2.93X.

Application of Whale Optimization Algorithm (WOA) on Quadratic Knapsack 0-1 Problem

<span lang="EN">Quadratic Knapsack Problem is a variation of the knapsack problem that aims to maximize an objective function. The objective function in this case is quadratic. While the constraints used are binary and linear capacity constraints. The Whale Optimization Algorithm is a metaheuristic algorithm that can solve this problem. Therefore, this paper aims to find out the best solution to solve the Knapsack 0-1 Quadratic Problem using the Whale Optimization Algorithm so that its effectiveness and efficiency are known. Based on the research has been done, the algorithm is said to be effective because, from each experiment, the algorithm is always converging or towards maximum profit. Also, with the right parameters, the algorithm can achieve optimal results. It is said to be efficient because getting optimal profit does not require more time and iteration. The combination of item parameters and maximum iteration dramatically Affect the total value of profit and its running time. However, the addition of item parameter combinations is faster to achieve optimal than the maximum iteration parameter.</span>

ON INCOMPLETE FACTORIZATION IMPLICIT TECHNIQUE FOR 2D ELLIPTIC FD EQUATIONS

A new variant of Incomplete Factorization Implicit (IFI) iterative technique for 2D elliptic finite-difference (FD) equations is suggested which is differed by applying the matrix tridiagonal algorithm. Its iteration parameter is shown be linked with the one for Alternating Direction Implicit method. An effective set of values for the parameter is suggested. A procedure for enhancing the set of iteration parameters for IFI is proposed. The technique is applied to a 5-point FD scheme, and to a 9-point FD scheme. It is suggested applying the solver for 5-point scheme to solving boundary-value problems for the 9-point scheme, too. The results of numerical experiment with Dirichlet and Neumann boundary-value problems for Poisson equation in a rectangle, and in a quasi-circle are presented. Mixed boundary-value problems in square are considered, too. The effectiveness of IFI is high, and weakly depends on the type of boundary conditions.

Research on Springback Control in Stretch Bending Based on Iterative Compensation Method

An unavoidable problem in the stretch-bending process is springback, which dictates the shape and dimensional accuracy of the product. This problem can be solved by adjusting the geometry of the die or through active process control. This study focuses on the design of the die shape to achieve the target product. Based on the fixed-point iterative method and displacement adjustment (DA) method, this paper proposes an iterative compensation method, which has a higher convergence rate, lower number of iterations, and higher precision compared to the DA method with only one control parameter. In addition, like the DA method, the proposed method does not depend on the material properties or mechanical model, but the difference is that it can quickly and effectively find out the iteration parameter, determine whether the parameter has convergence or not, and has no compensation factor. According to the deviation of iterative parameters between the value after stretch bending and the target value, the iterative compensation method can be used to calculate the compensation magnitude and compensation direction of the iterative parameter. For stretch-bending processes with invariable- and variable-curvature die shapes, the convergence of control parameters is verified mathematically with the convergence theorem of the method, and experiments are conducted to verify the iterative compensation method. The experimental results show that the target products can be obtained with a small number of iterations without knowing the specific material properties.

Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems

We discuss an efficient preconditioner and iterative numerical method to solve large complex linear algebraic systems of the form (W + iT)u = c, where W and T are symmetric matrices, and at least one of them is nonsingular. When the real part W is dominantly stronger or weaker than the imaginary part T, we propose a block multiplicative (BM) preconditioner or its variant (VBM), respectively. The BM and VBM preconditioned iteration methods are shown to be parameter-free, in terms of eigenvalue distributions of the preconditioned matrix. Furthermore, when the relationship between W and T is obscure, we propose a new preconditioned BM method (PBM) to overcome this difficulty. Both convergent properties of these new iteration methods and spectral properties of the corresponding preconditioned matrices are discussed. The optimal value of iteration parameter for the PBM method is determined. Numerical experiments involving the Helmholtz equation and some other applications show the effectiveness and robustness of the proposed preconditioners and corresponding iterative methods.

Compressive sensing for noisy solder joint imagery based on convex optimization

Purpose The purpose of this paper is to develop a compressive sensing (CS) algorithm for noisy solder joint imagery compression and recovery. A fast gradient-based compressive sensing (FGbCS) approach is proposed based on the convex optimization. The proposed algorithm is able to improve performance in terms of peak signal noise ratio (PSNR) and computational cost. Design/methodology/approach Unlike traditional CS methods, the authors first transformed a noise solder joint image to a sparse signal by a discrete cosine transform (DCT), so that the reconstruction of noisy solder joint imagery is changed to a convex optimization problem. Then, a so-called gradient-based method is utilized for solving the problem. To improve the method efficiency, the authors assume the problem to be convex with the Lipschitz gradient through the replacement of an iteration parameter by the Lipschitz constant. Moreover, a FGbCS algorithm is proposed to recover the noisy solder joint imagery under different parameters. Findings Experiments reveal that the proposed algorithm can achieve better results on PNSR with fewer computational costs than classical algorithms like Orthogonal Matching Pursuit (OMP), Greedy Basis Pursuit (GBP), Subspace Pursuit (SP), Compressive Sampling Matching Pursuit (CoSaMP) and Iterative Re-weighted Least Squares (IRLS). Convergence of the proposed algorithm is with a faster rate O(k*k) instead of O(1/k). Practical implications This paper provides a novel methodology for the CS of noisy solder joint imagery, and the proposed algorithm can also be used in other imagery compression and recovery. Originality/value According to the CS theory, a sparse or compressible signal can be represented by a fewer number of bases than those required by the Nyquist theorem. The new development might provide some fundamental guidelines for noisy imagery compression and recovering.

Numerical simulation of magnetohydrodynamic micropolar fluid flow and heat transfer in a channel with shrinking walls

We numerically study the steady hydromagnetic (magnetohydrodynamic) flow and heat transfer characteristics of a viscous incompressible electrically conducting micropolar fluid in a channel with shrinking walls. Unlike the classical shooting methodology, two distinct numerical techniques are employed to solve the transformed self-similar nonlinear ordinary differential equations (ODEs). One is the combination of a direct and an iterative method (successive over-relaxation with optimal relaxation parameter) for solving the sparse system of linear algebraic equations arising from the finite difference discretization of the linearised ODEs. For the second one, a pseudotransient method is used where time plays the role of an iteration parameter until the steady state is reached. The two approaches may be easily extended to other geometries (for example, sheets, disks, and cylinders) with possible wall conditions like slip, stretching, rotation, suction, and injection. Effects of some physical parameters on the flow and heat transfer are discussed and presented through tables and graphs. Detailed description of the computational procedure and the results of the study may be beneficial for the researchers in the flow and thermal control of polymeric processing.