Simple Iterative Method Research Articles

Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra

Quantum simulation is capable of simulating certain dynamical systems in continuous time—Schrödinger’s equations being the most direct and well known—more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger’s equations via a method called Schrödingerisation (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.

A Simple Iterative Method for Dynamic Analysis of MDOF Systems with Arbitrary Time-Varying Coefficients Based on Successive Differentiation

A Simple Iterative Method for Dynamic Analysis of MDOF Systems with Arbitrary Time-Varying Coefficients Based on Successive Differentiation

Решение оптимизационной задачи оценивания моделей полносвязной линейной регрессии

<p>This article is devoted to the problem of estimating fully connected linear regression models using the maximum likelihood method. Previously, a special numerical method was developed for this purpose, based on solving a nonlinear system using the method of simple iterations. At the same time, the issues of choosing initial approximations and fulfilling sufficient conditions for convergence were not studied. This article proposes a new method for solving the optimization problem of estimating fully connected regressions, similar to the method of estimating orthogonal regressions. It has been proven that, with equal error variances of interconnected variables, estimates of b-parameters of fully connected regression are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse covariance matrix. And if the ratios of the error variances of the variables are equal to the ratios of the variances of the variables, then the b-parameter estimates are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse correlation matrix, multiplied by the specific ratios of the standard deviations of the variables. A numerical experiment was carried out to confirm the correctness of the developed mathematical apparatus. The proposed method for solving the optimization problem of estimating fully connected regressions can be effectively used when solving problems of constructing multiple fully connected linear regressions.</p>

On the External Estimation of Reachable and Null-Controllable Limit Sets for Linear Discrete-Time Systems with a Summary Constraint on the Scalar Control

The problem of constructing reachable and null-controllable sets for stationary linear discrete-time systems with a summary constraint on the scalar control is considered. For the case of quadratic constraints and a diagonalizable matrix of the system, these sets are built explicitly in the form of ellipsoids. In the general case, the limit reachable and null-controllable sets are represented as fixed points of a contraction mapping in the metric space of compact sets. On the basis of the method of simple iteration, a convergent procedure for constructing their external estimates with an indication of the a priori approximation error is proposed. Examples are given.

A Block Hybrid Method with Equally Spaced Grid Points for Third-Order Initial Value Problems

In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature.

AN EFFICIENT ALGORITHM FOR THE NUMERICAL SOLUTION OF A THREE-DIMENSIONAL THERMAL CONDUCTIVITY ISSUE

A mathematical model of slab heating based on a numerical solution of a three-dimensional thermal conductivity problem has been created and programmatically implemented in the work. To solve a system of difference equations, a layer-by-layer method is proposed that allows using the run-through method to solve a system of difference equations of a three-dimensional problem within a rapidly converging iterative procedure. Comparative calculations of slab heating are carried out using the proposed layer-by-layer method and the method of simple iteration with different degrees of grinding of the spatial grid. As a result, it was found that with the grinding of the mesh, the effectiveness of the layered method in relation to the simple iteration method increases.

Generation a solution to the equations of elasticity theory for a layered strip basing on the principle of compressed mappings

A systematic presentation of the modified classical semi-inverse SaintVenant method as an iterative one is given on the example of generating a solution to the differential equations of elasticity theory for a long layered strip. The firstorder differential equations of the plane problem are reduced to the dimensionless form and replaced by integral equations with respect to the transverse coordinate, just as it is done in the Picard method of simple iterations. In this case, a small parameter appears in the integral equations before the integral sign as a multiplying factor, which is used to ensure convergence of solutions in accordance with the Banach’s principle of compressed mappings. The equations and elasticity relations are converted to a form that enables to calculate the unknowns consecutively, so that the unknowns being calculated in one equation are the inputs for the next equation, and etc. Fulfillment of the boundary conditions at the long edges leads to ordinary differential equations for slowly and rapidly changing singular components of the solution with sixteen effective stiffness coefficients that are defined by integrals from the given ones as a stepped function of Young's moduli for each layer. Integrating of these ordinary differential equations makes it possible to obtain the formulas for all the required unknowns of the problem, including transverse stresses that are not defined in the classical theory of the beam and solutions of the edge effect type, and to fulfill all the boundary conditions for the elasticity theory problem. The solution of three boundary value problems of the strip elasticity theory is provided such as for a two-layer strip with layers of the same thickness and different thicknesses, and a strip with an arbitrary number of layers. Formulas for all unknowns of the problem are obtained.

Magnetic feature and regression analysis of Reiner-Philippoff boundary layer flow

The design and optimization of magnetic fluid-based processes, such as magnetohydrodynamic power generation and magnetic drug targeting, have become significant in recent times. To understand the magnetic and radiation features, as well as the sensitivity of engineering physical quantities to each dimensionless parameter in fluid flow, the Reiner-Philippoff (RP) fluid is best suited in this instance, as it exhibits shear-thickening, shear-thinning, and Newtonian behavior. The non-similar transformation is used to transform the partial differential equations that describe the flow into two-variable differential equations. The transformed dynamical equations are solved numerically using a spectral-based numerical technique; namely, the bivariate simple iteration method (BSIM). The effects of magnetic field strength and radiation parameter on the stretching and shrinking sheets are examined. The investigation reveals that both the magnetic field strength and radiation parameter have a significant impact on the flow behavior and rate of heat transfer. For the radiation parameter R∈[0,0.5], the skin friction coefficient and Nusselt number increase by 22.6% and 5.98%, respectively. Additionally, an 100% increment in Prandtl parameter reduces the Nusselt number by 287.36%. Regression analysis is performed to identify the most significant parameter affecting the engineering quantities. The results show that the Bingham constant, γ, is the only parameter that is not significant on the skin friction coefficient, while all the parameters have significant impact on the Nusselt number. The findings in this study have important implications in the biomedical industry, the design of machines, and the fourth industrial revolution.

Efficient Feature Reconstruction via l 2,1-Norm Regularization for Few-Shot Classification

Few-shot classification, which aims to recognize novel classes with the aid of learned knowledge and a few labeled samples, remains challenging and draws emerging concerns in computer vision. Metric-based methods incorporate the idea of metric learning into tackling the few-shot classification problem, i.e., the class membership of query samples can be determined in the latent space. Several recent methods propose to infer such memberships by feature reconstruction, which, however either exploit attention mechanisms or closed-form solutions. In this paper, we revisit the essence of feature reconstruction applied to few-shot classification and introduce a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> -norm regularization to constraint it. As with the intention of attention mechanisms, we attempt to guide feature reconstruction towards focusing more on semantically rich target regions and diminishing the contribution of profitless features. We thereby obtain more discriminative query reconstructions for each class and perform classification based on these reconstruction errors. We formulate the proposal as a constrained optimization problem and design a simple and efficient iterative method as a solution. Furthermore, we provide the analysis on the convergence of proposed iterative method in theory. We conduct few-shot classification on two fine-grained and two general datasets. Extensive experimental results reveal that our method achieves new state-of-the-art performance.

Solving quasimonotone and non-monotone variational inequalities

AbstractWe present a simple iterative method for solving quasimonotone as well as classical variational inequalities without monotonicity. Strong convergence analysis is given under mild conditions and thus generalize the few existing results that only present weak convergence methods under restrictive assumptions. We give finite and infinite dimensional numerical examples to compare and illustrate the simplicity and computational advantages of the proposed scheme.

Integral Method for Determining the Stress State of an Anisotropic Reinforced Concrete Shell

The article deals with the issue of estimating the strength and stress-strain state of a reinforced concrete shell by the method of membrane theory and the theory of infinitesimal bending. A brief description of the methodology consists in the fact that the ground stress state of the shell is allocated to an independent problem, when, without introducing boundary effects, two of the four boundary conditions of the general moment theory are distinguished, which, together with the equations of the momentless theory, determine the ground stress state, and then boundary effects are superimposed. The equilibrium equations of the moment theory in forces and moments, geometric equations, components of tangential deformation and displacement, and the physical equations of state connecting them, expressing forces and moments through the components of deformation, are presented, since the characteristics of the stress-strain state of the shell depend not only on the variability of external influences and forces, but also on the length of the structure. The provisions laid down in the article shall be retained in the case of casings made of anisotropic material, provided that the presented elastic ratio is met. A system of equations of the generalized semi-membrane (semi-bending) state of an arbitrary shell of zero curvature, which determines the accuracy of this approach, is solved. The equations of the semi-torque theory for a circular cylindrical shell are presented, as well as describing the semi-membrane stress state of a long shell of zero curvature. The most important result of the study is the method of constructing integrals of the ground stress state of the shell based on the method of simple iterations, which expands the possibilities of the reliability theory, which makes it possible to build the foundations for the practical calculation of reinforced concrete shells according to the membrane theory and the theory of infinitesimal bending.

Complex Three-Dimensional Mathematical Model of the Ignition of a Coniferous Tree via a Cloud-to-Ground Lightning Discharge: Electrophysical, Thermophysical and Physico-Chemical Processes

Thunderstorms are the main natural source of forest fires. The ignition mechanism of trees begins with the impact of cloud-to-ground lightning discharge. A common drawback of all predicting systems is that they ignore the physical mechanism of forest fire as a result of thunderstorm activity. The purpose of this article is to develop a physically based mathematical model for the ignition of a coniferous tree via cloud-to-ground lightning discharge, taking into account thermophysical, electrophysical, and physicochemical processes. The novelty of the article is explained by the development of an improved mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge, taking into account the processes of soot formation caused by the thermal decomposition phase of dry organic matter. Mathematically, the process of tree ignition is described by a system of non-stationary nonlinear differential equations of heat conduction and diffusion. In this research, a locally one-dimensional method is used to solve three-dimensional partial differential equations. The finite difference method is used to solve one-dimensional heat conduction and diffusion equations. Difference analogues of the equations are solved using the marching method. To resolve nonlinearity, a simple iteration method is used. Temperature distributions in a structurally inhomogeneous trunk of a coniferous tree, as well as distributions of volume fractions of phases and concentrations of gas mixture components, are obtained. The conditions for tree trunk ignition under conditions of thunderstorm activity are determined. As a result, a complex three-dimensional mathematical model is developed, which makes it possible to identify the conditions for the ignition of a coniferous tree trunk via cloud-to-ground lightning discharge.

A Readily Achieved Potentiostatic Method in Density Functional Theory Calculation for Improved Prediction of the Performance for Electrocatalytic Nitrogen Reduction Reaction.

Accurate prediction of the catalytic performance of nitrogen reduction reaction catalysts based on density functional theory (DFT) calculation is of great significance for developing catalytic materials for nitrogen fixation. However, the applied electrode potential induced the fixation of Fermi level and solvation effect are commonly ignored in the current computational hydrogen electrode method, which leads to the large deviation between the calculation predicted limit potential and the experimentally measured limit potential. In this work, the simple external iteration method is proposed to simulate the Fermi level of the catalysts that are fixed by the applied electrode potential, along with the hybrid solvent model to describe the strong interaction, such as hydrogen bond, between the solvent molecules and the intermediates. This method allowed the theoretical and experimental limit potentials to be in good agreement, indicating the significant effect of the electrode potential and solvation in the DFT calculation. These results will guide the calculation-based prediction of other reaction systems in the field of electrocatalysis.

Inferring the Nominal Molecular Mass of an Analyte from Its Electron Ionization Mass Spectrum.

The performance of three algorithms for predicting nominal molecular mass from an analyte's electron ionization mass spectrum is presented. The Peak Interpretation Method (PIM) attempts to quantify the likelihood that a molecular ion peak is contained in the mass spectrum, whereas the Simple Search Hitlist Method (SS-HM) and iterative Hybrid Search Hitlist Method (iHS-HM) leverage results from mass spectral library searching. These predictions can be employed in combination (recommended) or independently. The methods were tested on two sets of query mass spectra searched against libraries that did not contain the reference mass spectra of the same compounds: 19,074 spectra of various organic molecules searched against the NIST17 mass spectral library and 162 spectra of small molecule drugs searched against SWGDRUG version 3.3. Individually, each molecular mass prediction method had computed precisions (the fraction of positive predictions that were correct) of 91, 89, and 74%, respectively. The methods become more valuable when predictions are taken together. When all three predictions were identical, which occurred in 33% of the test cases, the predicted molecular mass was almost always correct (>99%).

Methods of numerical analysis for some integral dynamical systems with delay arguments

The aim of this work is to construct direct and iterative numerical methods for solving functional equations with hereditary components. Such equations are a convenient tool for modeling dynamical systems. In particular, they are used in population models structured by age with a finite life span. Models based on integro-differential and integral equations with various kinds of delay arguments are considered. For nonlinear equations, the operators are linearized according to the modified Newton-Kantorovich scheme. Direct quadrature and simple iteration methods are used to discretize linear equations. These methods are constructed in the paper: an iterative method for solving a nonlinear integro-differential equation on the semiaxis (−∞,0] , a direct method for solving the signal recovery problem, and iterative methods for solving a nonlinear Volterra integral equation with a constant delay. Special quadrature formulas based on orthogonal Lagger polynomials are used to approximate improper integrals on the semiaxis. The results of numerical experiments confirm the convergence of suggested methods. The proposed approaches can also be applied to other classes of nonlinear equations with delays.

A New Quotient and Iterative Detection Method in an Affine Krylov Subspace for Solving Eigenvalue Problems

The paper solves the eigenvalues of a symmetric matrix by using three novel algorithms developed in the m-dimensional affine Krylov subspace. The n-dimensional eigenvector is superposed by a constant shifting vector and an m-vector. In the first algorithm, the m-vector is derived as a function of eigenvalue by maximizing the Rayleigh quotient to generate the first characteristic equation, which, however, is not easy to determine the eigenvalues since its roots are not of simple ones, exhibiting turning points, spikes, and even no intersecting point to the zero line. To overcome that difficulty by the first algorithm, we propose the second characteristic equation through a new quotient with the inner product of the shifting vector to the eigen-equation. The Newton method and the fictitious time integration method are convergent very fast due to the simple roots of the second characteristic equation. For both symmetric and nonsymmetric eigenvalue problems solved by the third algorithm, we develop a simple iterative detection method to maximize the Euclidean norm of the eigenvector in terms of the eigen-parameter, of which the peaks of the response curve correspond to the eigenvalues. Through a few finer tunings to the smaller intervals sequentially, a very accurate eigenvalue and eigenvector can be obtained. The efficiency and accuracy of the proposed iterative algorithms are verified and compared to the Lanczos algorithm and the Rayleigh quotient iteration method.

An Iterative Method Based on a 3-DOF Model for Optimal Seismic Design of Coal-Fired Thermal Power Plant

AbstractAs a key infrastructure in the lifeline system, the safety of the Coal-Fired Thermal Power Plants (CFTPPs) under earthquake loading is not properly considered since a systematic optimization method for seismic design of CFTPPs is scarce. This paper presents a simple iterative method for globally optimizing the dynamic parameters of the typical CFTPPs, which can be considered a special type of tuned mass damping. In this paper, the fixed-point theory is first applied to a 2-DOF model of an actual CFTPP as a benchmark. Because the optimized parameters obtained from the fixed-point theory are roughly and cannot be used in practice, an iterative method based on the frequency domain analysis of a 3-DOF model of the CFTPP is then presented and used to obtain more detailed and optimized parameters, where the basic idea is to make extremum values of frequency response curve as small as possible. To show the potential of the proposed method, an illustrative example is introduced, and the results show that the presented method is effective in alleviating the seismic responses of CFTPPs. Compared to the classic fixed-point theory, the presented iterative method can handle multi-DOF models under different conditions but needs no complicated calculations, and this makes it possible to lead to a more efficient and precise seismic design of CFTPPs.

Numerical simulation of nanofluid flow due to a stretchable rotating disk

In this study, a steady magnetohydrodynamic (MHD) flow due to stretchable rotating disk in the presence of gyrotactic microorganisms is investigated. The governing equations modeling the flow are solved numerically using the newly introduced simple iteration method (SIM) that seeks to linearize a system using relaxation technique that effectively decouples the system. To verify the convergence and accuracy of the method, solution error and residual error analysis are carried out, respectively. The obtained results suggest that the SIM is a highly efficient method that produces convergent and highly accurate solutions. The effects of various parameters as well as combined parameter effects on the solution profiles are also investigated. An increase in the Hall and permeability parameters leads to a corresponding rise in the microorganism?s density and nanoparticle volume fraction.

Effective algorithms for solving functional equations with superposition on the example of the Feigenbaum equation

Purpose. New algorithms were consider for functional equations solving using the Feigenbaum equation as an example. This equation is of great interest in the theory of deterministic chaos and is a good illustrative example in the class of functional equations with superposition. Methods. The article proposes three new effective methods for solving functional equations — the method of successive approximations, the method of successive approximations using the fast Fourier transform and the numerical-analytical method using a small parameter. Results. Three new methods for solving functional equations were presented, considered on the example of the Feigenbaum equation. For each of them, the features of their application were investigated, as well as the complexity of the resulting algorithms was estimated. The methods previously used by researchers to solve functional equations are compared with those described in this article. In the description of the latter, the numerical-analytical method, several coefficients of expansions of the universal Feigenbaum constants were written out. Conclusion. The obtained algorithms, based on simple iteration methods, allow solving functional equations with superposition without the need to reverse the Jacobi matrix. This feature greatly simplifies the use of computer memory and gives a gain in the operating time of the algorithms in question, compared with previously used ones. Also, the latter, numerically-analytical method made it possible to obtain sequentially the coefficients of expansions of the universal Feigenbaum constants, which in fact can be an analytical representation of these constants.

Jumpdiff: A Python Library for Statistical Inference of Jump-Diffusion Processes in Observational or Experimental Data Sets

We introduce a Python library, called jumpdiff, which includes all necessary functions to assess jump-diffusion processes. This library includes functions which compute a set of non-parametric estimators of all contributions composing a jump-diffusion process, namely the drift, the diffusion, and the stochastic jump strengths. Having a set of measurements from a jump-diffusion process, jumpdiff is able to retrieve the evolution equation producing data series statistically equivalent to the series of measurements. The back-end calculations are based on second-order corrections of the conditional moments expressed from the series of Kramers-Moyal coefficients. Additionally, the library is also able to test if stochastic jump contributions are present in the dynamics underlying a set of measurements. Finally, we introduce a simple iterative method for deriving secondorder corrections of any Kramers-Moyal coefficient.