Methods For Systems Of Equations Research Articles

Correction: A Bregman–Kaczmarz method for nonlinear systems of equations

Correction: A Bregman–Kaczmarz method for nonlinear systems of equations

Методы расчета растекания опасных и загрязняющих веществ по рельефу

The study considers and compares several methods of calculation of the process of stable liquid spread over a solid impermeable surface: the SPH method and the VOF method for the three-dimensional system of equations; the Lax-Wendroff method and the finite volume method for the two-dimensional «shallow water» system of equations. The «shallow water» systems of equations are obtained by averaging the full system of equations over depth assuming the absence of a vertical velocity component. The procedure of numerical calculation by the Lax-Wendroff method is provided in detail. The emphasis is made on considering shallow water approximation as such approximation enables obtaining a three-dimensional flow pattern in the framework of a two-dimensional model. The «shallow water» system of equations is provided taking into account the action of friction forces. All four calculation methods have been verified by using the data of a water dam breaking experiment in a 122-meter canal. The experiments recorded the dynamics of liquid propagation and water rise at different locations and moments. The comparison implies that the SPH method requires significant computational costs due to the need to specify a large number of particles in order to ensure acceptable accuracy. The VOF calculation method blurs the air-water interface, which hampers the determination of spill configuration and the free surface position. The dam breaking problem has been solved most accurately by two methods, namely the Lax-Wendroff method and the finite volume method, which has been confirmed by the quantitative assessment of error in calculations by various methods. It has been demonstrated that the shallow water approximation implemented by the Lax-Wendroff numerical method used in the study ensures accuracy sufficient to solve problems of spill over the terrain surface.

A class of pseudoinverse-free greedy block nonlinear Kaczmarz methods for nonlinear systems of equations

<abstract><p>In this research, we constructed a class of nonlinear greedy average block Kaczmarz methods to solve nonlinear problems without computing the Moore-Penrose pseudoinverse of the Jacobian matrix. These kinds of methods adopt the average technique of the Gaussian Kaczmarz method and combine the greedy strategy, which greatly reduces the amount of computation. The local convergence analysis and numerical experiments of the proposed methods are given. The numerical results show the effectiveness of the proposed methods.</p></abstract>

A Bregman–Kaczmarz method for nonlinear systems of equations

A Bregman–Kaczmarz method for nonlinear systems of equations

A priori error estimates of a three-step two-level finite element Galerkin method for a 2D-Boussinesq system of equations

A priori error estimates of a three-step two-level finite element Galerkin method for a 2D-Boussinesq system of equations

Inexact free derivative quasi-Newton method for large-scale nonlinear system of equations

In this work, we propose a free derivative quasi-Newton method for solving large-scale nonlinear systems of equation. We introduce a two-stage linear search direction and develop its global convergence theory. Besides, we prove that the method enjoys superlinear convergence rate. Finally, numerical experiments illustrate that the proposed method is competitive with respect to Newton-Krylov methods and other well-known methods for solving large-scale nonlinear systems of equations.

On the convergence of iterative schemes for solving a piecewise linear system of equations

This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first provide a negative answer via a counterexample to a conjecture on the global and finite convergence of the Newton iteration for symmetric and positive definite matrices. Additionally, we discuss some surprising features of the semi-smooth Newton iteration in low dimensions and its behavior in higher dimensions. Secondly, we present two iterative schemes inspired by the classical Jacobi and Gauss-Seidel methods for linear systems of equations for finding a solution to the problem. We study sufficient conditions for the convergence of both proposed procedures, which are also sufficient for the existence and uniqueness of solutions to the problem. Lastly, we perform some computational experiments designed to illustrate the behavior (in terms of CPU time) of the proposed iterations versus the semi-smooth Newton method for dense and sparse large-scale problems. Moreover, we included the numerical solution of a discretization of the Boussinesq PDE modeling a two-dimensional flow in a homogeneous phreatic aquifer.

Single step iterative method for linear system of equations with complex symmetric positive semi-definite coefficient matrices

In this study, we propose a new single-step iterative method for solving complex linear systems Az≡(W+iT)z=f, where z,f∈Rn, W∈Rn×n and T∈Rn×n are symmetric positive semi-definite matrices such that null(W)∩null(T)={0}. The convergence of the new method is analyzed in detail and discussion on the obtaining the optimal parameter is given. From Wang et al. (2017)[36] we can write W=PTDWP,T=PTDTP, where DW=Diag(μ1,…,μn),DT=Diag(λ1,…,λn), and P∈Rn×n is a nonsingular matrix and λk, μk satisfy μk+λk=1,0≤λk,μk≤1,k=1,…,n. Then we show that under some conditions on μmax=max{μk}k=1n, the new method has faster convergence rate in comparison with recently introduced methods. Finally, some numerical examples are given to demonstrate the efficiency of the new procedure in actual computation.

Adjoint shape optimization of a duct for a wall jet film cooling setup

The paper presents the results of optimization of the geometric parameters of the simplified wall jet cooling system using a modified Adjoint Shape optimization method for algebraic systems of equations (Discrete Adjoint Optimization). The modification consists in using a linearized discrete system of equations with the replacement of derivatives by their finite-volume approximations. The jet flowed through a duct and out from a nozzle. The duct was inclined at an angle of 35 degrees to the cooled wall. The mean velocity ratio between the jet and the main flow was set to 2. The total heat flux on the cooled wall was taken as a cost function. The problem was considered in a two-dimensional stationary turbulent formulation (RANS). As a result of optimization, the shape of the duct changed significantly, affecting the flow inside it. The optimization led to the disappearance of the recirculation zone and reattaching of the jet to the cooled wall. As a result of the optimization performed, the heat flux at the wall increased by 20%.

On the order of convergence of Broyden\u2019s method

We present two theoretical results and two surprising conjectures concerning convergence properties of Broyden’s method for smooth nonlinear systems of equations. First, we show that when Broyden’s method is applied to a nonlinear mapping F:mathbb {R}^nrightarrow mathbb {R}^n with n-d affine component functions and the initial matrix B_0 is chosen suitably, then the generated sequence (u^k,F(u^k),B_k)_{kge 1} can be identified with a lower-dimensional sequence that is also generated by Broyden’s method. This property enables us to prove, second, that for such mixed linear–nonlinear systems of equations a proper choice of B_0 ensures 2d-step q-quadratic convergence, which improves upon the previously known 2n steps. Numerical experiments of high precision confirm the faster convergence and show that it is not available if B_0 deviates from the correct choice. In addition, the experiments suggest two surprising possibilities: It seems that Broyden’s method is (2d-1)-step q-quadratically convergent for d>1 and that it admits a q-order of convergence of 2^{1/(2d)}. These conjectures are new even for d=n.

Improved conjugate gradient method for nonlinear system of equations

In this paper, we propose a hybrid conjugate gradient (CG) method based on the approach of convex combination of Fletcher–Reeves (FR) and Polak–Ribiere–Polyak (PRP) parameters, and Quasi-Newton’s update. This is made possible by using self-scaling memory-less Broyden’s update together with a hybrid direction consisting of two CG parameters. However, an important property of the new algorithm is that, it generates a descent search direction via non-monotone type line search. The global convergence of the algorithm is established under appropriate conditions. Finally, numerical experiments on some benchmark test problems, demonstrate the effectiveness of the proposed algorithm over some existing alternatives.

Mathematical Model of Temperature Changes in Heating Media

Climate conditions on the territory of Russia require heating of residential, public and industrial buildings. The regional variety of temperature modes distinguishes it from most countries of the world. Heat supply in terms of primary fuel and energy resources consumption is the largest segment in the country’s energy supply. Russia ranks first in the world in terms of the scale of heat supply: the volume of heat production, the development of heating, the length of heat networks, and fuel consumption for heat production. Development of modern, energy-efficient heat generating plants is required. Heat supply to agriculture in the Russian Federation lags behind the developed countries by 2-3 times. Various methods and equipment are being developed to reduce energy consumption in thermal processes. The most important task of developing and modernizing energy system and electrifying agriculture is to increase the energy efficiency of energy supply systems based on rational and reliable energy supply. A promising solution in the field of energy saving in agriculture is the use of heat pumps. (Research purpose) The research purpose is in creating a mathematical model of temperature changes in heat carriers. (Materials and methods) Authors used the Newton method for nonlinear systems of equations. The article presents the derivatives of the obtained functions of the system. (Results and discussion) The article presents the graph of the temperature distribution during heat transfer and a graph of changes in the temperature of heat carriers along the heat exchange surface. (Conclusions) The article presents a mathematical model suitable for studying the temperature distribution during heat transfer between two or more heat carriers, which can be recommended for engineering calculations. The convergence of the tangent method is achieved in no more than six iterations, regardless of the initial conditions; the accuracy of calculations is 0.1 percent. A static error is of 5-10 percent in mathematical modeling.

A space time conservation element and solution element method for solving two-species chemotaxis model

This work is related to the numerical investigation of two species chemotaxis models in both one and two dimensions. This system in its simpler form is a set of non-linear partial differential equations. First equation represents the dynamics of cell densities and the others are responsible for chemoattractant concentration. The system is known to produce delta-type singularities in finite time. Therefore, a less order accurate numerical schemes are not capable of handling such a complicated spiky solution. A Space Time Conservation Element and Solution Element (CE/SE) method is proposed for solving such systems. This method has distinct attributes which includes treatment of space and time variables in unified fashion. Furthermore, numerical diffusion is reduced on the basis that both conserved quantities and derivatives are anonymous. Due to this feature, diffusion in numerical schemes is inherently reduced. The Nessyahu–Tadmor (NT) central scheme is also implemented for validation and comparison. Several one and two dimensional case studies are carried out. Numerical results obtained through proposed numerical method are analyzed in comparison to Nessyahu–Tadmor (NT) central scheme. Moreover, effects of the different densities and concentration functions are explored to see generic applicability of the proposed method for current system of equations. It is observed that the CE/SE and NT central method has the capability to capture the delta type singularities in the solution profile. However, CE/SE resolves the solution peaks better as compare to NT central scheme.

Stress State of an Elastic Layer with a Cylindrical Cavity on a Rigid Foundation

The third basic problem of elasticity for a layer with a cylindrical cavity parallel to its surfaces is solved. Stresses on the cavity and the upper boundary of the layer are specified. On the lower boundary, the displacements are given. The solution to the spatial problem of elasticity is obtained by the generalized Fourier method for the Lame system of equations. The infinite systems of linear algebraic equations obtained by satisfying the boundary conditions are solved by the reduction method. As a result, the displacements and stresses are obtained at different points of the elastic body. We carried out a comparative analysis for the stress–strain state of the layer with a cylindrical cavity and without a cavity as well as for different distances to the lower boundary of the layer.

Hydraulic losses in the initial section of a flow parts of at aggregates of liquid rocket engines

The article deals with the need for accurate hydraulic losses determination along the length of the flow part in the process of designing units for supplying liquid rocket engines (LRE). The dependencies that are usually used, as a rule, are of a criterion-empirical nature and are not suitable for their application, taking into account the tendency of an increase in the rotating speed of turbo-pumping LRE supply units to 100,000 r.p.m. The analysis necessary to select the friction resistance laws for the flow part elements of the rocket engine supply units was carried out. A numerical integration equation system method to determine the variation in the characteristic thicknesses of the spatial boundary layer along the channel length in the initial section of the flow and hydraulic losses (taking into account the inertial component of the flow core velocity) was considered and proposed, depending on the current flow regimes in the flow path elements of the LRE supply units.

Preconditioned Subspace Descent Method for Nonlinear Systems of Equations

AbstractNonlinear least squares iterative solver is considered for real-valued sufficiently smooth functions. The algorithm is based on successive solution of orthogonal projections of the linearized equation on a sequence of appropriately chosen low-dimensional subspaces. The bases of the latter are constructed using only the first-order derivatives of the function. The technique based on the concept of the limiting stepsize along normalized direction (developed earlier by the author) is used to guarantee the monotone decrease of the nonlinear residual norm. Under rather mild conditions, the convergence to zero is proved for the gradient and residual norms. The results of numerical testing are presented, including not only small-sized standard test problems, but also larger and harder examples, such as algebraic problems associated with canonical decomposition of dense and sparse 3D tensors as well as finite-difference discretizations of 2D nonlinear boundary problems for 2nd order partial differential equations.

The Fretting Contact of Coated Bodies. Part I – Contact Parameters

In case of a mismatch between the elastic properties of the contacting bodies, the solution of the fretting contact can only be achieved numerically due to the connection between the contact pressure and the shear tractions. The numerical treatment of the contact processes employs an iterative strategy, and therefore requires the computation of the displacement response of the elastic body, assumed as an elastic half-space, subjected to general loadings. Unlike the case of homogeneous bodies with known Green’s functions, a closed-form expression of the response of layered semi-infinite solids to unit point loads has only been attained in the frequency domain. The latter formulas are used in this paper to assess the displacement due to arbitrary normal and shear surface tractions, thus empowering the application of a trial-and-error approach in finding the parameters of a fretting contact: the contact area, the slip and the stick regions, and the distributions of pressure and shear tractions. The contact parameters are assessed in a nested loop strategy, involving three levels of iterations. The inner level, based on the conjugate gradient method for linear systems of equations, finds the pressure when solving the contact problem along the normal direction, disconnected from any tangential effects, or the shear tractions when considering the contact equations in the tangential direction, disconnected from the pressure influence. The middle level stabilizes the pressure with respect to the shear tractions, thus assuring that the global instantaneous contact solution is achieved. The outer level manages the reproduction of the loading history in a fretting loop, by load incrementation. The proposed strategy proves itself as a robust tool for the prediction of the fretting contact process involving coatings.

A Multilayer Nonlinear Elimination Preconditioned Inexact Newton Method for Steady-State Incompressible Flow Problems in Three Dimensions

We develop a multilayer nonlinear elimination preconditioned inexact Newton method for a nonlinear algebraic system of equations, and a target application is the three-dimensional steady-state incompressible Navier--Stokes equations at high Reynolds numbers. Nonlinear steady-state problems are often more difficult to solve than time-dependent problems because the Jacobian matrix is less diagonally dominant, and a good initial guess from the previous time step is not available. For such problems, Newton-like methods may suffer from slow convergence or stagnation even with globalization techniques such as line search. In this paper, we introduce a cascadic multilayer nonlinear elimination approach based on feedback from intermediate solutions to improve the convergence of Newton iteration. Numerical experiments show that the proposed algorithm is superior to the classical inexact Newton method and other single layer nonlinear elimination approaches in terms of the robustness and efficiency. Using the proposed nonlinear preconditioner with a highly parallel domain decomposition framework, we demonstrate that steady solutions of the Navier--Stokes equations with Reynolds numbers as large as 7,500 can be obtained for the lid-driven cavity flow problem in three dimensions without the use of any continuation methods.

Newton-type multilevel optimization method

Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impressive performance of multilevel optimization methods is an empirical observation, and no theoretical explanation has so far been proposed. In order to address this issue, we study the convergence properties of a multilevel method that is motivated by second-order methods. We take the first step toward establishing how the structure of an optimization problem is related to the convergence rate of multilevel algorithms.

A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations

The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method for the reaction-diffusion systems of equations with space described by the fractional Laplacian is developed. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and an optimal L2-error estimate in space is proved. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model.