Sparse Systems Of Nonlinear Equations Research Articles

Multilevel field-split preconditioners with domain decomposition for steady and unsteady flow problems

With the incremental applications of Newton–Krylov methods for solving large sparse nonlinear systems of equations, the design of robust and scalable linear preconditioners plays an essential role for the whole solver. In this paper, we investigate the family of field-split (FS) preconditioners with different combinations of physics-based and domain decomposition methods, applied to the two typical fluid problems, i.e., the unsteady flow through fractured porous media and the steady buoyancy driven flow. In the implementation, several new versions of FS preconditioners are considered under the framework of the domain decomposition technique: additive FS, multiplicative FS, Schur-complement FS, and the constrained pressure residual (CPR) method, where the inverse of corresponding matrices is approximated by using the restricted additive Schwarz (RAS) algorithm. Rigorous eigenvalue analysis for various FS preconditioners is also provided for facilitating the design of algorithms. In particular, our approach further enhances the numerical performance by presenting a family of multilevel field-split methods for efficiently preconditioning. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the proposed preconditioning strategies for both standard benchmarks as well as realistic flow problems on a supercomputer.

Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems

In this paper we adapt the Newton-Raphson and Potra-Pták algorithms by combining them with the modified Newton-Raphson method by inserting a condition. Problems of systems of sparse nonlinear equations are solved the algorithms implemented in Matlab® environment. In addition, the methods are adapted and applied to space trusses problems with geometric nonlinear behavior. Structures are discretized by the Finite Element Positional Method, and nonlinear responses are obtained in an incremental and iterative process using the Linear Arc-Length path-following technique. For the studied problems, the proposed algorithms had good computational performance reaching the solution with shorter processing time and fewer iterations until convergence to a given tolerance, when compared to the standard algorithms of the Newton-Raphson and Potra-Pták methods.

The Efficient 3D Gravity Focusing Density Inversion Based on Preconditioned JFNK Method under Undulating Terrain: A Case Study from Huayangchuan, Shaanxi Province, China

Since polymetallic ores show higher anomalies in gravity exploration methods, we usually obtain the position and range of ore bodies by density inversion of gravity data. The three-dimensional (3D) gravity focusing density inversion is a common interpretation method in mineral exploration, which can directly and quantitatively obtain the density distribution of subsurface targets. However, in actual cases, it is computation inefficient. We proposed the preconditioned Jacobian-free Newton-Krylov (JFNK) method to accomplish the focusing inversion. The JFNK method is an efficient algorithm in solving large sparse systems of nonlinear equations, and we further accelerate the inversion process by the preconditioned technique. In the actual area, the gravity anomalies are distributed on the naturally undulating surface. Nowadays, the gravity inversion under undulating terrain was mainly achieved by discretizing the ground into unstructured meshes, but it is complicated and time-consuming. To improve the practicality, we presented an equivalent-dimensional method that incorporates unstructured meshes with structured meshes in gravity inversion, and the horizontal size is determined by the gradient of observed gravity and terrain data. The small size meshes are adopted at the position where the terrain or gravity gradient is large. We used synthetic data with undulating-terrain to test our new method. The results indicated that the recovered model obtained by this method was similar to the inversion method of unstructured meshes, and the new method computes faster. We also applied the method to field data in Huayangchuan, Shaanxi Province. The survey area has complicated terrain conditions and contains multiple polymetallic ores. Based on the high-density characteristics of polymetallic ore bodies in the area, we calculate the field data into 3D density models of the subsurface by the preconditioned JFNK method and infer six polymetallic ores.

Modified Newton-PHSS method for solving nonlinear systems with positive definite Jacobian matrices

By making use of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration as the inner solver for the modified Newton method, we establish the modified Newton-PHSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Moreover, the local convergence theorem is proved under proper conditions. Numerical results are given to show its feasibility and effectiveness. In addition, the advantages of the modified Newton-PHSS method over some other methods are shown by solving two systems of nonlinear equations.

A manifold-based approach to sparse global constraint satisfaction problems

We consider square, sparse nonlinear systems of equations whose Jacobian is structurally nonsingular, with reasonable bound constraints on all variables. We propose an algorithm for finding good approximations to all well-separated solutions of such systems. We assume that the input system is ordered such that its Jacobian is in bordered block lower triangular form with small diagonal blocks and with small border width; this can be performed fully automatically with off-the-shelf decomposition methods. Five decades of numerical experience show that models of technical systems tend to decompose favorably in practice. Once the block decomposition is available, we reduce the task of solving the large nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The proposed method resolves this issue with the novel backsolve step. We study the effect of the decomposition on a sequence of challenging problems. Beyond a certain problem size, the computational effort of multistart (no decomposition) grows exponentially. In contrast, thanks to the decomposition, for the proposed method the computational effort grows only linearly with the problem size. It depends on the problem size and on the hyperparameter settings whether the decomposition and the more sophisticated algorithm pay off. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size hyperparameter improves the robustness of the proposed method.

Modified Newton–SHSS method for a class of systems of nonlinear equations

It has been proved that Newton–HSS method is efficient and robust for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. In this paper, by utilizing the single-step Hermitian and skew-Hermitian splitting (SHSS) iteration technique, which performs efficiently under certain conditions, as the inner solver of the modified Newton method, we propose a class of modified Newton–SHSS methods. Subsequently, the local and semilocal convergence properties of our method will be discussed under some reasonable assumptions. Furthermore, we introduce the modified Newton–SHSS method with a backtracking strategy and analyze its basic global convergence theorem. Finally, several typical instances are used to illustrate the advantages of our methods when the Hermitian part of the Jacobian matrices are dominant.

Numerical methods and initial estimates for the simulation of steady-state reactive distillation columns with an algorithm based on tearing equations methodology

The solution of phenomenological models for steady-state reactive distillation columns involves the solution of a set of nonlinear algebraic equations. In this work was employed a method for generating initial estimates and an algorithm that uses the “tearing equations” methodology on the steady-state reactive distillation mathematical model, i.e., the large and sparse nonlinear system of equations were tired into several subsets of equations, wherein each subset of equations is solved in sequential steps to obtain model unknowns (temperatures, flow rates, extents of reaction, mole fractions, etc.). Among these steps, the most complex step is, the only nonlinear subset of equations, that is solved for obtaining the extents of reaction by using a method for the solution of a set of nonlinear algebraic equations. For this step were tested the Newton-Raphson and homotopy continuation methods. This methodology was used in two cases of study, one case with a less complex solution, the production of ethylene glycol, and another case with a little more difficult solution (metathesis of 2-pentene). In the simpler case of study, the methodology presented convergence using both Newton-Raphson and homotopy continuation methods. In the metathesis of 2-pentene case, convergence was obtained only when the homotopy continuation method was used.

Semilocal Convergence Analysis for MMN-HSS Methods under Hölder Conditions

AbstractMulti-step modified Newton-HSS (MMN-HSS) methods, which are variants of inexact Newton methods, have been shown to be competitive for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices. Previously, we established these MMN-HSS methods under Lipschitz conditions, and we now present a semilocal convergence theorem assuming the nonlinear operator satisfies milder Hölder continuity conditions. Some numerical examples demonstrate our theoretical analysis.

Design and evaluation of homotopies for efficient and robust continuation

Homotopy continuation, in combination with a quasi-Newton method, can be an efficient and robust technique for solving large sparse systems of nonlinear equations. The homotopy itself is pivotal in determining the efficiency and robustness of the continuation algorithm. As the homotopy is defined implicitly by a nonlinear system of equations to which the analytical solution is by assumption unknown, many properties of the homotopy can only be studied using numerical methods. The properties of a given homotopy which have the greatest impact on the corresponding continuation algorithm are traceability and linear solver performance. Metrics are presented for the analysis and characterization of these properties. Several homotopies are presented and studied using these metrics in the context of a parallel implicit three-dimensional Newton–Krylov–Schur flow solver for computational fluid dynamics. Several geometries, grids, and flow types are investigated in the study. Additional studies include the impact of grid refinement and the application of a coordinate transformation to the homotopy as measured through the traceability and linear solver performance metrics.

Modified Newton-NSS method for solving systems of nonlinear equations

By making use of the normal and skew-Hermitian splitting (NSS) method as the inner solver for the modified Newton method, we establish a class of modified Newton-NSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Under proper conditions, the local convergence theorem is proved. Furthermore, the successive-overrelaxation (SOR) technique has been proved quite successfully in accelerating the convergence rate of the NSS or the Hermitian and skew-Hermitian splitting (HSS) iteration method, so we employ the SOR method in the NSS iteration, and we get a new method, which is called modified Newton SNSS method. Numerical results are given to examine its feasibility and effectiveness.

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart–Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.

Robust viscous-inviscid interaction scheme for application on unstructured meshes

A coupled viscous-inviscid interaction scheme combining the continuity equation for potential flow with the three-dimensional integral boundary layer equations is presented. The inviscid problem is discretized by a finite-element approach whereas an upwind-biased finite-volume scheme is employed for the boundary layer equations. The discretization is applicable to unstructured tetrahedral-triangular meshes and results in a sparse system of non-linear equations which is solved by a Newton-type method. The mathematical reasons for the singularities commonly associated with the integral boundary layer equations in separated flow regions are analyzed and the connection between the mathematical singularities and the numerical ill-conditioning is discussed. It is shown that, by a suitable choice of closure relations, it is possible to obtain a boundary layer model free from numerical ill-conditioning in separated flow regions. The accuracy of the coupled viscous-inviscid model is investigated in a number of test cases including transitional and mildly separated flow over two different natural laminar flow airfoils and three-dimensional flow over a swept wing. It is concluded that the coupled method is able to provide reasonably accurate predictions of viscous and inviscid flow field quantities for the investigated cases.

Multi-step modified Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iterative method for solving large sparse non-Hermitian positive definite system of linear equations. In this paper, we establish a class of multi-step modified Newton-HSS (MMN-HSS) methods for solving large sparse system of nonlinear equations with positive definite Jacobian matrices. The MMN-HSS methods use the multi-step modified Newton methods to solve the nonlinear equations, and the HSS method to approximately solve the Newton equation. Local and semilocal convergence theorems are proved under proper conditions. Also, we present the global multi-step modified Newton-HSS methods with a backtracking strategy and analyse its global convergence. Finally, numerical results are given to confirm the feasibility and effectiveness of our method.

A monolithic homotopy continuation algorithm with application to computational fluid dynamics

A new class of homotopy continuation methods is developed suitable for globalizing quasi-Newton methods for large sparse nonlinear systems of equations. The new continuation methods, described as monolithic homotopy continuation, differ from the classical predictor–corrector algorithm in that the predictor and corrector phases are replaced with a single phase which includes both a predictor and corrector component. Conditional convergence and stability are proved analytically. Using a Laplacian-like operator to construct the homotopy, the new algorithm is shown to be more efficient than the predictor–corrector homotopy continuation algorithm as well as an implementation of the widely-used pseudo-transient continuation algorithm for some inviscid and turbulent, subsonic and transonic external aerodynamic flows over the ONERA M6 wing and the NACA 0012 airfoil using a parallel implicit Newton–Krylov finite-difference flow solver.

A Fully Implicit Method for Lattice Boltzmann Equations

Existing approaches for solving the lattice Boltzmann equations with finite difference methods are explicit and semi-implicit; both have certain stability constraints on the time step size. In this work, a fully implicit second-order finite difference scheme is developed. We focus on a parallel, highly scalable, Newton--Krylov--RAS algorithm for the solution of a large sparse nonlinear system of equations arising at each time step. Here, RAS is a restricted additive Schwarz preconditioner based on a first-order spatial discretization. We show numerically that by using the fully implicit method the time step size is no longer constrained by the CFL condition, and the Newton--Krylov--RAS algorithm is scalable on a supercomputer with more than ten thousand processors. Moreover, to calculate the steady state solution we investigate an adaptive time stepping strategy. The total compute time required by the implicit method with adaptive time stepping is much smaller than that of an explicit method for several t...

A parallel adaptive nonlinear elimination preconditioned inexact Newton method for transonic full potential equation

We propose and study a right-preconditioned inexact Newton method for the numerical solution of large sparse nonlinear system of equations. The target applications are nonlinear problems whose derivatives have some local discontinuities such that the traditional inexact Newton method suffers from slow or no convergence even with globalization techniques. The proposed adaptive nonlinear elimination preconditioned inexact Newton method consists of three major ingredients: a subspace correction, a global update, and an adaptive partitioning strategy. The key idea is to remove the local high nonlinearity before performing the global Newton update. The partition used to define the subspace nonlinear problem is chosen adaptively based on the information derived from the intermediate Newton solution. Some numerical experiments are presented to demonstrate the robustness and efficiency of the algorithm compared to the classical inexact Newton method. Some parallel performance results obtained on a cluster of PCs are reported.

Convergence analysis of the modified Newton–HSS method under the Hölder continuous condition

The modified Newton–HSS method, which is constructed by employing the Hermitian and skew-Hermitian splitting methods as the inner iteration process at each step of the outer modified Newton’s iteration, has been proved to be a competitive method for solving large sparse systems of nonlinear equations with non-Hermitian positive-definite Jacobian matrices. In this paper, under the hypotheses that the derivative is continuous and the derivative satisfies the Hölder continuous condition, two local convergence theorems are established for the modified Newton–HSS method. Furthermore, the rate of convergence of the modified Newton–HSS method is also characterized in terms of the rate of convergence of the matrix ‖T(α;x)‖. The numerical example is given to confirm the concrete applications of the results of our paper.

A Fully Implicit Domain Decomposition Algorithm for Discrete-velocity BGK Equation

Popular approaches for solving discrete-velocity BGK equation are explicit and semiimplicit methods, and both have certain constraints on the time step size. In this work, a fully implicit finite difference scheme is developed for the discrete-velocity BGK equation. We focus on a parallel, fully coupled, Newton-Krylov-RAS algorithm which imposes no limit on the time step size but requires the solution of a large sparse nonlinear system of equations at every time step. Here, RAS is a restricted additive Schwarz method. We show numerically that with such a preconditioned, nonlinearly implicit method the time step size is no longer constrained by the CFL condition, and scalable in terms of both strong and weak scalabilities on a supercomputer with thousands of processors.

Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices

Modified Hermitian and skew-Hermitian splitting (MHSS) method is an unconditionally convergent iterative method for solving large sparse complex symmetric systems of linear equations. By making use of the MHSS iteration as the inner solver for the inexact Newton method, we establish a class of inexact Newton-MHSS methods for solving large sparse systems of nonlinear equations with complex symmetric Jacobian matrices at the solution points. The local and semi-local convergence properties are analyzed under some proper assumptions. Moreover, by introducing a backtracking linear search technique, a kind of global convergence inexact Newton-MHSS methods are also presented and analyzed. Numerical results are given to examine the feasibility and effectiveness of the inexact Newton-MHSS methods.

Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations

Newton-HSS methods, which are variants of inexact Newton methods different from the Newton–Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive-definite Jacobian matrices (J. Comp. Math. 2010; 28:235–260). In that paper, only local convergence was proved. In this paper, we prove a Kantorovich-type semilocal convergence. Then we introduce Newton-HSS methods with a backtracking strategy and analyse their global convergence. Finally, these globally convergent Newton-HSS methods are shown to work well on several typical examples using different forcing terms to stop the inner iterations. Copyright © 2010 John Wiley & Sons, Ltd.