Large Systems Of Nonlinear Equations Research Articles

Parallelization of the time integration for time‐periodic flow problems

AbstractThe solution of a time‐periodic flow problem is expanded into a discrete Fourier series. This yields a large nonlinear system of equations which can be solved in parallel. We show for a one‐dimensional model problem that this method for the parallelization of the time domain converges in a robust and efficient manner. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Preconditioned Implicit Solvers for Nonlinear PDEs in Monument Conservation

Mathematical models for the description, in a quantitative way, of the damages induced on monuments by the action of specific pollutants are often systems of nonlinear, possibly degenerate, parabolic equations. Although some of the asymptotic properties of the solutions are known, one needs a numerical approximation scheme in order to have a quantitative forecast at any time of interest. In this paper an implicit numerical method is proposed, analyzed, and numerically tested for parabolic equations of porous media type and on a systems of two PDEs that models the sulfation of marble in monuments. Due to the nonlinearity inside the differential operator of the underlying mathematical model, the use of an iterative solver for a large system of nonlinear equations is required, and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures. Numerical experiments for the validation of the analysis complement this contribution.

BB: AnRPackage for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function

We discuss <code>R</code> package <b>BB</b>, in particular, its capabilities for solving a nonlinear system of equations. The function <code>BBsolve</code> in <b>BB</b> can be used for this purpose. We demonstrate the utility of these functions for solving: (a) large systems of nonlinear equations, (b) smooth, nonlinear estimating equations in statistical modeling, and (c) non-smooth estimating equations arising in rank-based regression modeling of censored failure time data. The function <code>BBoptim</code> can be used to solve smooth, box-constrained optimization problems. A main strength of <b>BB</b> is that, due to its low memory and storage requirements, it is ideally suited for solving high-dimensional problems with thousands of variables.

An Efficient Numerical Technique for the Solution of the Monodomain and Bidomain Equations

Most numerical schemes for solving the monodomain or bidomain equations use a forward approximation to some or all of the time derivatives. This approach, however, constrains the maximum timestep that may be used by stability considerations as well as accuracy considerations. Stability may be ensured by using a backward approximation to all time derivatives, although this approach requires the solution of a very large system of nonlinear equations at each timestep which is computationally prohibitive. In this paper we propose a semi-implicit algorithm that ensures stability. A linear system is solved on each timestep to update the transmembrane potential and, if the bidomain equations are being used, the extracellular potential. The remainder of the equations to be solved uncouple into small systems of ordinary differential equations. The backward Euler method may be used to solve these systems and guarantee numerical stability: as these systems are small, only the solution of small nonlinear systems are required. Simulations are carried out to show that the use of this algorithm allows much larger timesteps to be used with only a minimal loss of accuracy. As a result of using these longer timesteps the computation time may be reduced substantially.

The Gauss–Seidel–quasi-Newton method: A hybrid algorithm for solving dynamic economic models

The Jacobi matrix required for second-order iterations, such as standard Newton–Raphson procedures, is often too costly to determine for large systems of non-linear equations of a dynamic economic model. In such circumstances, first-order iterative methods are commonly adopted. The problem then, however, is that baseline first-order iterations may not converge. This paper provides a solution to this problem by developing a hybrid method of first- and second-order iterations for solving large-scale dynamic models. The modified algorithm is robust and fast and relative running times increase with the size and complexity of the economic model. As it is easy to implement – only using standard numerical procedures to augment conventional and intuitive first-order iterations – the algorithm is particularly attractive.

Adaptive Timestepping for Elastohydrodynamic Lubrication Solvers

The application of variable-step time-integration techniques in a method of lines approach for the numerical solution of transient elastohydrodynamic lubrication calculations is described. Spatial discretization of the coupled integro-differential equations leads to a system of differential-algebraic equations in time. At each timestep a large system of nonlinear equations of 104--107 variables is solved by using multigrid methods and multilevel multi-integration. The application of temporal error control methods is considered with reference to both the differential-algebraic formulation of the system to be solved and the multigrid-based nonlinear equation solver to be used. The error control method used takes account of the magnitude of the spatial error present by using a local time error control based on the magnitude of this error. This approach is shown to be beneficial in reducing the computational work required. Experimental results on a variety of lubrication problems, including a challenging rough surface problem, are used to display the effectiveness of the methods.

Globally Convergent Algorithm for Solving Large Nonlinear Systems of Equations

An algorithm for solving large nonlinear systems of equations is introduced. The algorithm is a variant of the inexact Newton method where an approximate Newton direction is taken from a space of small dimension (Krylov subspace). The algorithm is based on the implicitly restarted Arnoldi method to construct a basis for the Krylov subspace in conjunction with a line search strategy to control the length of the step. It should cope quite well with large-scale systems of equations. A convergence theory for this algorithm is presented. It is shown that this algorithm is globally convergent. This theory is sufficiently general that it holds for any algorithm that projects the problem on a lower dimensional subspace.

Parallel Construction of Nonlinear Force-Free Fields

A numerical approach to calculating nonlinear force-free fields is presented. The approach is similar to Sakurai (1981) being a current-field iteration scheme using the integral solution to Ampere's law (the Biot–Savart law). However, the method of solution presented here is simpler than Sakurai's approach, in that the field is directly constructed on a grid without the intermediate solution of a large system of nonlinear equations. The method also permits straightforward implementation on parallel computers. Results of applying the method to a number test cases, including boundary conditions with substantial currents, are presented.

Explicit inversion formulas for a model in diffuse tomography

Optical, or diffuse tomography, refers to the use of low energy probes to obtain images of highly scattering media. This represents an emerging area for the application of new mathematical tools to important problems in medical imaging. It is our contention that it poses important mathematical challenges dealing with the solution of large systems of nonlinear equations. In this paper we present a case for which a complete solution is possible; however the consideration of any larger system presents at the moment a number of unresolved difficulties. A variety of mathematical models of the underlying physics are reviewed in [1] including the ones considered in [4,7–10], where we have discussed both the “direct”, as well as the more interesting “inverse problem” for a discrete model. The time evolution of the system is governed by a Markov chain with discrete state space and discrete time. The one step transition probability matrix is denoted by P . Some of the states are “incoming”, some are “outgoing”, and the rest are “hidden” states. The first two types correspond to sources and detectors respectively, and the hidden ones represent the state of a photon while it moves in the interior of the object.

Inexact overlapped block Broyden methods for solving nonlinear equations

In this paper a parallelizable overlapped block Broyden method is presented for solving large systems of nonlinear equations. The basic idea is to perform the block Broyden iteration described in [SIAM J. Sci. Comput. 18 (1997) 1367] for the overlapped blocks and then assemble the overlapping solutions in a weighted average manner at each iteration. A family of nonlinear overlapped solvers could be generated combining it with some iterative or direct linear solvers. The conditions under which the algorithm is locally convergent are studied and many useful techniques regarding the implementation are also considered.

Modelling the three-way catalytic converter with mechanistic kinetics using the Newton–Krylov method on a parallel computer

A mathematical model for an automotive three-way catalytic converter based on experimental mechanistic kinetics is developed. The transient model includes a one-space dimension discretization for the gas phase and a two-dimensional discretization for the solid phase is comprised of the washcoat and substrate. Axi-symmetry is assumed. The combination of the complex kinetic model and the associated transport equations generates a large system of non-linear equations that is solved using the Newton–Krylov method based on a pre-conditioned GMRES algorithm. Simulated light-off curves for cold start operation illustrate the importance of including dynamic adsorption process in the model for this type of operating condition. Diffusion limitation in the washcoat is shown to be very significant even at relatively low operating temperatures. From the numerical standpoint, the importance of the choice of pre-conditioner is demonstrated. The use of parallel computing at a fine grain level on vector–vector and vector–matrix operation is shown to provide a large degree of speedup, which increases as the number of grid points increases.

The Newton-arithmetic mean method for the solution of systems of nonlinear equations

This paper is concerned with the development of the Newton-arithmetic mean method for large systems of nonlinear equations with block-partitioned Jacobian matrix. This method is well suited for implementation on a parallel computer; its degree of decomposition is very high. The convergence of the method is analysed for the class of systems whose Jacobian matrix satisfies an affine invariant Lipschitz condition. An estimation of the radius of the attraction ball is given. Special attention is reserved to the case of weakly nonlinear systems. A numerical example highlights some peculiar properties of the method.

Tracing the buckling of a rectangular plate with the Block GMRES method

We investigate bifurcation scenario of the von Kármán equations with partially clamped boundary conditions defined in rectangular domains. First, we study how the (preconditioned) Block GMRES method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations. Next, we discuss linear stabilities of the von Kármán equations with partially clamped boundary conditions. In particular, we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von Kármán equations via computer algebra. The Block GMRES method is used to solve linear systems and to detect singularity along solution paths of the discrete problem. Sample numerical results are reported.

Synchronous and Asynchronous Interval Newton-Schwarz Methods for a Class of Large Systems of Nonlinear Equations

We introduce a class of parallel interval arithmetic iteration methods for nonlinear systems of equations, especially of the type Ax+ϕ(x) = f, ϕ diagonal, in R N . These methods combine enclosure and global convergence properties of Newton-like interval methods with the computational efficiency of parallel block iteration methods: algebraic forms of Schwarz-type methods which generalize both the well-known additive algebraic Schwarz Alternating Procedure and multisplitting methods. We discuss both synchronous and asynchronous variants. Besides enclosure and convergence properties, we present numerical results from a CRAY T3E.

CFD methods for three-dimensional computation of complex flow problems

Abstract This paper provides an overview of some of the CFD methods developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) [http://www.mems.rice.edu/TAFSM/]. The paper also provides many examples of three-dimensional flow simulations carried out with these CFD methods and advanced parallel supercomputers. The methods and tools described in this paper include: stabilized finite element formulations; formulations for flows with moving boundaries and interfaces; mesh update methods; iterative solution techniques for large nonlinear equation systems; and parallel implementation of these methods. Our target is to be able to address effectively certain classes of flow simulation problems. These include: unsteady flows with interfaces; fluid–object interactions; fluid–structure interactions; airdrop systems; aerodynamics of complex shapes; and contaminant dispersion.

Construction of Multiscaling Functions with Approximation and Symmetry

This paper presents a new and efficient way to create multiscaling functions with given approximation order, regularity, symmetry, and short support. Previous techniques were operating in time domain and required the solution of large systems of nonlinear equations. By switching to the frequency domain and employing the latest results of the multiwavelet theory we are able to elaborate asimple and efficient method of construction of multiscaling functions. Our algorithm is based on a recently found factorization of the refinement mask through the two-scale similarity transform (TST). Theoretical results and new examples are presented.

Accelerated Inexact Newton Schemes for Large Systems of Nonlinear Equations

Classical iteration methods for systems, such as Jacobi iteration, can be accelerated considerably by Krylov subspace methods like GMRES @. In this paper, we describe how inexact Newton methods for problems can be accelerated in a similar way and how this leads to a general framework that includes many well-known techniques for solving and systems, as well as new ones. Inexact Newton methods are frequently used in practice to avoid the expensive exact solution of the large system arising in the (possibly also inexact) linearization step of Newton's process. Our framework includes acceleration techniques for the linear steps as well as for the nonlinear steps in Newton's process. The described class of methods, the accelerated inexact Newton (AIN) methods, contains methods like GMRES and GMRESR for systems, Arnoldi and JacDav{} for eigenproblems, and many variants of Newton's method, like damped Newton, for general problems. As numerical experiments suggest, the AIN{} approach may be useful for the construction of efficient schemes for solving problems.

Verified construction of fully symmetric cubature formulae of degree 4k‐1 with few nodes for fully symmetric planar regions using invariant theory

AbstractTraditionally, the construction of a cubature formula requires solving a large system of nonlinear equations to determine the nodes and weights. Using invariant theory this system is easy to solve due to the smart choice of an appropriate basis for the vector space spanned by the polynomials which are invariant with respect to the symmetry group of the square. The presented method for the verified construction of two‐dimensional fully symmetric cubature formulae of degree 4k ‐‐ 1 has the great advantage that even for high degrees the hardest computation to perform is solving a linear system and a system of two polynomial equations of relatively low degree in two unknowns.Enclosures of the nodes and weights are obtained by means of interval arithmetic and verification algorithms for solving linear and nonlinear systems of equations. In order to compute enclosures of high accuracy the optimal scalar product and istaggered arithmetic are employed.The described method constructs formulae with fewer nodes than the Gauss‐product formulae of the same degree, if existing, and it even allows one to obtain some new formulae in some particular cases.

Tensor-GMRES Method for Large Systems of Nonlinear Equations

This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large systems of nonlinear equations. Krylov subspace projection techniques for asymmetric systems of linear equations are coupled with a tensor model formation and solution technique for nonlinear equations. Similar to traditional tensor methods, the new tensor method is shown to have significant computational advantages over the analogous Newton counterpart on a set of nonsingular and singular problems. For example, an application to the Euler equations for the flow through a nozzle with a given area ratio shows that the tensor-GMRES method can be much more efficient than the analogous Newton-GMRES method. The new tensor method is also consistent with preconditioning and matrix-free implementation.

Load-flow parallel processing system. Conjugate-gradient neural network architecture

Whilst very many previously published papers have been devoted to the non-linear mapping functions of neural networks, the possible scope of neural networks in numerical analysis applications has not so far been the subject of correspondingly extensive investigation. In that acknowledgment, research has been undertaken in the area of solving by neural networks the large non-linear equation systems that arise in power network system analysis. From this research, the paper reports the development of neural network architectures for Newton-Raphson load-flow analysis. Load-flow analysis is interpreted as an unconstrained minimisation formulation. The linearised load-flow equations at each Newton iteration are transformed to a scalar objective function of quadratic form. Using standard minimisation procedures for minimising this objective function requires only the multiplications and additions which arise in matrix/vector operations. It is here where neural networks can excel. The paper shows how a massively parallel processing structure can be achieved in which very many multiplications are carried out at the same time. The neural network architecture developed, therefore, achieves ultra-high-speed load-flow analysis. The computing time to achieve a converged load-flow solution for a 1000 node power network is less than about 20 ms. Reference is made to validation studies in which load-flow solutions from the new neural network architecture are compared with those from a standard sequential processor Newton-Raphson load-flow program.