Newton-Krylov Method Research Articles

A parallel multi-domain solution methodology applied to nonlinear thermal transport problems in nuclear fuel pins

This paper describes an efficient and nonlinearly consistent parallel solution methodology for solving coupled nonlinear thermal transport problems that occur in nuclear reactor applications over hundreds of individual 3D physical subdomains. Efficiency is obtained by leveraging knowledge of the physical domains, the physics on individual domains, and the couplings between them for preconditioning within a Jacobian Free Newton Krylov method. Details of the computational infrastructure that enabled this work, namely the open source Advanced Multi-Physics (AMP) package developed by the authors is described. Details of verification and validation experiments, and parallel performance analysis in weak and strong scaling studies demonstrating the achieved efficiency of the algorithm are presented. Furthermore, numerical experiments demonstrate that the preconditioner developed is independent of the number of fuel subdomains in a fuel rod, which is particularly important when simulating different types of fuel rods. Finally, we demonstrate the power of the coupling methodology by considering problems with couplings between surface and volume physics and coupling of nonlinear thermal transport in fuel rods to an external radiation transport code.

A Jacobian‐free Newton–Krylov method for thermalhydraulics simulations

SummaryThe current paper is focused on investigating a Jacobian‐free Newton–Krylov (JFNK) method to obtain a fully implicit solution for two‐phase flows. In the JFNK formulation, the Jacobian matrix is not directly evaluated, potentially leading to major computational savings compared with a simple Newton's solver. The objectives of the present paper are as follows: (i) application of the JFNK method to two‐fluid models; (ii) investigation of the advantages and disadvantages of the fully implicit JFNK method compared with commonly used explicit formulations and implicit Newton–Krylov calculations using the determination of the Jacobian matrix; and (iii) comparison of the numerical predictions with those obtained by the Canadian Algorithm for Thermaulhydraulics Network Analysis 4. Two well‐known benchmarks are considered, the water faucet and the oscillating manometer.An isentropic two‐fluid model is selected. Time discretization is performed using a backward Euler scheme. A Crank–Nicolson scheme is also implemented to check the effect of temporal discretization on the predictions. Advection Upstream Splitting Method+ is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. One explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented. A detailed grid and model parameter sensitivity analysis is performed.For both cases, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL numbers up to 200 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared with the calculations requiring the determination of the Jacobian matrix. Copyright © 2015 John Wiley & Sons, Ltd.

Composing Scalable Nonlinear Algebraic Solvers

Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.

PRECONDITIONING ISSUES IN THE NUMERICAL SOLUTION OF NONLINEAR EQUATIONS AND NONLINEAR LEAST SQUARES

Second order methods for optimization call for the solution of sequences of linear systems. In this survey we will discuss several issues related to the preconditioning of such sequences. Covered topics include both techniques for building updates of factorized preconditioners and quasi-Newton approaches. Sequences of unsymmetric linear systems arising in Newton-Krylov methods will be considered as well as symmetric positive definite sequences arising in the solution of nonlinear least-squares by Truncated Gauss-Newton methods.

CFD high-order accurate scheme Jacobian-Free Newton Krylov method

High-order accurate scheme for Computational Fluid Dynamics (CFD) finite difference method can provide more exact flow field solution than second order accurate scheme, but it is hard to get Jacobian matrix for lower–upper symmetric Gauss–Seidel (LU-SGS) method because of its complicated computing stencil, which lead to the poor convergence speed of LU-SGS. A Jacobian-Free Newton–Krylov (JFNK) method of high-order accurate scheme was developed, and a nonlinear type of preconditioner was applied based on traditional 7 diagonals matrix, which was solved with LU-SGS method. In cylinder steady flow case, JFNK method was better than original LU-SGS method, nearly one half wall time was saved.

Extending Summation Precision for Network Reduction Operations

Double precision summation is at the core of numerous important algorithms such as Newton–Krylov methods and other operations involving inner products, such as matrix multiplication and dot products. However, the effectiveness of summation is limited by the accumulation of rounding errors due to compressed representations, which are an increasing problem with the scaling of modern HPC systems and data sets that can easily perform summations with millions or billions of operands. To reduce the impact of precision loss, researchers have proposed increased- and arbitrary-precision libraries that provide reproducible error or even bounded error accumulation for large sums. However, such libraries increase computation and communication time significantly, and do not always guarantee an exact result. In this article, we propose fixed-point representations of double precision variables that enable arbitrarily large summations without error and provide exact and reproducible results. We call this format big integer (BigInt). Even though such formats have been studied for local processor computations, we make the case that using fixed-point representation for distributed computation over a system-wide network is feasible with performance comparable to that of double-precision floating point summation. This is possible by the inclusion of simple and inexpensive logic into modern NICs, or by using the programmable logic found in many modern NICs, in order to accelerate performance on large-scale systems in order to avoid waking up processors.

Asynchronous partial update of the Restricted Additive Schwarz preconditioner to solve nonlinear CFD problems

This paper presents a method that speeds up the solution of unsteady nonlinear problems. When a Newton–Krylov method is used, the most time-consuming part of the numerical simulation is the solution of the successive linear systems. Generally, there are only slight changes between two consecutive Jacobian matrices. Thus, in order to save computations, it is possible to use the same preconditioning matrix for a few successive linear systems, but the convergence of the Krylov method is slowed down. In the case of a preconditioner based on domain decomposition, one can update only the parts of the preconditioner associated to certain subdomains, keeping the other ones constant. In order to achieve an efficient parallel implementation, we propose to add processes dedicated to the asynchronous update of some parts of the preconditioner. Numerical results are provided for a reaction–diffusion problem and for a model problem based on the lid-driven cavity. They show that this addition of processes can speed up the computation and improve the robustness.

Efficient fixed point and Newton–Krylov solvers for FFT-based homogenization of elasticity at large deformations

In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton–Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton’s method using linear fixed point solvers and roughly $$100$$ times less iterations than the nonlinear fixed point method. However, the Newton–Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant–Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources

We present a parallel matrix‐free implicit finite volume scheme for the solution of unsteady three‐dimensional advection‐diffusion‐reaction equations with smooth and Dirac‐Delta source terms. The scheme is formally second order in space and a Newton–Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix‐vector product required is hardcoded without any approximations, obtaining a matrix‐free method that needs little storage and is well‐suited for parallel implementation. We describe the matrix‐free implementation of the method in detail and give numerical evidence of its second‐order convergence in the presence of smooth source terms. For nonsmooth source terms, the convergence order drops to one half. Furthermore, we demonstrate the method's applicability for the long‐time simulation of calcium flow in heart cells and show its parallel scaling. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 143–167, 2015

Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton–Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

Scalable Parallel Approach for High-Fidelity Steady-State Aeroelastic Analysis and Adjoint Derivative Computations

Aeroelastic systems achieve the best performance when the aerodynamic shape and structural sizing are optimized concurrently, but such an optimization is challenging when high-fidelity aerodynamic and structural models are required. This paper addresses this challenge through several significant improvements. Fully coupled Newton–Krylov methods are presented for the solution of aerostructural systems and for the corresponding adjoint systems. The coupled adjoint method presented can compute gradients with respect to thousands of multidisciplinary design variables accurately and efficiently. This is enabled by several improvements in the computation of the multidisciplinary terms in the coupled adjoint. The parallel scalability of the methods is demonstrated for a full aircraft configuration using an Euler computational fluid dynamics model with more than state variables and a detailed structural finite element model of the wing with more than degrees of freedom. The coupled Newton–Krylov methods are shown to improve the convergence rate of both the aerostructural solution and the coupled adjoint derivative computations. Gradient computations of aerodynamic and structural functions with respect to both aerodynamic shape and structural sizing variables are verified, and scaling is demonstrated to variables. The accuracy and scalability of the presented methods make it possible to perform aerostructural optimizations of full aircraft configurations with respect to hundreds of external shape and structural sizing design variables, leading to optimal aeroelastic tailoring.

Dynamic implicit 3D adaptive mesh refinement for non-equilibrium radiation diffusion

The time dependent non-equilibrium radiation diffusion equations are important for solving the transport of energy through radiation in optically thick regimes and find applications in several fields including astrophysics and inertial confinement fusion. The associated initial boundary value problems that are encountered often exhibit a wide range of scales in space and time and are extremely challenging to solve. To efficiently and accurately simulate these systems we describe our research on combining techniques that will also find use more broadly for long term time integration of nonlinear multi-physics systems: implicit time integration for efficient long term time integration of stiff multi-physics systems, local control theory based step size control to minimize the required global number of time steps while controlling accuracy, dynamic 3D adaptive mesh refinement (AMR) to minimize memory and computational costs, Jacobian Free Newton–Krylov methods on AMR grids for efficient nonlinear solution, and optimal multilevel preconditioner components that provide level independent solver convergence.

The preconditioned Jacobian-free Newton-Krylov methods for nonequilibrium radiation diffusion equations

In general, it is difficult to use the Newton-Krylov methods to solve the large-scale multi-variable nonequilibrium reaction-diffusion systems. In this paper, by employing two new semi-implicit dis...

Matrix-free second-order methods in implicit time integration for compressible flows using automatic differentiation

In the numerical simulation of inviscid and viscous compressible fluid flow, implicit time integration schemes based on Newton-Krylov methods are frequently used. A crucial ingredient of Krylov subspace methods is the evaluation of the product of the Jacobian matrix of the spatial operator, e.g., fluxes, and a Krylov vector. In this article, we consider a matrix-free, consistently second-order accurate implementation of the Jacobian-vector product within the flow solver QUADFLOW using automatic differentiation. The convergences of the non-linear iteration using first-and second-order accurate Jacobian-vector products are compared. Different algorithmic parameters such as the CFL number and the choice of the preconditioner are taken into account. Investigations of inviscid flow around an airfoil and a wing of an aircraft, as well as the viscous laminar flow over a flat plate demonstrate that the convergence of the Newton-Krylov solver can significantly be improved using consistently second-order accurate matrix-vector products, compared to standard first-order approximations. It has been shown that a hybrid strategy using a blend between first-order and second-order accurate matrix-vector products is the best choice to obtain an efficient and robust algorithm.

Noise Reduction in Coarse Bifurcation Analysis of Stochastic Agent-Based Models: An Example of Consumer Lock-In

We investigate coarse equilibrium states of a fine-scale, stochastic agent-based model of consumer lock-in in a duopolistic market. In the model, agents decide on their next purchase based on a combination of their personal preference and their neighbours' opinions. For agents with independent identically-distributed parameters and all-to-all coupling, we derive an analytic approximate coarse evolution-map for the expected average purchase. We then study the emergence of coarse fronts when spatial segregation is present in the relative perceived quality of products. We develop a novel Newton-Krylov method that is able to compute accurately and efficiently coarse fixed points when the underlying fine-scale dynamics is stochastic. The main novelty of the algorithm is in the elimination of the noise that is generated when estimating Jacobian-vector products using time-integration of perturbed initial conditions. We present numerical results that demonstrate the convergence properties of the numerical method, and use the method to show that macroscopic fronts in this model destabilise at a coarse symmetry-breaking bifurcation.

Fluid preconditioning for Newton–Krylov-based, fully implicit, electrostatic particle-in-cell simulations

A recent proof-of-principle study proposes an energy- and charge-conserving, nonlinearly implicit electrostatic particle-in-cell (PIC) algorithm in one dimension [9]. The algorithm in the reference employs an unpreconditioned Jacobian-free Newton–Krylov method, which ensures nonlinear convergence at every timestep (resolving the dynamical timescale of interest). Kinetic enslavement, which is one key component of the algorithm, not only enables fully implicit PIC as a practical approach, but also allows preconditioning the kinetic solver with a fluid approximation. This study proposes such a preconditioner, in which the linearized moment equations are closed with moments computed from particles. Effective acceleration of the linear GMRES solve is demonstrated, on both uniform and non-uniform meshes. The algorithm performance is largely insensitive to the electron–ion mass ratio. Numerical experiments are performed on a 1D multi-scale ion acoustic wave test problem.

Towards Faster Solution of Large Power Flow Problems

Current and future developments in the power system industry demand fast power flow solvers for larger power flow problems. The established methods are no longer viable for such problems, as they are not scalable in the problem size. In this paper, the use of Newton-Krylov power flow methods is proposed, and a multitude of preconditioning techniques for such methods are discussed and compared. It is shown that incomplete factorizations can perform very well as preconditioner, resulting in a solver that scales in the problem size. It is further shown that using a preconditioned inner-outer Krylov method has no significant advantage over applying the preconditioner directly to the outer iterations. Finally, algebraic multigrid is demonstrated as a preconditioner for Newton-Krylov power flow and argued to be the method of choice in some scenarios.

DIFFUSION PIECEWISE HOMOGENIZATION VIA FLUX DISCONTINUITY RATIOS

We analyze piecewise homogenization with flux-weighted cross sections and preservation of averaged currents at the boundary of the homogenized domain. Introduction of a set of flux discontinuity ratios (FDR) that preserve reference interface currents leads to preservation of averaged region reaction rates and fluxes. We consider the class of numerical discretizations with one degree of freedom per volume and per surface and prove that when the homogenization and computing meshes are equal there is a unique solution for the FDRs which exactly preserve interface currents. For diffusion submeshing we introduce a Jacobian-Free Newton-Krylov method and for all cases considered obtain an ‘exact’ numerical solution (eight digits for the interface currents). The homogenization is completed by extending the familiar full assembly homogenization via flux discontinuity factors to the sides of regions laying on the boundary of the piecewise homogenized domain. Finally, for the familiar nodal discretization we numerically find that the FDRs obtained with no submesh (nearly at no cost) can be effectively used for whole-core diffusion calculations with submesh. This is not the case, however, for cell-centered finite differences.

Developments on the Broyden procedure to solve nonlinear problems arising in CFD

This paper is devoted to the solution of nonlinear time-dependant partial differential equations arising in CFD using Broyden’s method in the parallel computing framework. We first use Broyden’s method in the context of the domain decomposition: we propose to update the Restricted Additive Schwarz preconditioner from one Newton iteration to another when a Newton–Krylov method is used. We also investigate a time-pipelining method where Broyden’s method is used as a solver of the nonlinear problem of each time step.

Inexact Newton and quasi-Newton methods for the output feedback pole assignment problem

The pole assignment problem (PAP) is a special algebraic inverse eigenvalue problem. In this paper, we present two types of algorithms, namely a quasi-Newton method with line search and some variants of the inexact Newton methods to tackle that problem. For a nonmonotone version of inexact Newton–Krylov method, we give local convergence under the assumptions of semismoothness and \(BD\)-regularity at the solution and global convergence under a nonmonotonic backtracking strategy. For a quasi-Newton method with line search, under suitable assumptions, we show local Q-superlinear convergence. Also, we consider a proximal point quasi-Newton algorithm for solving PAP. Moreover, we modify these methods to tackle the PAP where the corresponding control system is with time delay. Numerical results illustrate the performance of the proposed methods.