Nonlinear Preconditioner Research Articles

S2SA preconditioning for the Sn equations with strictly nonnegative spatial discretization

Preconditioners based upon transport sweeps and diffusion-synthetic acceleration have been constructed and applied to the zeroth and first spatial moments of the 1-D Sn transport equation using a strictly nonnegative nonlinear spatial closure. Linear and nonlinear preconditioners have been derived and analyzed. The effectiveness of various combinations of these preconditioners are compared using the source iteration, matrix-free Picard Krylov, and nonlinear Krylov acceleration methods. In one dimension, preconditioning with a linear S2SA diffusion equation is found to be essentially equivalent to using a nonlinear diffusion equation. The ability to use a linear diffusion equation has important implications for preconditioning the Sn equations with a strictly nonnegative spatial discretization in multiple dimensions.

A Parallel Adaptive Nonlinear Elimination Preconditioned Inexact Newton for Transonic Full Potential Flow Problems

The inexact Newton method (INB) is the one of the most commonly-used methods for solving large spare nonlinear systems of equations arising from the discretization of partial differential equations. The method is quite robust and efficient for the smooth nonlinear systems, but for the case that the solution to the nonlinear system or its derivatives with local discontinuity, the convergence rate of INB degrades, at worst, it fails to converge even using some globalization technique. Such local discontinuity causes the nonlinear system unbalanced and the problem often appear in a variety of applications in computational fluid mechanics, such as the shock wave in the transonic compressible flows, the boundary layers or singularity around the corner in the high-Reynolds number incompressible flows.The recent research efforts for the INB-based nonlinear solver are to develop the effective nonlinear preconditioning to enhance its robustness and efficiency. As for linear systems, a nonlinear preconditioner can be applied to the nonlinear system on the left (referred as left preconditioning) or on the right (referred as right nonlinear preconditioning). The basic idea of left nonlinear preconditioning is to reformulate implicitly the original system as a new balanced system by some mean and then solve the new easier nonlinear system by INB. The additive Schwarz preconditioned inexact Newton algorithm (ASPIN) is belong to this class, which constructs the nonlinear preconditioner based on the Schwarz framework.On the other hand, the philosophy of right preconditioning is different, in which the nonlinear function does not need to be changed. For example, Hwang et. al successfully employ nonlinear elimination as right nonlinear preconditioner for INB with application in quasi one- dimensional shocked duct flow calculation. The basic idea of NE is to implicitly remove those components causing troubles and obtain a better balanced intermediate solution for which INB can be applied. In that paper, the assumption made is the components to be eliminated are known in advance based on the physical knowledge for the problem, which corresponds to the region near the nozzle. However, in practice, it might not be always possible to determine the bad components need to be eliminated, especially for the multi-dimensional applications.In this work, we propose a new algorithm, namely a parallel adaptive nonlinear elimination inexact Newton method (INB-adaptive NE) that relaxes such impractical assumption. We use the global information obtained by an intermediate INB solution to define adaptively the to-be- eliminated components. Taking the case of transonic flows passing the airfoil as a example, we report the numerical results obtained by a cluster of PCs that through the information exchange between the global and local data, the efficiency and robustness of INB can be greatly improved. Furthermore, INB-adaptive NE is nonlinearly scalable with respect to the number of processors similar to ASPIN. Due to the flexibility for the selection of the global Jacobian solver, the linear scalability of the method can be easily taken care by multilevel Schwarz Krylov subspace method.

Steepest descent preconditioning for nonlinear GMRES optimization

SUMMARYSteepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N‐GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, whereas the second employs a predefined small step. A simple global convergence proof is provided for the N‐GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N‐GMRES optimization is also motivated by relating it to standard non‐preconditioned GMRES for linear systems in the case of a standard quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N‐GMRES optimization algorithm is able to very significantly accelerate convergence of stand‐alone steepest descent optimization. Moreover, performance of steepest‐descent preconditioned N‐GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited‐memory Broyden–Fletcher–Goldfarb–Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest‐descent preconditioned N‐GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared with established techniques. In addition, it is argued that the real potential of the N‐GMRES optimization framework lies in the fact that it can make use of problem‐dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N‐GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example. Copyright © 2012 John Wiley & Sons, Ltd.

Parallel nonlinear preconditioners on multicore architectures

Parallel nonlinear preconditioners, for solving mildly nonlinear systems, are proposed. These algorithms are based on both the Fletcher---Reeves version of the nonlinear conjugate gradient method and a polynomial preconditioner type based on block two-stage methods. The behavior of these algorithms is analyzed when incomplete LU factorizations are used in order to obtain the inner splittings of the block two-stage method. As our illustrative example we have considered a nonlinear elliptic partial differential equation, known as the Bratu problem. The reported experiments show the performance of the algorithms designed in this work on two multicore architectures.

On nonlinear preconditioners in Newton–Krylov methods for unsteady flows

AbstractThe application of nonlinear schemes like dual time stepping as preconditioners in matrix‐free Newton–Krylov‐solvers is considered and analyzed, with a special emphasis on unsteady viscous flows. We provide a novel formulation of the left preconditioned operator that says it is in fact linear in the matrix‐free sense, but changes the Newton scheme. This allows to get some insight in the convergence properties of these schemes, which is demonstrated through numerical results. Copyright © 2009 John Wiley & Sons, Ltd.

An element agglomeration nonlinear additive Schwarz preconditioned Newton method for unstructured finite element problems

This paper extends previous results on nonlinear Schwarz preconditioning (Cai and Keyes 2002) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The nonlocal finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in Jones and Vassilevski 2001. Then, the algebraic construction from Jones, Vassilevski and Woodward 2003 of the corresponding non-linear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. A numerical illustration is also provided.

A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier–Stokes equations

A nonlinear additive Schwarz preconditioned inexact Newton method (ASPIN) was introduced recently for solving large sparse highly nonlinear systems of equations obtained from the discretization of nonlinear partial differential equations. In this paper, we discuss some extensions of ASPIN for solving steady-state incompressible Navier–Stokes equations with high Reynolds numbers in the velocity–pressure formulation. The key idea of ASPIN is to find the solution of the original system by solving a nonlinearly preconditioned system that has the same solution as the original system, but with more balanced nonlinearities. Our parallel nonlinear preconditioner is constructed using a nonlinear overlapping additive Schwarz method. To show the robustness and scalability of the algorithm, we present some numerical results obtained on a parallel computer for two benchmark problems: a driven cavity flow problem and a backward-facing step problem with high Reynolds numbers. The sparse nonlinear system is obtained by applying a Q 1 − Q 1 Galerkin least squares finite element discretization on two-dimensional unstructured meshes. We compare our approach with an inexact Newton method using different choices of forcing terms. Our numerical results show that ASPIN has good convergence and is more robust than the traditional inexact Newton method with respect to certain parameters such as the Reynolds number, the mesh size, and the number of processors.

Nonlinearly Preconditioned Inexact Newton Algorithms

Inexact Newton algorithms are commonly used for solving large sparse nonlinear system of equations $F(u^{\ast})=0$ arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of $\|F\|$, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution $u^{\ast}$, one may want to solve instead an equivalent nonlinearly preconditioned system ${\cal F}(u^{\ast})=0$ whose nonlinearities are more balanced. In this paper, we propose and study a nonlinear additive Schwarz-based parallel nonlinear preconditioner and show numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails.