165 research outputs found
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.Comment: 29 pages, 14 figures, 13 table
Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method
For linear problems, domain decomposition methods can be used directly as
iterative solvers, but also as preconditioners for Krylov methods. In practice,
Krylov acceleration is almost always used, since the Krylov method finds a much
better residual polynomial than the stationary iteration, and thus converges
much faster. We show in this paper that also for non-linear problems, domain
decomposition methods can either be used directly as iterative solvers, or one
can use them as preconditioners for Newton's method. For the concrete case of
the parallel Schwarz method, we show that we obtain a preconditioner we call
RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is
similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all
components directly defined by the iterative method. This has the advantage
that RASPEN already converges when used as an iterative solver, in contrast to
ASPIN, and we thus get a substantially better preconditioner for Newton's
method. The iterative construction also allows us to naturally define a coarse
correction using the multigrid full approximation scheme, which leads to a
convergent two level non-linear iterative domain decomposition method and a two
level RASPEN non-linear preconditioner. We illustrate our findings with
numerical results on the Forchheimer equation and a non-linear diffusion
problem
Additive and Hybrid Nonlinear Two-Level Schwarz Methods and Energy Minimizing Coarse Spaces for Unstructured Grids
Nonlinear domain decomposition (DD) methods, such as, e.g., ASPIN (Additive Schwarz Preconditioned Inexact Newton), RASPEN (Restricted Additive Schwarz Preconditioned Inexact Newton), Nonlinear-FETI-DP, or Nonlinear-BDDC methods, can be reasonable alternatives to classical Newton-Krylov-DD methods for the solution of sparse nonlinear systems of equations, e.g., arising from a discretization of a nonlinear partial differential equation. These nonlinear DD approaches are often able to effectively tackle unevenly distributed nonlinearities and outperform Newton’s method with respect to convergence speed as well as global convergence behavior. Furthermore, they often improve parallel scalability due to a superior ratio of local to global work.
Nonetheless, as for linear DD methods, it is often necessary to incorporate an appropriate coarse space in a second level to obtain numerical scalability for increasing numbers of subdomains. In addition to that, an appropriate coarse space can also improve the nonlinear convergence of nonlinear DD methods.
In this paper, four variants how to integrate coarse spaces in nonlinear Schwarz methods in an additive or multiplicative way using Galerkin projections are introduced. These new variants can be interpreted as natural nonlinear equivalents to well-known linear additive and hybrid two-level Schwarz preconditioners. Furthermore, they facilitate the use of various coarse spaces, e.g., coarse spaces based on energy-minimizing extensions, which can easily be used for irregular domain decompositions, as, e.g., obtained by graph partitioners. In particular, Multiscale Finite Element Method (MsFEM) type coarse spaces are considered, and it is shown that they outperform classical approaches for certain heterogeneous nonlinear problems.
The new approaches are then compared with classical Newton-Krylov-DD and nonlinear one-level Schwarz approaches for different homogeneous and heterogeneous model problems based on the p-Laplace operator
An efficient and effective nonlinear solver in a parallel software for large scale petroleum reservoir simulation
Abstract. We study a parallel Newton-Krylov-Schwarz (NKS) based algorithm for solving large sparse systems resulting from a fully implicit discretization of partial differential equations arising from petroleum reservoir simulations. Our NKS algorithm is designed by combining an inexact Newton method with a rank-2 updated quasi-Newton method. In order to improve the computational efficiency, both DDM and SPMD parallelism strategies are adopted. The effectiveness of the overall algorithm depends heavily on the performance of the linear preconditioner, which is made of a combination of several preconditioning components including AMG, relaxed ILU, up scaling, additive Schwarz, CRPlike(constraint residual preconditioning), Watts correction, Shur complement
Newton additive and multiplicative Schwarz iterative methods
Convergence properties are presented for Newton additive and multiplicative Schwarz (AS and MS) iterative methods for the solution of nonlinear systems in several variables. These methods consist of approximate solutions of the linear Newton step using either AS or MS iterations, where overlap between subdomains can be used. Restricted versions of these methods are also considered. These Schwarz methods
can also be used to precondition a Krylov subspace method for the solution of the linear Newton
steps. Numerical experiments on parallel computers are presented, indicating the effectiveness of these
methods.The Spanish Ministry of Science and Education (TIN2005-09037-C02-02); Universidad de Alicante
(VIGROB-020); the U.S. Department of Energy (DE-FG02-05ER25672)
Nonlinear Schwarz preconditioning for nonlinear optimization problems with bound constraints
We propose a nonlinear additive Schwarz method for solving nonlinear
optimization problems with bound constraints. Our method is used as a
"right-preconditioner" for solving the first-order optimality system arising
within the sequential quadratic programming (SQP) framework using Newton's
method. The algorithmic scalability of this preconditioner is enhanced by
incorporating a solution-dependent coarse space, which takes into account the
restricted constraints from the fine level. By means of numerical examples, we
demonstrate that the proposed preconditioned Newton methods outperform standard
active-set methods considered in the literature
Solution strategies for nonlinear conservation laws
Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation
A Unifying Theory for Nonlinear Additively and Multiplicatively Preconditioned Globalization Strategies : Convergence Results and Examples From the Field of Nonlinear Elastostatics and Elastodynamics
Nonlinear right preconditioned globalization strategies for the solution of nonlinear programming problems of the following kind where is a convex set of admissible solutions, , and , sufficiently smooth, are presented. Preconditioned globalization strategies are traditional Linesearch or Trust-Region strategies in combination with a nonlinear update operator which results from a nonlinear solution process for smaller, but related, nonlinear programming problems. We will formulate conditions on this abstract operator, in order to ensure global convergence, i.e., convergence to first-order critical points, of the resulting method. In addition, we introduce particular implementations of this abstract operator, i.e., nonlinear multiplicatively preconditioned Trust-Region (MPTS) and Linesearch strategies (MPLS), as well as nonlinear additively preconditioned Trust-Region (APTS) and Linesearch (APLS) strategies. As it turns out, these additive strategies are novel parallel, locally adaptive and robust solution methods for nonlinear programming problems. Moreover, the MPTS strategy generalizes the RMTR concepts in [GK08] in order to allow also for the application of alternating nonlinear domain decomposition methods. On the other hand, the MPLS method simplifies and generalizes the concepts in [WG08] giving rise to a novel solution strategy for pointwise constrained nonlinear programming problems. The respective nonlinear solution strategies are analyzed and global convergence is shown. In addition, global convergence is also shown for combined nonlinear additively and multiplicatively preconditioned Trust-Region and Linesearch strategies. Moreover, we show the efficiency and reliability of these methods in the context of problems arising from the field of nonlinear elasticity in 3d. Particular emphasis has been placed on the formulation and analysis of the resulting minimization problems. Here, we show that these problems satisfy the assumptions stated to show convergence of the respective preconditioned globalization strategies. Moreover, various elasto-static and elasto-dynamic examples are presented in order to compare the convergence rates and runtimes of the different strategies
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Scalability of preconditioners as a strategy for parallel computation of compressible fluid flow
Parallel implementations of a Newton-Krylov-Schwarz algorithm are used to solve a model problem representing low Mach number compressible fluid flow over a backward-facing step. The Mach number is specifically selected to result in a numerically {open_quote}stiff{close_quotes} matrix problem, based on an implicit finite volume discretization of the compressible 2D Navier-Stokes/energy equations using primitive variables. Newton`s method is used to linearize the discrete system, and a preconditioned Krylov projection technique is used to solve the resulting linear system. Domain decomposition enables the development of a global preconditioner via the parallel construction of contributions derived from subdomains. Formation of the global preconditioner is based upon additive and multiplicative Schwarz algorithms, with and without subdomain overlap. The degree of parallelism of this technique is further enhanced with the use of a matrix-free approximation for the Jacobian used in the Krylov technique (in this case, GMRES(k)). Of paramount interest to this study is the implementation and optimization of these techniques on parallel shared-memory hardware, namely the Cray C90 and SGI Challenge architectures. These architectures were chosen as representative and commonly available to researchers interested in the solution of problems of this type. The Newton-Krylov-Schwarz solution technique is increasingly being investigated for computational fluid dynamics (CFD) applications due to the advantages of full coupling of all variables and equations, rapid non-linear convergence, and moderate memory requirements. A parallel version of this method that scales effectively on the above architectures would be extremely attractive to practitioners, resulting in efficient, cost-effective, parallel solutions exhibiting the benefits of the solution technique
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