The Experts below are selected from a list of 14373 Experts worldwide ranked by ideXlab platform
W J Rider - One of the best experts on this subject based on the ideXlab platform.
-
physics based preconditioning and the newton Krylov Method for non equilibrium radiation diffusion
Journal of Computational Physics, 2000Co-Authors: Vincent A Mousseau, D.a. Knoll, W J RiderAbstract:An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical Methods, Jacobian-free Newton?Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's Method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, we employ a Jacobian?free implementation of Newton's Method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov Method used to solve the linear systems that come from the iterations of Newton's Method. The preconditioner in this solution Method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid Methods. The preconditioner also addresses the strong coupling between equations with local 2×2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements.
-
a multigrid preconditioned newton Krylov Method
SIAM Journal on Scientific Computing, 1999Co-Authors: D.a. Knoll, W J RiderAbstract:We study multigrid preconditioning of matrix-free Newton--Krylov Methods as a means of developing more efficient nonlinear iterative Methods for large scale simulation. Newton--Krylov Methods have proven dependable in solving nonlinear systems while not requiring the explicit formation or storage of the complete Jacobian. However, the standard algorithmic scaling of Krylov Methods is nonoptimal, with increasing linear system dimension. This motivates our use of multigrid-based preconditioning. It is demonstrated that a simple multigrid-based preconditioner can effectively limit the growth of Krylov iterations as the dimension of the linear system is increased. Different performance aspects of the proposed algorithm are investigated on three nonlinear, nonsymmetric, boundary value problems. Our goal is to develop a hybrid Methodology which has Newton--Krylov nonlinear convergence properties and multigrid-like linear convergence scaling for large scale simulation.
-
a multigrid newton Krylov Method for multimaterial equilibrium radiation diffusion
Journal of Computational Physics, 1999Co-Authors: W J Rider, D.a. Knoll, Gordon L OlsonAbstract:We focus on a fully implicit, nonlinearly converged, solution of multimaterial equilibrium radiation diffusion problems. The nonlinear Method of solution is a Newton?Krylov (generalized minimum residual, GMRES) Method preconditioned by a multigrid Method. The multigrid iteration matrix results from a Picard-type linearization of the governing equations. The governing equation is highly nonlinear with the principal forms of nonlinearity found in the fourth-order dependence of the radiation energy on temperature, the temperature dependence of the opacity, and flux limiting. The efficiency of both the linear and nonlinear iterative techniques is investigated. With the realistic time step control the solution of the linear system does not scale linearly with multigrid as might be expected from theory. In contrast, we find that the use of multigrid to precondition a Newton?Krylov (GMRES) Method provides a robust, scalable solution for the nonlinear system. Also only through converging the nonlinearities within a time step does the solution Method achieve its design accuracy.
D.a. Knoll - One of the best experts on this subject based on the ideXlab platform.
-
implementation of the jacobian free newton Krylov Method for solving the first order ice sheet momentum balance
Journal of Computational Physics, 2011Co-Authors: Jeanfrancois Lemieux, D.a. Knoll, Stephen Price, Katherine J Evans, Andrew G Salinger, David M Holland, A J PayneAbstract:We have implemented the Jacobian-free Newton-Krylov (JFNK) Method for solving the first-order ice sheet momentum equation in order to improve the numerical performance of the Glimmer-Community Ice Sheet Model (Glimmer-CISM), the land ice component of the Community Earth System Model (CESM). Our JFNK implementation is based on significant re-use of existing code. For example, our physics-based preconditioner uses the original Picard linear solver in Glimmer-CISM. For several test cases spanning a range of geometries and boundary conditions, our JFNK implementation is 1.8-3.6 times more efficient than the standard Picard solver in Glimmer-CISM. Importantly, this computational gain of JFNK over the Picard solver increases when refining the grid. Global convergence of the JFNK solver has been significantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact Newton Method. While a diverse set of test cases show that our JFNK implementation is usually robust, for some problems it may fail to converge with increasing resolution (as does the Picard solver). Globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of Picard and JFNK and the use of homotopy Methods.
-
acceleration of k eigenvalue criticality calculations using the jacobian free newton Krylov Method
Nuclear Science and Engineering, 2011Co-Authors: D.a. Knoll, Hyeongkae Park, Christopher K NewmanAbstract:We present a new approach for the $k$--eigenvalue problem using a combination of classical power iteration and the Jacobian--free Newton--Krylov Method (JFNK). The Method poses the $k$--eigenvalue problem as a fully coupled nonlinear system, which is solved by JFNK with an effective block preconditioning consisting of the power iteration and algebraic multigrid. We demonstrate effectiveness and algorithmic scalability of the Method on a 1-D, one group problem and two 2-D two group problems and provide comparison to other efforts using silmilar algorithmic approaches.
-
tightly coupled multiphysics algorithms for pebble bed reactors
Nuclear Science and Engineering, 2010Co-Authors: Hyeongkae Park, D.a. Knoll, Derek Gaston, Richard C MartineauAbstract:We have developed a tightly coupled multiphysics simulation tool for the pebble-bed reactor (PBR) concept, a type of Very High-Temperature gas-cooled Reactor (VHTR). The simulation tool, PRONGHORN, takes advantages of the Multiphysics Object-Oriented Simulation Environment library, and is capable of solving multidimensional thermal-fluid and neutronics problems implicitly with a Newton-based approach. Expensive Jacobian matrix formation is alleviated via the Jacobian-free Newton-Krylov Method, and physics-based preconditioning is applied to minimize Krylov iterations. Motivation for the work is provided via analysis and numerical experiments on simpler multiphysics reactor models. We then provide detail of the physical models and numerical Methods in PRONGHORN. Finally, PRONGHORN's algorithmic capability is demonstrated on a number of PBR test cases.
-
tightly coupled multi physics simulations for pebble bed reactors
2009Co-Authors: Hyeongkae Park, D.a. Knoll, D Gaston, Samet Y Kadioglu, R Martineau, William Taitano, D LebrungrandieAbstract:We have developed a simulation tool for Pebble-bed reactor (PBR), a type of Very High-Temperature gas-cooled Reactor (VHTR). The simulation tool, PRONGHORN, takes advantages of the Multiphysics Object-Oriented Simulation library, and is capable of solving multidimensional thermal-fluid and neutronics problems implicitly in parallel. Expensive Jacobian matrix formation is alleviated by the Jacobian-free Newton-Krylov Method, and physics-based preconditioning is applied to improve the convergence. Validity of the thermal-fluid solver has been tested and the comparison has revealed a limitation of the current flow model (Darcy’s flow) near boundaries, while the neutronics solver agrees well with other codes on the PBMR400 benchmark problem.
-
physics based preconditioning and the newton Krylov Method for non equilibrium radiation diffusion
Journal of Computational Physics, 2000Co-Authors: Vincent A Mousseau, D.a. Knoll, W J RiderAbstract:An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical Methods, Jacobian-free Newton?Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's Method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, we employ a Jacobian?free implementation of Newton's Method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov Method used to solve the linear systems that come from the iterations of Newton's Method. The preconditioner in this solution Method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid Methods. The preconditioner also addresses the strong coupling between equations with local 2×2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements.
Gordon L Olson - One of the best experts on this subject based on the ideXlab platform.
-
a multigrid newton Krylov Method for multimaterial equilibrium radiation diffusion
Journal of Computational Physics, 1999Co-Authors: W J Rider, D.a. Knoll, Gordon L OlsonAbstract:We focus on a fully implicit, nonlinearly converged, solution of multimaterial equilibrium radiation diffusion problems. The nonlinear Method of solution is a Newton?Krylov (generalized minimum residual, GMRES) Method preconditioned by a multigrid Method. The multigrid iteration matrix results from a Picard-type linearization of the governing equations. The governing equation is highly nonlinear with the principal forms of nonlinearity found in the fourth-order dependence of the radiation energy on temperature, the temperature dependence of the opacity, and flux limiting. The efficiency of both the linear and nonlinear iterative techniques is investigated. With the realistic time step control the solution of the linear system does not scale linearly with multigrid as might be expected from theory. In contrast, we find that the use of multigrid to precondition a Newton?Krylov (GMRES) Method provides a robust, scalable solution for the nonlinear system. Also only through converging the nonlinearities within a time step does the solution Method achieve its design accuracy.
Martin Stoll - One of the best experts on this subject based on the ideXlab platform.
-
a low rank inexact newton Krylov Method for stochastic eigenvalue problems
Computational methods in applied mathematics, 2019Co-Authors: Peter Benner, Akwum Onwunta, Martin StollAbstract:This paper aims at the efficient numerical solution of stochastic eigenvalue problems. Such problems often lead to prohibitively high dimensional systems with tensor product structure when discretized with the stochastic Galerkin Method. Here, we exploit this inherent tensor product structure to develop a globalized low-rank inexact Newton Method with which we tackle the stochastic eigenproblem. We illustrate the effectiveness of our solver with numerical experiments.
Vincent A Mousseau - One of the best experts on this subject based on the ideXlab platform.
-
physics based preconditioning and the newton Krylov Method for non equilibrium radiation diffusion
Journal of Computational Physics, 2000Co-Authors: Vincent A Mousseau, D.a. Knoll, W J RiderAbstract:An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical Methods, Jacobian-free Newton?Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's Method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, we employ a Jacobian?free implementation of Newton's Method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov Method used to solve the linear systems that come from the iterations of Newton's Method. The preconditioner in this solution Method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid Methods. The preconditioner also addresses the strong coupling between equations with local 2×2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements.
