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A Matrix Free Newton /Krylov Method For Coupling Complex Multi-Physics Subsystems

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##### A Matrix Free Newton /Krylov Method For Coupling Complex Multi-Physics Subsystems

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**A Matrix Free Newton /Krylov Method For Coupling Complex**Multi-Physics Subsystems Yunlin XuSchool of Nuclear EngineeringPurdue UniversityOctober 23, 2006**Content**• Introduction • MFNK and Optimal Perturbation Size Fixed Point Iteration (FPI) for coupling subsystems • A Matrix Free Newton/Krylov method based on FPI • Local Convergence analysis of MFNK • Truncation and Round-off Error • Estimation of Optimal Finite Difference Perturbation • Global Convergence strategies • Line search • Model trust region • Numerical Examples • Summary**Features of Multi-Physics Subsystems**• Multiple nonlinear subsystems are coupled together: The solution of each subsystem depends on some external variables which come from the other system : internal variables : external variables • Each subsystem can be solved with reliable methods as long as they remain decoupled**Two General Approaches for Coupling Subsystems**• Analytic Approach: reformulate the coupled system into a larger system of equations • Standard Newton-type methods can be applied • Synthetic Approach: combine the subsystem solvers for the coupled system • utilize the well-tested and reliable solutions of each of the subsystems because: • It may be too expensive to reformulate the coupled system and forego the significant investment in developing reliable solvers for each of the subsystems. • One of the subsystems may be solved using commercial software that prevents access to the source code which makes it impossible to reformulate the coupled equations for the analytic approach**step n**step n+1 N N N Y Y Y TRAC-E PARCS Converged? Converged? Converged? step n+1 step n TRAC-E PARCS Time Advancement:Marching vs Nest Scheme • The time steps must be kept small for accuracy and stability concerns • Marching • Nested • Computational cost for each time step increased • Numerical Stability and accuracy can be improved • Time step size may be extended**128 Chans**Ringhals BWR Stability • 48 hours on 2 GHz machine for initialization!**Synthetic Approaches**• Nested Iteration • Subsystems are chained in block Gauss-Seidel or block Jacobi iteration • Convergence is not guaranteed. • Matrix Free Newton/Krylov Method • Approximate Mat-Vec by quotient: • The system Jacobian is not constructed • Local Convergence guaranteed • Problems with Direct Application of MFNK for Coupling of Subsystems • Solvers for the subsystems are not fully utilized • Difficult to find a good preconditioner for MFNK • In some cases, it is not possible to obtain residuals for a subsystem if the solver of subsystem is commercial software which can be used only as a “black box”.**Objectives of Research**• Propose a general approach to implement efficient matrix free Newton/Krylov methods for coupling complex subsystems with their respective solvers • Identify and address specific issues which arise in implementing MFNK for practical applications • Local convergence analysis of the matrix free Newton/Krylov method • Optimal perturbation size for the finite difference approximation in MFNK • Globally convergent strategies**Fixed Point Iteration for Coupling Subsystems**• Block Iteration • Block Iteration for coupling subsystems are fixed point iterations • The condition for convergence of FPI is ||Φ(x*)||<1. • If ||Φ(x*)||>1, then the FPI may diverge**Matrix Free Newton Krylov Method Based on a Fixed Point**iteration • Define a nonlinear system: F(x)= Φ(x) -x =0 • The solution of this system is the fixed point of function Φ(x), which is also the solution of original coupled nonlinear system • MFNK algorithm:**Local Convergence of INM**• Inexact Newton Method (INM) • Local Convergence of INM • The convergence of INM depends on the inner residual, assume • If p2, the INM has local q-quadratic convergence. • If 1<p<2, INM converges with q-order at least p. • If p=1 and , the INM has local q-linear convergence.**Local Convergence of MFNK**• MFNK is an INM The inner residual consists with: • iterative residual • finite difference residual • There are two conflicting sources of error in finite difference: • Truncation error • Round off error**Local Convergence of MFNK (cont.)**• The optimal should balance the round-off error and truncation error • In theory, MFNK has local q-linear convergence, if • In practice, MFNK can achieve q-quadratic convergence, if**Optimal vs Empirical Perturbation Size**• The norm of the Jacobian and can be estimated with information provided by the MFNK algorithm: or • An empirical prescription was proposed attempt to balance the truncation and round-off errors (Dennis)**Global Convergence Strategies**• Solution x* of system of nonlinear equation: F(x)=0 is also the global minimizer of optimization problem: • Newton step sN is the step from current solution to global minimizer of model problem: • f(xc+sN) may be larger than f(xc), due to big stepsN such that m(xc+sN) is no longer a good approximation off(xc+sN). In this case, we need globally convergent strategy to force f(xc+sN)<f(xc)**Descent Direction**• Newton step is descent direction of both objective function and its model: • For any descent direction pk, there exist λ satisfies: (1) α-condition β-condition • A sequence {xk} generated by xk+1=xk+λkpk satisfying previous condition will converge to a minimizer of f(x). (2) (1),(2) proofs can be found in Dennis & Schnabel’s book**Line Search**• Take MF Newton step as descent direction, and select λ to minimize a model of following function • Quadratic model • λ predicted from quadratic model**Information Requiredin Quadratic Model**• Two function values: • One Gradient • Approximations for Gradient in MFNK or**xc+s(c)**xN sN xc c Model Trust Region • Minimize model function in neighborhood, trust region subject to**C.P.**xN N sN xc c Double Dogleg Curve • Approximate optimal path with double dogleg curve • Step along double dogleg curve.**Cauchy Point**• Cauchy Point is minimizer in steepest descent direction • Projection of Step for Cauchy Point on Krylov subspace (Brown & Saad)**Example Problem I: Polynomials**• Two dimensional second order polynomials • Solution • Jacobian • Nonlinear level**PLY 1 Step Sizes**Optimal Empirical**PLY 2 Stepsizes**Optimal Empirical**Numerical Examples: Navier-Stokes-Like Problem (Goyon)**• PDE Diffusion Convection Non-physical Force function • Boundary Condition • Force function • Goyon, Precoditioned Newton Methods using Incremental Unknowns Methods for Resolution of a Steady-State Navier-Stokes-Like Problem. Applied Mathematic and Computation, 87(1997), pp. 289-311.**Structure of the Matrices**Jacobian Diag block**NSL1: 50X50 meshes, =0.0015**• Solving (u,v) as One Nonlinear System, w0=(1-810-3) w***NSL1: Newton Iterative error and residual**Error Residual**NSL1: Coupling Subsystem**• Solving u and v as two subsystem, and coupled by FPI or MFNK**NSL1: Global convergence**w0=(1-810-3) w***NSL1: First Backtracking**The first backtracking occurred after the first Newton iteration. The L2 norm of residualLambda before 1.65150No GS 2.42510 LS 1.07990 0.316829 MTR 1.03694 0.316829**NSL2: Residuals for FPI and MFR**FPI MFR**NSL 2: Optimal vs Empirical Finite Difference**Optimal Empirical**Summary**• A general approach, MFNK, was presented here for coupling subsystems with respective solvers. • Based on any FPI, a corresponding MFNK method can be constructed. • MFNK provides a more efficient method than FPI for coupling subsystems. • MFNK can converge for several cases in which the corresponding FPI diverges. • Locally, MFNK converges at least q-linearly and in many cases q-quadratically. • A more sophisticated FPI scheme provides a more efficient nonlinear system for the corresponding MFNK.**Summary (Cont.)**• A method was proposed to estimate the optimal perturbation size for matrix free Newton/Krylov methods. • The method was based on an analysis of the truncation error and the round-off error introduced by the finite difference approximation. • The optimal perturbation size can be accurately estimated in the MFNK algorithm with almost no additional computational cost. • Numerical examples shows that the optimum perturbation size leads to improved convergence of the MFNK method compared to the perturbation determined by empirical formulas.**Summary (Cont.)**• Line Search and Model Trust Region, were implemented within the framework of MFNK • For the line search method, a quadratic or higher order model was used with an approximation for the gradient • For the model trust region strategy, a double dogleg approach was implemented using the projection of a Newton step and Cauchy point within the Krylov subspace • the model trust region strategy showed better local performance than the line search strategy • Peer-to-Peer parallel MFNK algorithm was implemented