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1 A Matrix Free Newton /Krylov Method For Coupling Complex Multi-Physics Subsystems Yunlin Xu School of Nuclear Engineering Purdue University October 23, 2006
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2 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
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3 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
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4 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
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5 Coupled Subsystem Example: Nuclear Reactor Simulation
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6 Time Advancement: Marching vs Nest Scheme TRAC-E PARCS step n step n+1 –The time steps must be kept small for accuracy and stability concerns Marching Nested TRAC-E PARCS step n step n+1 Converged ? N Y N Y N Y –Computational cost for each time step increased –Numerical Stability and accuracy can be improved –Time step size may be extended
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7 Ringhals BWR Stability 48 hours on 2 GHz machine for initialization! 128 Chans
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8 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”.
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9 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
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10 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
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11 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:
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12 Local Convergence of INM The convergence of INM depends on the inner residual, assume –If p 2, 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. Inexact Newton Method (INM) Local Convergence of INM
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13 Local Convergence of MFNK The inner residual consists with: MFNK is an INM –iterative residual –finite difference residual There are two conflicting sources of error in finite difference: –Truncation error –Round off error
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14 Local Convergence of MFNK (cont.) In theory, MFNK has local q-linear convergence, if In practice, MFNK can achieve q-quadratic convergence, if The optimal should balance the round-off error and truncation error
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15 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)
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16 Global Convergence Strategies Solution x* of system of nonlinear equation: F ( x )=0 is also the global minimizer of optimization problem: Newton step s N is the step from current solution to global minimizer of model problem: f(x c +s N ) may be larger than f(x c ), due to big step s N such that m(x c +s N ) is no longer a good approximation of f(x c +s N ). In this case, we need globally convergent strategy to force f(x c +s N )<f(x c )
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17 Descent Direction Newton step is descent direction of both objective function and its model: For any descent direction p k, there exist λ satisfies: (1) α-condition β-condition A sequence { x k } generated by x k+1 = x k +λ k p k satisfying previous condition will converge to a minimizer of f ( x ). (2) (1),(2) proofs can be found in Dennis & Schnabel’s book
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18 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
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19 Information Required in Quadratic Model Two function values: One Gradient Approximations for Gradient in MFNK or
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20 Model Trust Region Minimize model function in neighborhood, trust region xcxc xNxN cc x c +s( c ) sNsN subject to
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21 Double Dogleg Curve Approximate optimal path with double dogleg curve C.P. xcxc xNxN N cc sNsN Step along double dogleg curve.
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22 Cauchy Point Cauchy Point is minimizer in steepest descent direction Projection of Step for Cauchy Point on Krylov subspace (Brown & Saad)
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23 Example Problem I: Polynomials Two dimensional second order polynomials Solution Jacobian Nonlinear level
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24 PLY 1 Truncation Error Dominated Case
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25 PLY 1 Errors
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26 PLY 1 Step Sizes Optimal Empirical
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27 PLY 1 Step Size Parametric
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28 PLY 2 Round Off Error Dominated Case
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29 PLY 2 Errors
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30 PLY 2 Stepsizes Optimal Empirical
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31 PLY 2 Stepsize Parametric
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32 Numerical Examples: Navier-Stokes-Like Problem (Goyon) PDE DiffusionConvectionNon-physicalForce 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.
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33 The Finite difference Equations
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34 Structure of the Matrices JacobianDiag block
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35 NSL1: 50X50 meshes, =0.0015 Solving (u,v) as One Nonlinear System, w 0 =(1-8 10 -3 ) w*
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36 NSL1: Newton Iterative error and residual ErrorResidual
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37 NSL1: Coupling Subsystem Solving u and v as two subsystem, and coupled by FPI or MFNK
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38 NSL1: Global convergence w 0 =(1-8 10 -3 ) w*
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39 NSL1 Residual
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40 NSL1: First Backtracking The first backtracking occurred after the first Newton iteration. The L2 norm of residual Lambda before 1.65150 No GS 2.42510 LS 1.07990 0.316829 MTR1.03694 0.316829
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41 NSL2: 1000X1000 meshes
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42 NSL2: Residuals for FPI and MFR FPIMFR
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43 NSL 2: Optimal vs Empirical Finite Difference Optimal Empirical
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44 NSL2 Step Sizes
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45 NSL2 Step Size parametric
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46 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.
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47 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.
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48 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
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