Quasi-Newton methods in preCICE for bounded variables

Quasi-Newton methods in preCICE for bounded variables Jun Chen1Miriam Schulte1 1Institute for Parallel and Distributed Systems University of Stuttgart September 11, 2025 Multi-secant quasi-Newton methods •Problem: solve a fixed-point problem x=S(F(x)) on an interface •Motivation: fixed-point iterations can converge slowly by strong coupling •Limitation: only the sequence of iterates and residuals on the interface are available •Idea: approximate the inverse of the Jacobian from history of the residual to accelerate convergence Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 2/13 Multi-secant quasi-Newton methods •Idea: approximate the inverse of the Jacobian from history of the residual to accelerate convergence •Minimization problem: ∥c Je R −1(˜xk)∥F for IQN-ILS or ∥c J˜ R −1(˜xk)− \ J˜ Rprev −1∥Ffor IQN-IMVJ subject to the secant condition: c Je R −1(˜xk)Vk≈Wk •xk+1 =˜ xk+J−1 ˜ R(˜xk)(−R(˜xk)) Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 3/13 Quasi-Newton methods in preCICE From tutorials/breaking-dam-2d/precice-config.xml 1<acceleration:IQN -ILS > 2<data name ="Displacement" mesh="Solid - Mesh " /> 3<data name =" Force " mesh="Solid -Mesh" /> 4<preconditioner type=" residual - sum" /> 5<filter type =" QR2" limit ="1e -1 " /> 6<initial - relaxation value =" 0.1" /> 7<max -used - iterations value =" 100" /> 8<time - windows - reused value =" 10" /> 9</ acceleration:IQN -ILS > Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 4/13 Quasi-Newton methods in preCICE From tutorials/elastic-tube-3d/precice-config.xml 1<acceleration:IQN -IMVJ > 2.... 3<imvj -restart -mode type ="RS -SVD " chunk -size="8" reused -time - windows -at - restart ="8" truncation - threshold =" 0.0001 "/> 4</ acceleration:IQN - IMVJ > Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 5/13 Acceleration A full iteration: xkFixed-Point −−−−−−→ ˜xkQN −−→ xk+1 Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 6/13 Overshooting problem •Overshooting: QN step towards certain direction may be too large, leading to values outside the physical bound •Scenario: Problems with temperature, concentration fields etc. •Visualisation: compute x(k+1) =x(k)−[b J(k)]−1r(x(k))of a 2-DoF system (x1= sin(x1x2+ 0.12) x2= sin(x1x2 2+ 0.15) Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 7/13 Overshooting problem Range of y-axis: [-0.2, 0.6] In literature: Iteration process of JFNK method and physical constraints, Article in Chinese Journal of Computational Physics, September 2012 Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 8/13 Approaches to prevent overshooting Assumptions: •The solvers alway give output in the physical bound •The bound is simple Design goals: •Feasible each iterate •Minimal overhead •Modular implementation •Retain convergence speed Jun Chen Quasi-Newton methods in preCICE for bounded variables September 11, 2025 9/13