2014.10.dartmouth
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This document discusses computational simulations of chaotic systems and the challenges of sensitivity analysis and optimization for such systems. It introduces the concept of Least Squares Shadowing as a solution, which formulates the problem as a least squares problem without an initial condition to avoid the divergence of solutions seen in traditional sensitivity analysis of chaotic systems. Algorithms for solving the Least Squares Shadowing problem are also presented.
2014.10.dartmouth
- 1. Adjoin Chaos ”Least Squares Shadowing” sensitivity analysis of chaotic high fidelity simulations Qiqi Wang Assistant Professor of Aero & Astro, MIT We gratefully acknowledge: U Mass Dartmouth, Oct 29, 2014
- 2. 2 Use of Computational Simulations Analysis Simulation on manually picked geometry and condition Design Beyond single simulation, towards optimization and parametric study
- 3. 3 Use of Computational Simulations Analysis Simulation on manually picked geometry and condition Design Beyond single simulation, towards optimization and parametric study Low fidelity simulation Potential flow solver, RANS, URANS High fidelity Simulation Large Eddy Simulation (LES), Detached Eddy Simulation (DES), Unsteady Multi-physics Simulations
- 4. 4 Use of Computational Simulations Analysis Simulation on manually picked geometry and condition Design Beyond single simulation, towards optimization and parametric study Low fidelity simulation Potential flow solver, RANS, URANS THE FRONTIER High fidelity Simulation Large Eddy Simulation (LES), Detached Eddy Simulation (DES), Unsteady Multi-physics Simulations THE FRONTIER ESTABLISHED
- 5. 5 Use of Computational Simulations Analysis Simulation on manually picked geometry and condition Design Beyond single simulation, towards optimization and parametric study Low fidelity simulation Potential flow solver, RANS, URANS THE FRONTIER High fidelity Simulation Large Eddy Simulation (LES), Detached Eddy Simulation (DES), Unsteady Multi-physics Simulations THE FRONTIER ESTABLISHED
- 6. • Reduce time and resources used in physical testing • 1970s – Boeing 757, 77 wings tested in wind tunnel. • 1990s – Boeing 777, 11 wings tested in wind tunnel. • Simulation-based design limited by the fidelity of simulation • Unsteady effects (noise, vibration) • Complex turbulent flows (separation, mixing) • 2000s – Boeing 787, 11 wings tested in wind tunnel. • High fidelity design can lead to faster, better next generation aerospace vehicles. Why Design using High Fidelity Simulations?
- 7. 7 From top500.org 10 years 1000 times In 10 years, you will think about a 100,000-core computer approximately the way you think about a 100-core computer today Total GFLOPS, top500 Fastest supercomputer 500th supercomputer The hardware will get there How about the software and the math behind?
- 8. 8 What’s different in high fidelity simulations? Optimal Control Jet Engine Combustor (Stanford / DOE) Launch abort system (NASA) Maneuvering F18 (Airforce) Landing gear noise Design optimization Rotorcraft wake (NASA) • Many high fidelity simulations in important aerospace applications are unsteady and chaotic. • Design in Chaos: High fidelity, unsteady simulation and design.
- 9. Towards Optimization with LES
- 10. Towards Optimization: Parameterized LES of turbine blade – baseline
- 11. Towards Optimization: Parameterized LES of turbine blade – Optimized
- 12. A chaotic optimization 12 min 𝑠 1 𝑇 0 𝑇 𝐽 𝑢 𝑑𝑡 𝑠. 𝑡. 𝑑𝑢 𝑑𝑡 = 𝑓(𝑢, 𝑠) • Cost function is averaged over long T • f is chaotic Time DragCoefficient LiftCoefficient
- 13. What is chaos? Why does it matter? 13 Cylinder in Re = 500 flow, Wang and Gao 2013 How to perform sensitivity analysis and adjoint? Divergence of sensitivity, a.k.a. the “butterfly effect”, poses an interesting mathematical question.
- 14. Chaos: solutions diverge under small perturbations in the governing equation 14
- 15. What is chaos? Why does it matter? 15 Time DragCoefficient LiftCoefficient Chaos is not periodic. Predicting time averaged quantity with small sampling error requires very long simulation.
- 16. What is Optimization in Chaos? 16 min 𝑠 1 𝑇 0 𝑇 𝐽 𝑢 𝑑𝑡 𝑠. 𝑡. 𝑑𝑢 𝑑𝑡 = 𝑓(𝑢, 𝑠) • The cost function is averaged over a long T • f governs chaotic dynamics Time DragCoefficient LiftCoefficient
- 17. But how do I know? This could be YOUR objective function in a chaotic simulation This could be YOUR objective function in a periodic simulation A prime indicator: Divergence Of Derivative
- 18. An objective function produced by a chaotic simulation Time averaged objective in chaos and its derivative Derivatives computed with traditional adjoint method
- 19. 19 Divergence Of Derivative Lea et al. Sensitivity analysis of the climate of a chaotic system,Tellus A 2000 Lorenz system Output z: rate of heat transfer Input ρ: temperature difference Time averaged output Input Input Derivative of time averaged output
- 20. 20 Divergent solution of linearized equation in choatic 2D cylinder wake Credit: T. Barth, ASA Ames. Re=3900 Re=10000
- 21. 21 Divergent of Derivative in chaotic vortex shedding over stalled airfoil FUN3D, NASA Langley Re=10000 Magnitude of the adjoint gradient
- 22. 22 Divergent of Derivative in 3D turbulent cylinder wake Adjoint solution for a circular cylinder Re=500. z-vorticity CDP, Stanford CTR
- 23. Code NASA Ames (Barth) CTR (Stanford) NASA Langley (FUN3D) Numerical scheme Density-based Discontinuous Galerkin Pressure-based finite volume Density-based finite volume Flow field 2D cylinder wake 3D wake Jet in crossflow Stalled airfoil wake Solution to both linearized and adjoint Navier-Stokes equations diverge exponentially as flow develops unsteady, aperiodic (a.k.a. chaotic) behavior 23
- 24. 24 Derivative does not commute with time average Lea et al. Sensitivity analysis of the climate of a chaotic system,Tellus A 2000 Lorenz system Output z: rate of heat transfer Input ρ: temperature difference Time averaged output Input Input Derivative of time averaged output
- 25. Small perturbation leads to large difference in solution of initial value problems 25 Input ParameterInput Parameter Time averaged outputTime dependent output Time
- 26. In a chaotic problem with fixed initial condition, • Small perturbation leads to large difference in solution • Linearized (and adjoint) equations diverge • Sensitivity computation fails even for well- defined time average quantities. • A barrier to optimization, inference and uncertainty quantification using high fidelity simulations. 26
- 27. In a chaotic problem with fixed initial condition, • Small perturbation leads to large difference in solution • Linearized (and adjoint) equations diverge • Sensitivity computation fails even for well- defined time average quantities. • A barrier to optimization, inference and uncertainty quantification using high fidelity simulations. Initial value problem of chaos is ill-conditioned. 27
- 28. Is there a solution? Yes.
- 29. Is there a solution? Yes. min 𝑠 1 𝑇 0 𝑇 𝐽 𝑢 𝑑𝑡 𝑠. 𝑡. 𝑑𝑢 𝑑𝑡 = 𝑓(𝑢, 𝑠) Where f governs chaotic dynamics 𝑠. 𝑡. 𝑢 0 = 𝑢0 So that the solution is unique defined?
- 30. Is there a solution? Yes. min 𝑠 1 𝑇 0 𝑇 𝐽 𝑢 𝑑𝑡 𝑠. 𝑡. 𝑑𝑢 𝑑𝑡 = 𝑓(𝑢, 𝑠) Where f governs chaotic dynamics 𝑠. 𝑡. 𝑢 0 = 𝑢0 So that the solution is unique defined? Least Squares Shadowing (LSS): uniquely define the solution without incurring the “butterfly effect”. 10.1016/j.jcp.2014.03.002
- 31. What is least squares shadowing? • Initial Value Problem: 31
- 32. Solutions to similar initial value problems 32 • u(t) (ρ=30) and ur(t) (ρ=28) are very different after some time!
- 33. What is least squares shadowing? • Initial Value Problem: • Least Squares Problem: 33
- 34. 34 Time Transformation • Time transformation is required to keep the trajectories close in phase space for infinite time Without Time Transformation With Time Transformation
- 35. • Least squares problem with time transformation: • Assume Ergodicity: No initial condition for u! • “Integral phase condition” for finding homoclinic cycles in dynamical systems • Does using least squares instead of an initial condition remove the ill-conditioning of chaos? Least Squares Problem Reference solution 35
- 36. 36 • u(t) (ρ=30) and ur(t) (ρ=28) are very different after some time! Solutions to similar initial value problems
- 37. Solutions to similar Least Squares Shadowing Problems 37 • u(τ) (ρ=30) shadows ur(t) (ρ=28)!
- 38. 38 Least squares shadowing Condition number O(1) Initial value problem Condition number O(eλT) Input ParameterInput Parameter Time Time Time dependent output Time dependent output
- 39. New approach: Least Squares Shadowing • Linearize the initial value problem • Linearize the regularized problem without initial condition • Compute sensitivity of time averaged quantity from linearized solution • Wang, Hu and Blonigan. Sensitivity computation of chaotic limit cycle oscillations. Journal of Computational Physics. 2014. arXiv:1204.0159 39
- 40. where is solution to the linearized initial value problem regularized problem without initial condition “Least Squares Shadowing” 40
- 41. “Least Squares Shadowing” Works Computed derivative of the output to the input (Initial value problem) 41 Computed derivative of the output to the input (Least squares problem) • Linearize the regularized problem without initial condition • Compute sensitivity of time averaged quantity from linearized solution
- 42. An objective function produced by a chaotic simulation Time averaged objective in chaos and its derivative Derivatives computed with traditional adjoint method Derivatives computed with Least Squares Shadowing
- 43. 43 Shadowing Lemma Consider a system governed by: For any δ>0 there exists ε>0, such that for every “ε- pseudo-solution” u satisfying ǁdu/dt−f(u)ǁ<ε, there exists a true solution u satisfying du/dτ−f(u)=0 under a time transformation τ(t), such that ǁu(τ)−u(t)ǁ<δ, |1−dτ/dt|<δ Pilyugin SY. Shadowing in dynamical systems. Volume 1706, Lecture Notes in Mathematics, Springer: Berlin, 1999.
- 44. Collection of 100 Lorenz System trajectories Each color represents a shadowing trajectory at a different parameter value. 44
- 45. Parallel space-time multigrid enables least squares sensitivity computation for isotropic homogeneous turbulent flows 45 Taylor micro-scale Reynolds number = 33
- 46. How to compute the Least Squares Shadowing Solution 46 Algorithm I: Shooting • Search for initial condition, u(τ(T0)) Algorithm II: Full trajectory Design • Search for entire trajectory, u(τ(t)), T0≤t ≤T1 Algorithm III: Multiple shooting • Search for trajectory checkpoints, u(τ(t)), t = t0, t1,…,tK
- 47. • Solve for Lagrange multipliers, w: • Operator is 2nd order and coercive • 2nd order Boundary Value Problem in time. • Solved with Multigrid in space and time. Full Trajectory Design 47
- 48. • Guess u(t),w(t) at each checkpoint, solve in segment: • Use mismatch in u(τ)-ur(t) and w(t) at checkpoints to update guess: Algorithm III, Multiple shooting 48 0 2 4 6 8 10 -5 0 5 10 15 t u-ur 1 Iteration
- 49. • Guess u(t),w(t) at each checkpoint, solve in segment: • Use mismatch in u(τ)-ur(t) and w(t) at checkpoints to update guess: 49 0 2 4 6 8 10 -5 0 5 10 15 t u-ur 1 Iteration Algorithm III, Checkpoint Design
- 50. • Guess u(t),w(t) at each checkpoint, solve in segment: • Use mismatch in u(τ)-ur(t) and w(t) at checkpoints to update guess: 50 0 2 4 6 8 10 -5 0 5 10 15 t u-ur 0 Iterations Algorithm III, Checkpoint Design
- 51. 0 2 4 6 8 10 -0.5 0 0.5 1 1.5 2 t u-ur • Guess u(t),w(t) at each checkpoint, solve in segment: • Use mismatch in u(τ)-ur(t) and w(t) at checkpoints to update guess: 51 2 Iterations Algorithm III, Checkpoint Design
- 52. 0 2 4 6 8 10 0 0.5 1 1.5 2 t u-ur • Guess u(t),w(t) at each checkpoint, solve in segment: • Use mismatch in u(τ)-ur(t) and w(t) at checkpoints to update guess: 52 10 Iterations Algorithm III, Checkpoint Design
- 53. Lower memory, easier implementation 53 • Solve a n(2K-1) by n(2K-1) symmetric linear system • Solver can be built using existing components: • Non-linear solver for . • Forward (tangent) solver for . • Adjoint solver for Lagrange multipliers . • Vector dot product (i.e. ).
- 54. Optimization in Chaos • Traditional sensitivity analysis fails for chaotic and turbulent fluid flows. • Least Squared Shadowing has shown great promise for these flows, especially using multiple shooting algorithm. • Currently working on Least Squares Shadowing for wall bounded turbulent flow with complex geometries. • Least Squares Shadowing will enable design and uncertainty quantification of chaotic and turbulent fluid flows. We gratefully acknowledge: