Structure-Preserving Model Order Reduction of Hamiltonian Systems

Structure-Preserving Model Order Reduction of Hamiltonian Systems

International Mathematical Union via YouTube Direct link

Intro

1 of 25

1 of 25

Intro

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Structure-Preserving Model Order Reduction of Hamiltonian Systems

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  1. 1 Intro
  2. 2 Motivation
  3. 3 Reduced order models
  4. 4 Subspace selection Option 1 - by snapshots and SVD-POO
  5. 5 When can we expect this to work? For this to be successful there must be some structure to the solution under parameter variation
  6. 6 Consider an example
  7. 7 A few observations
  8. 8 Hamiltonian problems To understand how to address these problems, let us consider Hamitonian problems Equations of evolution
  9. 9 Model order reduction Definition: A € R is a symplectic basis transformation
  10. 10 Symplectic transformations
  11. 11 Model order reduction Suppose for a symplectic subspace
  12. 12 The greedy method - algorithm
  13. 13 Symplectic Empirical Interpolation Nonlinear case
  14. 14 General Hamiltonian problems Now consider the general state dependent Hamiltonian problem
  15. 15 Constant degenerate Poisson structure EP
  16. 16 Example:The KdV equation
  17. 17 State-dependent Poisson structure The complication now is that the Darboux map evolves and is unknown a priori We evolve the map
  18. 18 Towards a local basis While the methods work well, the size of the basis is generaly very large This is a classic challenge associated with transport dominated problems which often has a slowly decay…
  19. 19 Error estimator To adapt the rank we need to consider two actions Decrease basis size - this is handled by rank condition of 2 and reduction in U Increase basis size - this requires both an error est…
  20. 20 Rank adaptation We use as condition for adaptation the growth
  21. 21 Example: Shallow water equations
  22. 22 Examples Similar example for 2d shallow water equation (26)
  23. 23 To summarize cost
  24. 24 To summarize The development of reduced order stable methods for time-dependent nonlinear problems is more complex than for traditional reduced models The Hamiltonian model offers access to a number …
  25. 25 References

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