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General Hamiltonian problems Now consider the general state dependent Hamiltonian problem
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Classroom Contents
Structure-Preserving Model Order Reduction of Hamiltonian Systems
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- 1 Intro
- 2 Motivation
- 3 Reduced order models
- 4 Subspace selection Option 1 - by snapshots and SVD-POO
- 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 Consider an example
- 7 A few observations
- 8 Hamiltonian problems To understand how to address these problems, let us consider Hamitonian problems Equations of evolution
- 9 Model order reduction Definition: A € R is a symplectic basis transformation
- 10 Symplectic transformations
- 11 Model order reduction Suppose for a symplectic subspace
- 12 The greedy method - algorithm
- 13 Symplectic Empirical Interpolation Nonlinear case
- 14 General Hamiltonian problems Now consider the general state dependent Hamiltonian problem
- 15 Constant degenerate Poisson structure EP
- 16 Example:The KdV equation
- 17 State-dependent Poisson structure The complication now is that the Darboux map evolves and is unknown a priori We evolve the map
- 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 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 Rank adaptation We use as condition for adaptation the growth
- 21 Example: Shallow water equations
- 22 Examples Similar example for 2d shallow water equation (26)
- 23 To summarize cost
- 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 References
