Completed
Hamiltonian problems To understand how to address these problems, let us consider Hamitonian problems Equations of evolution
Class Central Classrooms beta
YouTube videos curated by Class Central.
Classroom Contents
Structure-Preserving Model Order Reduction of Hamiltonian Systems
Automatically move to the next video in the Classroom when playback concludes
- 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
