This repository contains development of earlier work by [1] and the Symplectic Capacities group from the Computational Symplectic Topology workshop at Tel-Aviv University and ICERM [2], of which I was a participant. Work by the Symplectic Capacities group is available in MATLAB code in the github repo [3].
Hofer-Zehnder capacities of convex sets in R^2n are an example of an invariant of symplectic manifolds (another example is Gromov width). These invariants are generally very difficult to compute or even estimate, and there are many open conjectures, even about the Hofer-Zehnder capacity.
Defines a class that estimates the Hofer-Zehnder capacity of a strictly convex, bounded, open subset of R^2n with C^2 boundary. The algorithm is a vectorized numpy implementation of the constrained gradient-descent algorithm provided in [1]. The constraints are non-linear and the algorithm uses a version of projected gradients that solves the local KKT conditions. In the algorithm, smooth loops in R^2n are approximated by piecewise linear paths with m linear segments.
Usage of HZCapacityEstimator.py is demonstrated in the notebook Examples.ipynb
After importing HZCapacityEstimator, initialize an instance of the class by calling
estimator = HZCapacityEstimator(n,m,H,dH,dG = None)
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nis half the desired dimension. -
mis the number of segments in the PL paths. Note that m must be at least 3. For a good approximation, we recommend setting m to be approximately 500-1000. -
His the Hamiltonian function such that the level set H = 1 is the boundary of the convex body. This function must satisfy several conditions for the algorithm to converge. SeeExamples.ipynbor the thesis of G-J. -
dHis the gradient of H -
dGthis optional argument is the gradient of the Legendre transform of H. If dG is not entered (e.g. if it is not possible to solve for dG algebraically) then the algorithm will approximate dG numerically.
Once you have defined an estimator object, as above, you can run the gradient-descent algorithm by calling
estimator.estimate(iterations = 100, epsilon = 10.**(-12), VOCAL = True)
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iterationsthis optional argument sets the maximum number of gradient descent steps. -
epsilonthis optional argument is the tolerance for early stopping. -
VOCALthis optional argument can be used to mute log messages that the algorithm prints to screen.
The function returns the estimated HZ-capacity.
[1] G"oing-Jaeschke. Numerical Analysis in Hamiltonian Dynamics. PhD Thesis, ETH Zurich.
[2] http://www.math.tau.ac.il/~ostrover/Workshop/CST/Workshop-CST.html