| Abstract: |
| In this talk I will discuss the construction of $\lambda$-contractive gradient flows in abstract metric spaces by means of a semi discretization of second order in time.
In the smooth setting, our scheme is simply a variational formulation of the BDF2 method; in the metric setting, it can be considered as the natural second order analogue of the Minimizing Movement or JKO scheme.
In difference to the JKO method, our scheme does not necessarily decrease the energy of the discrete solution in each time step, but we can still prove a suitable \emph{almost diminishing} property. \
It is well-know that in smooth situations, the BDF2 method converges to order $\tau^2$.
We prove that our variational scheme converges at least to order $\tau^{1/2}$ in the general non-smooth setting, provided a certain convexity hypothesis is satisfied.
Specifically, that hypothesis is equivalent to $\lambda$-uniform convexity in the flat case, and is implied by $\lambda$-convexity along generalized geodesics in the $L^2$-Wasserstein case. \
In the special case of the Fokker-Planck equation seen as $L^2$-Wasserstein gradient flow an alternative approach can be used, which relies heavily on the differentiable structure of the underlying space.
Here, we can prove a stronger convergence result, though without an explicit rate. |
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