Entropic Regularization of NonGradient Systems

Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

“…The convergence of this new scheme is stated in Theorem 4.2, the proof of which is sketched since it only differs slightly to that of Theorem 2.3. The results and techniques of this section are similar to those appearing in [12,2]. The following assumption introduces a theoretical constraint on the scaling of the time step and strength of entropic regularisation.…”

“…The next Lemma provides uniform bounds in n and h for the second moments, free energy functionals, and positive part of the entropy functionals of the solutions from the scheme ( 9) - (11). The proof is inspired by the procedure found in [2,22,27], first obtaining bounds locally and then extending them over the full time interval.…”

“…Other works that also develop operator-splitting schemes for degenerate PDEs are [10,38], however these works only deal with a linear, local conservative dynamics and use a rather involved splitting mechanism. Several papers including [27,22,24,2] also develop JKO-type minimizing movement schemes for degenerate diffusion equations; however these papers use one-step schemes where the cost functions are often non-homogeneous, time-step dependent and do not induce a metric. We also mention recent works in which operator-splitting methods have been investigated for partial differential equations containing a Wasserstein gradient flow part and a non-Wasserstein part.…”

“…The well-posedness. The well-posedness of the regularised scheme relies on the well-posedness of the minimisation problem (62), the proof of which can be found in [2]*Section 3.…”

“…as the joint distribution with first marginal ρn+1 k , and second marginal ρ ε (y) := γ ε (x, y)dx. One can then calculate/express H(γ ε ), F(ρ ε ), (c h , γ ε ) in terms of ρn+1 k (see [2]*Lemma 4.3). Comparing the performance of ρ n+1 k against ρ ε in (61) we get, making use of the scaling (59) and that…”