On differentiability of the membrane-mediated mechanical interaction energy of discrete-continuum membrane-particle models

Free energy computation of particles with membrane-mediated interactions via Langevin dynamics

“…Note that contributions from H 2 0 (Ω q ) and W q to the solution u q can be decoupled by orthogonalization, cf. [37,Lemma 3.1]. This turns out to be beneficial both for further analysis and finite element approximation.…”

“…For the construction and analysis of finite element approximations of the minimization problem (7), and thus of M(q), we refer, e.g., to [22,37,42,43].…”

“…It has been shown by Kies [43] and Gräser and Kies [37,Lemma 4.6] that the geometric interaction potential M : D ⊂ R 3N → R defined in ( 9) is a smooth function.…”

“…To this end, we consider the directional derivative ∂ e M(q) at q ∈ D in an arbitrary direction e ∈ R 3N . Such a representation was recently provided by Kies [43] and Gräser and Kies [37,Lemma 4.8] utilizing the framework of shape calculus [44]. For each fixed direction…”

“…It is well-known, cf., e.g., Durmus et al [36], that gradient based Markov Chain Monte-Carlo (MCDC) methods such as Langevin-based samplers [30,31,32,33], typically have better convergence properties than gradient free Metropolis-type algorithms. Exploiting recent results on derivatives of the geometric interaction potential [37], we are finally able to introduce novel Langevin-based sampling methods for the computation of free energy differences of fluctuating particles in 'fast' membranes.…”