Abstract. This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented.

We propose a novel method for sampling and optimization tasks based on a stochastic interacting particle system. We explain how this method can be used for the following two goals: (i) generating approximate samples from a given target distribution and (ii) optimizing a given objective function. The approach is derivative-free and affine invariant, and is therefore well-suited for solving inverse problems defined by complex forward models: (i) allows generation of samples from the Bayesian posterior and (ii) allows determination of the maximum a posteriori estimator. We investigate the properties of the proposed family of methods in terms of various parameter choices, both analytically and by means of numerical simulations. The analysis and numerical simulation establish that the method has potential for general purpose optimization tasks over Euclidean space; contraction properties of the algorithm are established under suitable conditions, and computational experiments demonstrate wide basins of attraction for various specific problems. The analysis and experiments also demonstrate the potential for the sampling methodologyThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

We study the convergence to equilibrium of the mean field PDE associated with the derivative-free methodologies for solving inverse problems that are presented by Garbuno-Inigo et al (2020 SIAM J. Appl. Dyn. Syst. 19 412-41), Herty and Visconti (2018 arXiv:1811.09387). We show stability estimates in the Euclidean Wasserstein distance for the mean field PDE by using optimal transport arguments. As a consequence, this recovers the convergence towards equilibrium estimates by Garbuno-Inigo et al (2020 SIAM J. Appl. Dyn. Syst. 19 412-41) in the case of a linear forward model.

We propose a new numerical method to solve the Cahn-Hilliard equation coupled with nonlinear wetting boundary conditions. We show that the method is mass-conservative and that the discrete solution satisfies a discrete energy law similar to the one satisfied by the exact solution. We perform several tests inspired by realistic situations to verify the accuracy and performance of the method: wetting of a chemically heterogeneous substrate in three dimensions, wettingdriven nucleation in a complex two-dimensional domain and three-dimensional diffusion through a porous medium. this is precisely the Young-Dupré angle. When one of the two fluids moves against the other, the contact angle becomes a dynamic quantity, and when the problem is formulated in the framework of conventional hydrodynamics, the contact line motion relative to the solid boundary results in the notorious stress singularity there, as first noted in the pioneering studies by Moffat [43] and Huh and Scriven [34]. Since then there have been numerous analyses and discussions of the singularity over the years, see e.g. [61,32,15] and also recent studies in [58,57] (with the latter one revisiting the classical Cox-Hocking matched asymptotic analysis and providing a correction to it). Recently, it was shown [45] that mesoscopic approaches such as dynamic density functional theory (DDFT) can fix this singularity behavior.A popular model for interface dynamics is the Cahn-Hilliard (CH) equation [10,12], which belongs to the class of phase-field and diffuse interface models. Originally proposed to model spinodal decomposition, the mechanism by which a binary mixture can separate to form two coexisting phases due to, e.g., a change of temperature [12], it has been used in a wide spectrum of different contexts since, such as solidification phenomena [14] and Saffman-Taylor instabilities in Hele-Shaw flows [31]. To account for wetting phenomena and contact lines on solid boundaries, the CH equation can be coupled to a wall boundary condition [11]. Such CH model has been employed successfully in various situations, including microfluidic devices [18, 19, 49, 65], flow in porous media [4], rheological systems [8], and patterning of thin polymer films [38]. Other potential applications include micro-separators [50], fuel cells [3] and CPU chip cooling based on electro-wetting [44]. Many of these applications are characterized by the presence of chemically heterogeneous substrates and/or complex geometries, which make their numerical simulation challenging.The form of the wetting boundary condition is dictated by the form of the wall free energy. For liquid-gas problems linear forms have been adopted, e.g. in the pioneering study by Seppecher [51] and in [9,66]. But a cubic is the lowest-order polynomial required such that the wall free energy can be minimized for the bulk densities, and it prevents the formation of boundary layers on the wall. Cubic forms have been adopted for binary fluid problems, e.g. in [35,69], but also for liquid-gas ones, see [55,56]...

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