Homogenization of lateral diffusion on a random surface

We study the problem of lateral diffusion on a static, quasi-planar surface generated by a stationary, ergodic random field possessing rapid small-scale spatial fluctuations. The aim is to study the effective behaviour of a particle undergoing Brownian motion on the surface viewed as a projection on the underlying plane. By formulating the problem as a diffusion in a random medium, we are able to use known results from the theory of stochastic homogenization of SDEs to show that, in the limit of small scale fluctuations, the diffusion process behaves quantitatively like a Brownian motion with constant diffusion tensor $D$. While $D$ will not have a closed-form expression in general, we are able to derive variational bounds for the effective diffusion tensor, and using a duality transformation argument, obtain a closed form expression for $D$ in the special case where $D$ is isotropic. We also describe a numerical scheme for approximating the effective diffusion tensor and illustrate this scheme with two examples.Comment: 25 pages, 7 figure

Importance Sampling for Multiscale Diffusions

We construct importance sampling schemes for stochastic differential equations with small noise and fast oscillating coefficients. Standard Monte Carlo methods perform poorly for these problems in the small noise limit. With multiscale processes there are additional complications, and indeed the straightforward adaptation of methods for standard small noise diffusions will not produce efficient schemes. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Examples and simulation results are provided

Noise-induced transitions in rugged energy landscapes

We consider the problem of an overdamped Brownian particle moving in multiscale potential with N + 1 characteristic length scales: the macroscale and N separated microscales. We show that the coarse-grained dynamics is given by an overdamped Langevin equation with respect to the free energy and with a space-dependent diffusion tensor, the calculation of which requires the solution of N fully coupled Poisson equations. We study in detail the structure of the bifurcation diagram for one-dimensional problems, and we show that the multiscale structure in the potential leads to hysteresis effects and to noise-induced transitions. Furthermore, we obtain an explicit formula for the effective diffusion coefficient for a self-similar separable potential, and we investigate the limit of infinitely many small scales

Stochastic Dynamics of Bionanosystems: Multiscale Analysis and Specialized Ensembles

An approach for simulating bionanosystems, such as viruses and ribosomes, is presented. This calibration-free approach is based on an all-atom description for bionanosystems, a universal interatomic force field, and a multiscale perspective. The supramillion-atom nature of these bionanosystems prohibits the use of a direct molecular dynamics approach for phenomena like viral structural transitions or self-assembly that develop over milliseconds or longer. A key element of these multiscale systems is the cross-talk between, and consequent strong coupling of, processes over many scales in space and time. We elucidate the role of interscale cross-talk and overcome bionanosystem simulation difficulties with automated construction of order parameters (OPs) describing supra-nanometer scale structural features, construction of OP dependent ensembles describing the statistical properties of atomistic variables that ultimately contribute to the entropies driving the dynamics of the OPs, and the derivation of a rigorous equation for the stochastic dynamics of the OPs. Since the atomic scale features of the system are treated statistically, several ensembles are constructed that reflect various experimental conditions. The theory provides a basis for a practical, quantitative bionanosystem modeling approach that preserves the cross-talk between the atomic and nanoscale features. A method for integrating information from nanotechnical experimental data in the derivation of equations of stochastic OP dynamics is also introduced.Comment: 24 page

From constant to rough: A survey of continuous volatility modeling

In this paper, we present a comprehensive survey of continuous stochastic volatility models, discussing their historical development and the key stylized facts that have driven the field. Special attention is dedicated to fractional and rough methods: we outline the motivation behind them and characterize some landmark models. In addition, we briefly touch the problem of VIX modeling and recent advances in the SPX-VIX joint calibration puzzle

Multiscale phase change, heat and mass transfer in direct contact membrane distillation

The global water shortage has become a serious threat for the world and the most promising solution for the water issue is the desalination of seawater or brackish water. In this work, direct contact membrane distillation (DCMD) as one of thermal desalination technologies was numerically and exprimentally analyzed to study its performance. A large DCMD system with multiple membrane modules in a parallel arrangement running on the waste heat from a diesel power generator was numerically analyzed using a thermo-fluid network model to study the technical feasibility of the use of the low-grade engine waste heat and simulate the distillation performance of the DCMD system. Next, a small DCMD experimental apparatus was fabricated to test for the distillation performance for various operating conditions (inlet temperatures, flow rates of feed and permeate streams and NaCl concentration) and design variables (filament spacing of a screen spacer in the flow channels and flow configuration). In the DCMD, two different regimes were observed in the water flux behavior regarding the salinity of feed water. In the first regime, from low NaCl concentration to 90% saturated NaCl concentration, there was a gradual decrease in the water flux due to the suppression of vapor pressure at the feed water which is simulated by a CFD model. In the second regime, at higher 90% saturated NaCl concentration, there was a sharp drop in the water flux due to the deposition of NaCl crystals on the membrane surface which is simulated by an analytical model using the adjusting parameter from the experiment. Finally, a nanoscale DCMD using Carbon Nanotube (CNT) membrane was numerically analyzed using non-equilibrium molecular dynamics (NEMD) simulation for different diameters and lengths of the CNT and operating conditions such as system temperature, temperature difference between the feed (hot) and permeate (cold) reservoirs, and sodium chloride (NaCl) concentration in the feed reservoir. The distillation performance of the DCMD systems is enhanced by increasing system temperature, temperature difference between feed and permeate streams, and decreasing the NaCl concentration. The permeability of the CNT membrane (1.8 x 10-5 liter/m2-s-Pa) was found two orders-of-magnitudes higher than a Polytetrafluoroethylene (PTFE) membrane (1.7 x 10-7 liter/m2-s-Pa ) used in our experimental work.Includes bibliographical reference

Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization

The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities