Mass Transport Problem Research Articles

Conformal Wasserstein distances: Comparing surfaces in polynomial time

We present a constructive approach to surface comparison realizable by a polynomial-time algorithm. We determine the “similarity” of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global Möbius transformations. We present in detail the case where the surfaces to compare are disk-like; we also sketch how the approach can be generalized to other types of surfaces.

Copper(I)‐Containing Ionic Liquids for High‐Rate Electrodeposition

New metal-containing ionic liquids [Cu(CH(3)CN)(n)][Tf(2)N] (n=2, 4; Tf(2)N=bis(trifluoromethylsulfonyl)- amide) have been synthesised and used as a non-aqueous electrolyte for the electrodeposition of copper at current densities greater than 25 A dm(-2). The tetrahedral copper(I)-containing cation in [Cu(CH(3)CN)(4)][Tf(2)N] is structurally analogous to quaternary ammonium and phosphonium ionic liquids and overcomes problems of metal solubility and mass transport. Two CH(3)CN ligands are removed at elevated temperatures to give [Cu(CH(3)CN)(2)][Tf(2)N], which can be used as a concentrated non-aqueous electrolyte. The structural and electrochemical characterisation of these compounds is described herein.

A Monge–Kantorovich mass transport problem for a discrete distance

This paper is concerned with a Monge–Kantorovich mass transport problem in which in the transport cost we replace the Euclidean distance with a discrete distance. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination. For this problem we construct special Kantorovich potentials, and optimal transport plans via a nonlocal version of the PDE formulation given by Evans and Gangbo for the classical case with the Euclidean distance. We also study how these problems, when rescaling the step distance, approximate the classical problem. In particular we obtain, taking limits in the rescaled nonlocal formulation, the PDE formulation given by Evans–Gangbo for the classical problem.

Coupled Porohyperelastic Mass Transport (PHEXPT) Finite Element Models for Soft Tissues Using ABAQUS

Finite element models (FEMs) including characteristic large deformations in highly nonlinear materials (hyperelasticity and coupled diffusive/convective transport of neutral mobile species) will allow quantitative study of in vivo tissues. Such FEMs will provide basic understanding of normal and pathological tissue responses and lead to optimization of local drug delivery strategies. We present a coupled porohyperelastic mass transport (PHEXPT) finite element approach developed using a commercially available ABAQUS finite element software. The PHEXPT transient simulations are based on sequential solution of the porohyperelastic (PHE) and mass transport (XPT) problems where an Eulerian PHE FEM is coupled to a Lagrangian XPT FEM using a custom-written FORTRAN program. The PHEXPT theoretical background is derived in the context of porous media transport theory and extended to ABAQUS finite element formulations. The essential assumptions needed in order to use ABAQUS are clearly identified in the derivation. Representative benchmark finite element simulations are provided along with analytical solutions (when appropriate). These simulations demonstrate the differences in transient and steady state responses including finite deformations, total stress, fluid pressure, relative fluid, and mobile species flux. A detailed description of important model considerations (e.g., material property functions and jump discontinuities at material interfaces) is also presented in the context of finite deformations. The ABAQUS-based PHEXPT approach enables the use of the available ABAQUS capabilities (interactive FEM mesh generation, finite element libraries, nonlinear material laws, pre- and postprocessing, etc.). PHEXPT FEMs can be used to simulate the transport of a relatively large neutral species (negligible osmotic fluid flux) in highly deformable hydrated soft tissues and tissue-engineered materials.

Optimal mass transport for higher dimensional adaptive grid generation

In this work, we describe an approach for higher dimensional adaptive grid generation based on solving the L 2 Monge–Kantorovich problem (MKP) which is a special case of the classical optimal mass transportation problem. Two methods are developed for computing the coordinate transformation used to define the grid adaptation. For the first method, the transformation is determined by solving a parabolic Monge–Ampère equation for a steady state solution. For the second method, the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the L 2 MKP. Several numerical experiments are presented to demonstrate the performance of the MKP methods and to compare them with some related adaptive grid methods. The experimental results demonstrate that the MKP methods show promise as effective and reliable methods for higher dimensional adaptive grid generation.

Alpha divergences based mass transport models for image matching problems

Registration methods could be roughly divided into two groups:area-based methods and feature-based methods. In the literature, theMonge-Kantorovich (MK) mass transport problem has been applied toimage registration as an area-based method. In this paper, wepropose to use Monge-Kantorovich (MK) mass transportmodel as a feature-based method. This novel image matchingmodel is a coupling of the MK problem withthe well-known alpha divergence from the probability theory.The optimal matching scheme is the one which minimizes theweighted alpha divergence between two images. A primal-dualapproach is employed to analyze the existence anduniqueness/non-uniqueness of the optimal matching scheme. A blockcoordinate method, analogous to the Sinkhorn matrix balancingmethod, can be used to compute the optimal matching scheme.We also derive a distance function for image morphing.Similar to elastic distances proposed by Younes, thegeodesic under this distance function has an explicit expression.

Optimal transportation meshfree approximation schemes for fluid and plastic flows

AbstractWe develop an optimal transportation meshfree (OTM) method for simulating general solid and fluid flows, including fluid–structure interaction. The method combines concepts from optimal transportation theory with material‐point sampling and max‐ent meshfree interpolation. The proposed OTM method generalizes the Benamou–Brenier differential formulation of optimal mass transportation problems to problems including arbitrary geometries and constitutive behavior. The OTM method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities. The OTM method exactly conserves linear and angular momentum and its convergence characteristics are verified in standard benchmark problems. We illustrate the range and scope of the method by means of two examples of application: the bouncing of a gas‐filled balloon off a rigid wall; and the classical Taylor‐anvil benchmark test extended to the hypervelocity range. Copyright © 2010 John Wiley & Sons, Ltd.

Development and characterization of fluorine tin oxide electrodes modified with high area porous thin films containing gold nanoparticles

Different electrode materials are prepared using fluoride doped tin oxide (FTO) electrodes modified with high area porous thin films of metal oxides containing gold nanoparticles. Three different metal oxides (TiO 2, MgO and SnO 2) have been assayed to this end. The effect of the metal oxide nature and gold loading on the structure and performance of the modified electrodes was examined by Scanning Electron Microscopy, Transmission Electron Microscopy, X-Ray Diffraction (XRD), Diffuse Reflectance Spectroscopy and electrochemical techniques. XRD measurements reveal that MgO electrodes present the smallest gold nanoparticles after the sintering step however, the electrochemical response of these electrodes shows important problems of mass transport derived from the high porosity of these materials (Brunauer Emmett Teller area of 125 m 2/g). The excellent sintering properties of titania nanoparticles result in robust films attached to the FTO electrodes which allow more reliable and reproducible results from an electroanalytical point of view.

Solving Complex Thermal and Mass Transport Problems with the Lattice Monte Carlo Method

The Lattice Monte Carlo (LMC) method recently developed by the authors is an unusually powerful and flexible method in which a given phenomenological thermal or mass transport problem is mapped onto a fine-grained lattice which is then analyzed with discrete random walk methods. We provide an overview of the LMC method. For mass diffusion we highlight the addressing of diffusion with reversible reaction. For thermal transport we highlight a calculation of the effective thermal conductivity of sintered hollow sphere structures making use of CT scans of actual material as well as the determination of temperature profiles in phase-change composites.

Arbeitsgemeinschaft: Optimal Transport and Geometry

This Arbeitsgemeinschaft was devoted to recent developments in optimal transport with emphasis on links and applications to geometry. The topics reached from the origin of optimal transport as a variational problem, where one minimizes a transportation cost when transporting one density into another, over the introduction of a metric on the space of probability measures, leading to the Wasserstein-space and convex funtionals on it, to the connection between Ricci-curvature and the optimal mass transport problem.

On the Hessian of the optimal transport potential

We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove BrascampLieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality. Mathematics Subject Classification (2000): 49Q20 (primary); 52A40, 44A35 (secondary).

Approaches to Improve Durability and Special Applications of FC MEAs

Three approaches are presented to address the most commonly observed durability problems in low temperature and high temperature fuel cells: carbon corrosion. Carbon support corrosion problem is severe during shutdown/re-start-up periods when gas mixing occurs. Process mitigation is very costly. The first approach is using corrosion resistant carbon. Electrochemical carbon screening methodology is presented and successful examples are demonstrated. Unsupported catalyst is another attractive approach. Cell stack testing indicates the much better durability of gas diffusion electrode (GDEs) with unsupported catalysts (USC) in 3000-4000 hour experiments. GDEs with UNC also showed advantage in low humidity operation and fast break-in. Technology in optimizing the performance of GDEs with UNC is illustrated. Ion beam deposition (IBD) is demonstrated to afford anode GDEs with similar performance to conventional one with complete dry process and without incorporating ionomer for proton conductivity. Cathode IBAD GDEs exhibit mass transport problem. Roll-to-roll IBAD manufacture is demonstrated

Uncluttering Graph Layouts Using Anisotropic Diffusion and Mass Transport

Many graph layouts include very dense areas, making the layout difficult to understand. In this paper, we propose a technique for modifying an existing layout in order to reduce the clutter in dense areas. A physically inspired evolution process based on a modified heat equation is used to create an improved layout density image, making better use of available screen space. Using results from optimal mass transport problems, a warp to the improved density image is computed. The graph nodes are displaced according to the warp. The warp maintains the overall structure of the graph, thus limiting disturbances to the mental map, while reducing the clutter in dense areas of the layout. The complexity of the algorithm depends mainly on the resolution of the image visualizing the graph and is linear in the size of the graph. This allows scaling the computation according to required running times. It is demonstrated how the algorithm can be significantly accelerated using a graphics processing unit (GPU), resulting in the ability to handle large graphs in a matter of seconds. Results on several layout algorithms and applications are demonstrated.

The limit as $p → ∞$ for the $p$-Laplacian with mixed boundary conditions and the mass transport problem through a given window

In this paper we study the limit as p → ∞ in a PDE problem involving the p -Laplacian with a right hand side, − \mathrm{div}(|Du|^{p − 2}Du) = f , with mixed boundary conditions, u = 0 on Γ and |Du|^{p − 2} \frac{∂u}{∂ν} = 0 on ∂Ω \backslash Γ . We find that this limit is related to an optimal mass transport problem, where the total mass given by f is transported outside the domain through a given window on the boundary Γ .

Seismic stress stimulation mobilizes colloids trapped in a porous rock

Low-frequency (1–500 Hz) stress (seismic) waves can alter flow and mass transport behavior in saturated subsurface porous formations. Numerous physical mechanisms have been proposed to explain this phenomenon. Coupling of dynamic stress to in-situ particle (colloid) mobilization is one such mechanism that can produce changes in pore matrix permeability. This phenomenon has broad-ranging impacts on colloid dynamics and mass transport problems over a scale range of microns to kilometers. Porous-flow experiments in sandstone cores demonstrated that in-situ colloidal particles can be released from pores by applying dynamic mechanical stress stimulation at frequencies below 100 Hz. Due to the lower attenuation at these frequencies, relative to ultrasonic, significant dynamic stress in the Earth can penetrate to distances of a kilometer or more. Thus, seismic waves could affect particle mobility in distant oil formations, aquifers and fault systems. Laboratory results are shown for release of in-situ particles from Fontainebleau sandstone induced by applying stress stimulation at 26 Hz. Although enhanced particle release relative to that induced by flowing de-ionized water alone was observed, the permeability of the core was unchanged, indicating that pore throat fouling was insignificant. The behavior of the post-stimulation particle release was distinctly different than the pre-stimulation behavior in that a cyclical pattern with uniform periodicity was observed. This cyclical behavior was observed to be independent of particle size over the range of 50 to 800 nm and is attributed to stimulation causing a long-term change in the distribution of the rate coefficients for release of particles from the pore space. The rate change is likely not due to alteration of particle–wall interactions which are sensitive to particle size. Size-independent release mechanisms that can explain the stimulated rate change are 1) enhanced flushing or squeezing out of particles trapped in dead-end pores, and 2) forced particle detachment and exposing of new detachment sites on the pore walls. The observations presented here are unique in that they indicate sub-pore-scale particle mobility and transport can be influenced by long-wavelength seismic-band stress. Implications on possible field-scale effects and micro-scale physical mechanisms are discussed.

Measurability of optimal transportation and strong coupling of martingale measures

We consider the optimal mass transportation problem in $\mathbb{R}^d$ with measurably parameterized marginals under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with a specified covariance measure process. This is done by pushing forward the first martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.

From Knothe's Transport to Brenier's Map and a Continuation Method for Optimal Transport

A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called Knothe–Rosenblatt rearrangement, which consists of rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge–Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.

An Optimization Problem for Mass Transportation with Congested Dynamics

Starting from the work by Brenier [Extended Monge–Kantorovich theory, in Optimal Transportation and Applications (Martina Franca 2001), Lecture Notes in Math. 1813, Springer-Verlag, Berlin (2003), pp. 91–121], where a dynamic formulation of mass transportation problems was given, we consider a more general framework, where different kinds of cost functions are allowed. This seems relevant in some problems presenting congestion effects as, for instance, traffic on a highway, crowds moving in domains with obstacles, and, in general, in all cases where the transportation does not behave as in the classical Monge setting. We show some numerical computations obtained by generalizing to our framework the approximation scheme introduced in Benamou and Brenier [A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375–393].

A nonholonomic Moser theorem and optimal transport

We prove the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We describe formal solutions of the corresponding nonholonomic mass transport problem and present the Hamiltonian framework for both the Otto calculus and its nonholonomic counterpart as infinite-dimensional Hamiltonian reductions on diffeomorphism groups. Finally, we define a nonholonomic analog of the Wasserstein (or, Kantorovich) metric on the space of densities and prove that the subriemannian heat equation defines a gradient flow on the nonholonomic Wasserstein space with the potential given by the Boltzmann relative entropy functional.

Solid–liquid mass transfer analysis in a multi-phase tank reactor containing submerged coated inclined-plates: A computational fluid dynamics approach

Although there is a voluminous literature on the estimation of interphase transport parameters in conventional slurry bubble column reactors, these correlations are inadequate in photoreactors equipped with specialized internals to facilitate light harvesting efficiency of the photocatalyst. This is particularly germane to the present externally illuminated bubble column reactor containing titania-coated plates immersed in the liquid column at different angles of inclination. Thus, a computational fluid dynamics (CFD) procedure utilizing the Eulerian–Eulerian approach has been used to solve the governing differential equations for the solid liquid mass transport problem based on the standard k – ε model incorporating additional terms that take account of the interfacial turbulent momentum transfer. Mass transfer from the surface of the coated-quartz plates to the liquid phase was modeled using the Launder–Spalding wall functions. The plates were coated with benzoic acid as solid substrate with water and air as the liquid and gas phases, respectively. The increase in mass transfer due to reduction of the boundary layer thickness during air-induced liquid recirculation on either side of the submerged inclined plates was correlated with difference between turbulent and molecular Schmidt numbers via an adjustable parameter, A. CFD simulation using the Launder–Spalding wall function (with A = 1.08 ) gave better agreement with experimental transient concentration profiles than calculations based on FLUENT's enhanced wall function for the plate orientations ( θ = 0 ∘ , 22 . 5 ∘ , and 45 ∘ ) studied. The solid-to-liquid mass transfer was higher on the lower-side of the plate than the upper-side. In particular, mass transfer coefficient was higher with the inclined plate than with the vertical or horizontal orientation suggesting an added advantage for the application of the system as a solar photoreactor.