Diffusion Matrix Research Articles

The boundedness-by-entropy method for cross-diffusion systems

The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.

Low-Temperature Air/Solvent Injection for Heavy-Oil Recovery in Naturally Fractured Reservoirs

Summary Limited studies on oil recovery from naturally fractured reservoirs using low-temperature air injection show that the process is strongly dependent on oxygen (O2)-diffusion coefficient and matrix permeability, both of which are typically low. A new approach (i.e., the addition of hydrocarbon solvent gases into air) is expected to improve the diffusivity of the gas mixture and to accelerate the oxidation. To study this new idea, called low temperature air/solvent injection, laboratory tests were performed by soaking heavy-oil-saturated cores in air/solvent filled reactors to determine the critical parameters on recovery. Laboratory tests were complemented by conducting experiments using air at different O2 concentrations: zero (i.e., nitrogen), 21.0 mol% (air), and 37.3 mol% (O2-enriched air). For safety reasons, it is imperative that enough time be given for air diffusion before the injected air breaks through a highly permeable fracture network. This implies that the huff ‘n’ puff type of injection is a plausible option as opposed to the continuous injection of air. A high recovery factor was obtained by soaking a single matrix in an air/solvent chamber at static conditions rather than with air only. The period of pressure stabilization was faster for the air/solvent mixture than in 100% solvent. The asphaltene content was lower in the air/solvent case than in the 100%-air injection case. Instead of pure-hydrocarbon solvent, injection of an air/solvent mixture yields a better recovery with less asphaltene. This is expected to reduce the cost of the process compared with pure-solvent injection. At low temperatures (75C), O2 consumption in the matrix oil was low, while at high temperatures, the O2 was partially (150C) or totally (200C in the presence of propane) diffused and consumed in the matrix.

Equiaxed and columnar dendrite growth simulation in Al-7Si- Mg ternary alloys using cellular automaton method

In this paper, a modified cellular automaton (MCA) model allowing for the prediction of dendrite growth of Al-Si-Mg ternary alloys in two and three dimensions is presented. The growth kinetic of S/L interface is calculated based on the solute equilibrium approach. In order to describe the dendrite growth with arbitrarily crystallographic orientations, this model introduces a modified decentered octahedron algorithm for neighborhood tracking to eliminate the effect of mesh dependency on dendrite growth. The thermody namic and kinetic data needed for dendrite growth is obtained through coupling with Pandat software package in combination with thermodynamic/kinetic/equilibrium phase diagram calculation databases. The effect of interactions between various alloying elements on solute diffusion coefficient is considered in the model. This model has first been used to simulate Al-7Si (weight percent) binary dendrite growth followed by a validation using theoretical predictions. For ternary alloy, Al-7Si-0.5Mg dendrite simulation has been carried out and the effects of solute interactions on diffusion matrix as well as the differences of Si and Mg in solute distribution have been analyzed. For actual application, this model has been applied to simulate the equiaxed dendrite growth with various crystallographic orientations of Al-7Si-0.36Mg ternary alloy, and the predicted secondary dendrite arm spacing (SDAS) shows a reasonable agreement with the experimental ones. Furthermore, the columnar dendrite growth in directional solidification has also been simulated and the predicted primary dendrite arm spacing (PDAS) is in good agreement with experiments. The simulated results effectively demonstrate the abilities of the model in prediction of dendritic microstructure of Al-Si-Mg ternary alloy.

Uniqueness Problems for Degenerate Fokker–Planck–Kolmogorov Equations

We study the uniqueness of solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation with a singular diffusion matrix in the class of probability measures. A survey of known result and methods is given. In addition, we obtain new sufficient conditions for the uniqueness in the case of unbounded coefficients and a partially singular diffusion matrix and also in the case where the diffusion matrix is a squared Lipschitzian matrix.

Contribution to the benchmark for ternary mixtures: Measurement of diffusion and Soret coefficients in 1,2,3,4-tetrahydronaphthalene, isobutylbenzene, and dodecane onboard the ISS.

The paper is devoted to processing the data of DCMIX 1 space experiment. In this experiment, the Optical digital interferometry was used to measure the diffusion and Soret coefficients in the ternary mixture of 1,2,3,4-tetrahydronaphthalene, isobutylbenzene and n-dodecane at mass fractions of 0.8/0.1/0.1 and at 25°C. The raw interferometric images were processed to obtain the temporal and spatial evolution of refractive indices for two laser beams of different wavelengths. The method for extracting the diffusion and thermal diffusion coefficients originally developed for optical beam deflection was extended to optical digital interferometry allowing for the spatial variation of refractive index along the diffusion path. The method was validated and applied to processing the data for Soret and diffusion steps in 5 experimental runs. The obtained results for the Soret coefficients and one of the eigenvalues of diffusion matrix showed acceptable agreement within each step. The second eigenvalue was not determined with sufficient accuracy.

Benchmark values for the Soret, thermodiffusion and molecular diffusion coefficients of the ternary mixture tetralin+isobutylbenzene+n-dodecane with 0.8-0.1-0.1 mass fraction.

With the aim of providing reliable benchmark values, we have measured the Soret, thermodiffusion and molecular diffusion coefficients for the ternary mixture formed by 1,2,3,4-tetrahydronaphthalene, isobutylbenzene and n-dodecane for a mass fraction of 0.8-0.1-0.1 and at a temperature of 25°C. The experimental techniques used by the six participating laboratories are Optical Digital Interferometry, Taylor Dispersion technique, Open Ended Capillary, Optical Beam Deflection, Thermogravitational technique and Sliding Symmetric Tubes technique in ground conditions and Selectable Optical Diagnostic Instrument (SODI) in microgravity conditions. The measurements obtained in the SODI installation have been analyzed independently by four laboratories. Benchmark values are proposed for the thermodiffusion and Soret coefficients and for the eigenvalues of the diffusion matrix in ground conditions, and for Soret coefficients in microgravity conditions.

3D finite element modeling and simulation of industrial semiconductor devices including impact ionization

In this article we address the numerical study of 3D semiconductor devices for applications in electronics industry including charge generation phenomena due to impact ionization. With this aim, we propose two novel 3D finite element (FE) models (methods A and B), for electron and hole Drift-Diffusion (DD) current densities. Method A is based on a primal-mixed formulation of the DD model as a function of the quasi-Fermi potential gradient, while method B is a modification of the standard DD formula based on the introduction of an artificial diffusion matrix. Method A is a Galerkin FE approximation of the density current (written in generalized ohmic form) where the harmonic average of the electrical conductivity is used instead of the standard average while method B is a genuine 3D extension of the classic 1D Scharfetter-Gummel difference formula interpreted as a stabilized Galerkin FE approximation with the use of an ‘optimal’ artificial diffusion. The proposed methods are compared in the 3D simulation of a p-n junction diode and of a p-MOS transistor in the on-state regime. Results show that method A outperforms method B in physical accuracy and numerical stability. Method A is then used in the 3D simulation of a n-MOS transistor in the off-state regime including impact ionization. Results demonstrate that the model is able to accurately compute the I-V characteristic of the device until drain-to-bulk junction breakdown.

Bounded weak solutions to a matrix drift–diffusion model for spin-coherent electron transport in semiconductors

The global-in-time existence and uniqueness of bounded weak solutions to a spinorial matrix drift–diffusion model for semiconductors is proved. Developing the electron density matrix in the Pauli basis, the coefficients (charge density and spin-vector density) satisfy a parabolic 4 × 4 cross-diffusion system. The key idea of the existence proof is to work with different variables: the spin-up and spin-down densities as well as the parallel and perpendicular components of the spin-vector density with respect to the precession vector. In these variables, the diffusion matrix becomes diagonal. The proofs of the L∞ estimates are based on Stampacchia truncation as well as Moser- and Alikakos-type iteration arguments. The monotonicity of the entropy (or free energy) is also proved. Numerical experiments in one-space dimension using a finite-volume discretization indicate that the entropy decays exponentially fast to the equilibrium state.

Rotation of a scroll wave reveals the metric tensor for the associated geodesic filament.

This study considers a reaction-diffusion medium that supports a rotating scroll wave with stationary filament and diffusivity matrix D. The filament is a geodesic curve in three-space. The metric tensor g associated with that geodesic is obtained in terms of D, resulting in g=(detD)D-1. Here detD (the determinant of D) is not necessarily constant in space. The detD factor, although strongly indicated previously, has not yet been derived deductively, as it is here. We take the opportunity to outline, starting from the reaction-diffusion equations, a sequence of mathematical steps leading to both geodesic and metric.

What can be learned from optical two-color diffusion and thermodiffusion experiments on ternary fluid mixtures?

A number of optical techniques have been developed during the recent years for the investigation of diffusion and thermodiffusion in ternary fluid mixtures, both on ground and on-board the International Space Station. All these methods are based on the simultaneous measurement of refractive index changes at two different wavelengths. Here, we discuss and compare different techniques with the emphasis on optical beam deflection (OBD), optical digital interferometry, and thermal diffusion forced Rayleigh scattering (TDFRS). We suggest to formally split the data evaluation into a phenomenological parameterization of the measured transients and a subsequent transformation from the refractive index into the concentration space. In all experiments, the transients measured at two different detection wavelengths can be described by four amplitudes and two eigenvalues of the diffusion coefficient matrix. It turns out that these six parameters are subjected to large errors and cannot be determined reliably. Five good quantities, which can be determined with a high accuracy, are the stationary amplitudes, the initial slopes as defined in TDFRS experiments and by application of a heuristic criterion for similar curves, a certain mean diffusion coefficient. These amplitudes and slopes are directly linked to the Soret and thermodiffusion coefficients after transformation with the inverse contrast factor matrix, which is frequently ill-conditioned. Since only five out of six free parameters are reliably determined, including the single mean diffusion coefficient, the determination of the four entries of the diffusion matrix is not possible. We apply our results to new OBD measurements of the symmetric (mass fractions 0.33/0.33/0.33) ternary benchmark mixture n-dodecane/isobutylbenzene/1,2,3,4-tetrahydronaphthalene and existing literature data for the same system.

Parameter-invariant models for load balancing on heterogeneous networks

Several schemes for homogeneous networks have been generalized for load balancing (LB) on heterogeneous networks. However, LB parameters must be recalculated with the change of weight distribution among network nodes in known schemes and this increases the computational complexity and weakens numerical stability on the base of original schemes in which LB parameters are only decided by communication topology.The known load balancing strategies are redesigned to contain a diffusion matrix M⁎ with parameters independent of network heterogeneity. In the proposed strategies, LB parameters are only decided by network topology which is identical with the situation of homogeneous network. This leads to lower overhead of computing LB parameters and fewer LB steps than in the known schemes. Furthermore, the proposed schemes have little variation in the number of LB steps to distinct load distributions, so that they have an enhanced numerical stability. Theoretical and experimental results verify these improvements.

Synthesis of uniform α-Si3N4 nanospheres by RF induction thermal plasma and their application in high thermal conductive nanocomposites.

In this paper, single-crystalline α-Si3N4 nanospheres with uniform size of ∼50 nm are successfully synthesized by using a radio frequency (RF) thermal plasma system in a one-step and continuous way. All Si3N4 nanoparticles present nearly perfect spherical shape with a narrow size distribution, and the diameter is well-controlled by changing the feeding rate. Compact Si3N4/PR (PR = phenolic resin) composites with high thermal conductivity, excellent temperature stability, low dielectric loss tangent, and enhanced breakdown strength are obtained by incorporating the as-synthesized Si3N4 nanospheres. These enhanced properties are the results of good compatibility and strong interfacial adhesion between compact Si3N4 nanospheres and polymer matrix, as large amount of Si3N4 nanospheres can uniformly disperse in the polymer matrix and form thermal conductive networks for diffusion of heat flow.

A Regularity Result for Quasilinear Stochastic Partial Differential Equations of Parabolic Type

We consider a nondegenerate quasilinear parabolic stochastic partial differential equation (SPDE) with a uniformly elliptic diffusion matrix. It is driven by a nonlinear noise. We study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Holder continuous in time and possesses spatial regularity. Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problem whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs.

The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids

A finite volume scheme based on minimization of a certain cell functional is constructed for unstructured polygonal meshes. This new scheme has a local stencil, allows arbitrary diffusion tensors, leads to a symmetric positive definite diffusion matrix in case that edge unknowns are defined at the midpoints of edges, and is linearity-preserving, i.e. , preserves linear solutions. Under a very weak geometry condition, the stability result and discrete H 1 error estimate of the scheme is obtained through a discrete functional approach. Finally, numerical results on various mesh types (including a particular jigsaw puzzle mesh) demonstrate the good performance of the scheme and validate the theoretical analysis.

Viscosity methods giving uniqueness for martingale problems

Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes. We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $ D$ to be nonsmooth and the direction of reflection to be degenerate. Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix.

Gaussian filtering and variational approximations for Bayesian smoothing in continuous-discrete stochastic dynamic systems

The Bayesian smoothing equations are generally intractable for systems described by nonlinear stochastic differential equations and discrete-time measurements. Gaussian approximations are a computationally efficient way to approximate the true smoothing distribution. In this work, we present a comparison between two Gaussian approximation methods. The Gaussian filtering based Gaussian smoother uses a Gaussian approximation for the filtering distribution to form an approximation for the smoothing distribution. The variational Gaussian smoother is based on minimizing the Kullback–Leibler divergence of the approximate smoothing distribution with respect to the true distribution. The results suggest that for highly nonlinear systems, the variational Gaussian smoother can be used to iteratively improve the Gaussian filtering based smoothing solution. We also present linearization and sigma-point methods to approximate the intractable Gaussian expectations in the variational Gaussian smoothing equations. In addition, we extend the variational Gaussian smoother for certain class of systems with singular diffusion matrix.

Ergodic BSDEs and related PDEs with Neumann boundary conditions under weak dissipative assumptions

We study a class of ergodic BSDEs related to PDEs with Neumann boundary conditions. The randomness of the driver is given by a forward process under weakly dissipative assumptions with an invertible and bounded diffusion matrix. Furthermore, this forward process is reflected in a convex subset of Rd not necessarily bounded. We study the link of such EBSDEs with PDEs and we apply our results to an ergodic optimal control problem.

AMP-activated protein kinase (AMPK) activation inhibits nuclear translocation of Smad4 in mesangial cells and diabetic kidneys.

Diabetic nephropathy is characterized by diffuse mesangial matrix expansion and is largely dependent on the TGF-β/Smad signaling pathway. Smad4 is required for TGF-β signaling; however, its regulation has not been well characterized in diabetic kidney disease. Here, we report that high glucose is sufficient to stimulate nuclear translocation of Smad4 in mesangial cells and that stimulation of the major energy sensor AMP-activated protein kinase (AMPK) has a potent effect to block Smad4 nuclear translocation. Activation of AMPK by 5-aminoimidazole-4-carboxamide-1-β-d-ribofuranoside (AICAR) inhibited high glucose-induced and TGF-β stimulation of nuclear Smad4. To identify which of the catalytic α-subunits may be involved, small interfering (si) RNA-based inhibition of AMPK α1- or α2-subunit was employed. Inhibition of either subunit reduced overall AMPK activity and contributed to Smad4 nuclear accumulation. In an animal model of early diabetic kidney disease, induction of diabetes was found to markedly stimulate Smad4 protein levels and enhance nuclear accumulation. AMPK activation with AICAR completely prevented the upregulation of Smad4 and reduced mesangial matrix accumulation. We conclude that stimulation of Smad4 in cell culture and in in vivo models of early diabetic kidney disease is dependent on AMPK.

A PREDATOR–PREY INTERACTION MODEL WITH SELF- AND CROSS-DIFFUSION IN AQUATIC SYSTEMS

In this paper, the complex dynamics of a spatial aquatic system in the presence of self- and cross-diffusion are investigated. Criteria for local stability, instability and global stability are obtained. The effect of critical wavelength which can drive a system to instability is investigated. We noticed that cross-diffusion coefficient can be quite significant, even for small values of off-diagonal terms in the diffusion matrix. With the help of numerical simulation, we observed the Turing patterns (spots, strips, spot-strips mixture), regular spiral patterns and irregular patchy structures. The beauty and complexity of the Turing patterns are attributed to a large variety of symmetry properties realized by different values of predator's immunity, rate of fish predation and half saturation constant of predator population.

New Postprocessing Methods for Remote Sensing Image Classification: A Systematic Study

This paper develops several new strategies for remote sensing image classification postprocessing (CPP) and conducts a systematic study in this area. CPP is defined as a refinement of the labeling in a classified image in order to enhance its original classification accuracy. The current mainstream classification methods (preprocessing) extract additional spatial features in order to complement spectral information and enhance classification using spectral responses alone. On the other hand, however, the CPP methods, providing a new solution to improve classification accuracy by refining the initial result, have not received sufficient attention. They have potential for achieving comparable accuracy to the preprocessing methods but in a more direct and succinct way. In this paper, we consider four groups of CPP strategies: filtering; random field; object-based voting; and relearning. In addition to the state-of-the-art CPP algorithms, we also propose a series of new ones, e.g., anisotropic probability diffusion and primitive cooccurrence matrix. In experiments, a number of multisource remote sensing data sets are used for evaluation of the considered CPP algorithms. It is shown that all the CPP strategies are capable of providing more accurate results than the raw classification. Among them, the relearning approaches achieve the best results. In addition, our relearning algorithms are compared with the state-of-the-art spectral-spatial classification. The results obtained further verify the effectiveness of CPP in different remote sensing applications.