General Diffusion Model Research Articles

General Diffusion Model for Polymeric Systems Based on Microscopic Molecular Collisions and Random Walk Movement

Molecular diffusivity in polymeric systems is often the determining factor in performance and efficiency of devices and systems and thus is one of the most important dynamic properties. In polymeric systems, various penetrants, including gas, solvent, and solute molecules can diffuse, and therefore, a general model that can broadly provide their molecular diffusivity could strongly accelerate the designing of polymeric devices. On the other hand, from the microscopic viewpoint, the molecular diffusive motion can be regarded as a random walk movement subject to an enormous number of molecular collisions with neighboring molecules. Therefore, in the present study, both of the microscopic descriptions common for all penetrants in polymeric systems have been unified into one model. The resultant model is the first general diffusion model that can describe diffusivity of all penetrants, including gas, solvent, and solute molecules in rubbery polymeric systems. The applicability of the microscopic model is demonstrated to be acceptable, except in pure polymer near its glass transition temperature.

확산모형에 대한 누율생성함수의 근사와 가우도 추정법

Diffusion is a basic mathematical tool for modern financial engineering. The theory of the estimation methods for diffusion models is an important topic of the financial engineering. Many researches have been tried to apply the likelihood estimation method for estimating diffusion models. However, the likelihood estimation method for diffusion is complicated and needs much amount of computing. In this paper we develop the estimation methods which are simple enough to be compared to the Euler approximation method, and efficient enough statistically to be compared to the likelihood estimation method. We devise pseudo-likelihood and propose the maximum pseudo-likelihood estimation methods. The pseudo-likelihoods are obtained by approximating the transition density with normal distributions. The means and the variances of the distributions are obtained from the delta expansion suggested by Lee, Song and Lee (2012). We compare the newly suggested estimators with other existing estimators by simulation study. From the simulation study we find the maximum pseudo-likelihood estimator has very similar properties with the maximum likelihood estimator. Also the maximum pseudo-likelihood estimator is easy to apply to general diffusion models, and can be obtained by simple numerical steps.

Prediction of the reinforced concrete structure durability under the risk of carbonation and chloride aggression

Abstract The paper presents an idea of predicting durability of concrete structures with steel reinforcement under conditions of chloride and carbonation corrosion risk. Mechanisms of destruction due to steel corrosion in such conditions are shown. The recently elaborated model of carbonation and general model of chloride diffusion have been discussed. An algorithm of the rest service life time prognosis has been shown and options of its main stages realization have been done. An example of durability prognosis for pre-stressed hollow-core floor slab with upper layer of concrete and epoxy-resin coating has been given. This example confirms the usefulness of the described prognosis method and demonstrates its helpfulness in a structure management according to the standards of EN 1504 series

Coupling the SAEM algorithm and the extended Kalman filter for maximum likelihood estimation in mixed-effects diffusion models

We consider some general mixed-effects diffusion models, in which the observations are made at discrete time points and include measurement errors. In these models, the observed likelihood is generally not explicit, making maximum likelihood estimation of the parameters particularly complex. We propose a specific inference methodology for these models. In particular, we combine the SAEM algorithm with the extended Kalman filter to estimate the population parameters. We also provide some tools for estimating the individual parameters, for recovering the individual underlying diffusion trajectories and for evaluating the model. The methods are evaluated on simulations and applied to a pharmacokinetics example.

Time-Variant Strength Capacity Model for GFRP Bars Embedded in Concrete

Glass fiber-reinforced polymer (GFRP) concrete reinforcement exhibits high strength, is lightweight, can decrease time of construction, and is corrosion resistant. However, research has shown that chemical reactions deteriorate the GFRP reinforcing bars over time, resulting in a reduced tensile capacity. This paper develops a time-variant probabilistic model to predict the tensile capacity of GFRP bars embedded in concrete. The developed model is probabilistic to properly account for the relevant sources of uncertainties, including the statistical uncertainty in the estimation of the unknown model parameters (because of the finite sample size), the model error associated with the inexact model form (e.g., a linear expression is used when the actual and unknown relations are nonlinear), and missing variables (i.e., the model only includes a subset of the variables that influence the quantity of interest.) The proposed model is based on a general diffusion model, in which water or ions penetrate the GFRP bar matrix and degrade the glass fiber-resin interface. The model indicates that GFRP reinforcement bars with larger diameters exhibit lower rates of capacity loss. The proposed probabilistic model is used to assess the probability of not meeting the tensile strength requirement based on specifications over time and can be used to assess the safety and performance of GFRP reinforced systems. Sensitivity and importance analyses are carried out to explore the effect of the parameters and random variables on the probability estimates.

Estimating Residual Hedging Risk with Least-Squares Monte Carlo

Frequently, dynamic hedging strategies minimizing risk exposure are not given in closed form, but need to be approximated numerically. This makes it difficult to estimate residual hedging risk, also called basis risk, when only imperfect hedging instruments are at hand. We propose an easy to implement and computationally efficient least-squares Monte Carlo algorithm to estimate residual hedging risk. The algorithm approximates the variance minimal hedging strategy within general diffusion models. Moreover, the algorithm produces both high-biased and low-biased estimators for the residual hedging error variance, thus providing an intrinsic criterion for the quality of the approximation. In a number of examples we show that the algorithm delivers accurate hedging error characteristics within seconds.

A priori analysis of subgrid mass diffusion vectors in high pressure turbulent hydrogen/oxygen reacting shear layer flames

Direct numerical simulations of high pressure (100 atm), turbulent H2/O2 non-premixed temporally developing shear layer flames are conducted at initial Reynolds numbers ranging from 850 ⩾ Reδ0 ⩾ 4500, with up to ∼3/4 × 109 grid points. A real gas equation of state, real property models, detailed chemistry, and generalized diffusion models are included. The results of the simulations focus on the mass diffusion vectors and their subgrid contributions relevant to large eddy simulation of turbulent combustion. Comparisons of the actual filtered mass flux vectors with their corresponding forms evaluated with filtered primitive variables are shown through correlation coefficients, probability density functions (PDFs) of the ratios between the vector magnitudes, and PDFs of the angles between the vectors. The results show a reasonable correlation (⩾0.75) for all species and simulations when evaluated globally. However, the correlations weaken substantially in regions of large temperature subgrid scale (SGS) variance and filtered SGS scalar dissipation. Within these regions, the correlations are also shown to vary inversely with Reynolds number. Vector analysis indicates that evaluating the mass flux vector using only the filtered primitive variables accurately predicts the direction of the actual filtered flux; however, the magnitude is poorly predicted in the aforementioned regions. Comparisons of the subgrid mass fluxes to the subgrid scalar fluxes are also represented as PDFs of the ratios between the vector magnitudes. These results show the subgrid mass flux to be significantly smaller (≲5%) than the subgrid scalar flux, and this ratio is shown to diminish with increasing Reynolds number. A final comparison of the divergence of the subgrid mass fluxes to the subgrid scalar fluxes are also represented as PDFs of the ratios between the magnitudes. At lower Reynolds numbers, these ratios suggest comparable values between the subgrid diffusion and turbulent stirring terms in regions of large temperature SGS variance and filtered SGS scalar dissipation. However, this ratio is shown to diminish, but remain significant, with increasing Reynolds number.

Portfolios and risk premia for the long run

This paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk. In models with several assets and a single state variable, long-run portfolios and risk premia admit explicit formulas up the solution of an ordinary differential equation which characterizes the principal eigenvalue of an elliptic operator. Multiple state variables lead to a quasilinear partial differential equation which is solvable for many models of interest. The paper derives the long-run optimal portfolio and the long-run optimal pricing measures depending on relative risk aversion, as well as their finite-horizon performance.

Traffic Fluctuations on Weighted Networks

Traffic fluctuation has so far been studied on unweighted networks. However, many real traffic systems are better represented and understood as weighted networks, where nodes and links are assigned some weight values representing their physical properties such as capacity and delay. Here, we introduce a general random diffusion (GRD) model to investigate the traffic fluctuations on weighted networks, where a random walks choice of route is affected not only by the number of links a node has, but also by the weights of individual links. We obtain analytical solutions that characterize the relation between the average traffic and the fluctuations through nodes and links. Our analysis is supported by results of extensive numerical simulations. We observe that the value ranges of the average traffic and the fluctuations, through nodes or links, increase dramatically with the level of heterogeneity in link weights. This highlights the key role that link weight plays in traffic fluctuation and the necessity to study traffic fluctuations on weighted networks.

Roussos-Maragos Tensor-Driven Diffusion for Image Interpolation

Roussos and Maragos proposed a method for image interpolation in "Reversible interpolation of vectorial images by an anisotropic diffusion-projection PDE." An earlier version was also published in conference paper (Roussos and Maragos, "Vector-Valued Image Interpolation by an Anisotropic Diffusion-Projection PDE," 2007). Given a discretely sampled image v, the method finds an image u such that v(n) = (h * u)(n), for all n in Z2, where h is the (assumed known) point spread function and * denotes convolution. The method is inspired by tensor-driven diffusion works of Tschumperlé and Weickert. Roussos and Maragos propose interpolation by evolving a diffusion equation to steady state, dt u = P0(div(T gradient(u)), where T is a tensor determined from image structure tensor and the diffusion is orthogonally projected to agree with the observed data. This diffusion is based on the general anisotropic diffusion model proposed by Weickert. The method can be applied to grayscale, color, or general vector-valued images.

Generalized Einstein relation and the Metzler–Klafter conjecture in a composite-subdiffusive regime

In this paper, a generalized diffusion model driven by the composite-subdiffusive fractional Brownian motion (FBM) is employed. Based on this stochastic process, we derive a fractional Fokker–Planck equation (FFPE) and obtain its solution. It is proved that the Generalized Einstein Relation (GER) and the Metzler and Klafter conjecture on the asymptotic behavior of stretched Gaussian hold the FFPE in a composite-subdiffusive regime.

Nonlinear analysis of a fractional reaction diffusion model for tumour invasion

Mathematical models in general and Reaction Diffusion Models in particular have been rigorously studied and applied in different forms to explain situation in biomedical and allied sciences including the complex tumour microenvironment. They have been proven to be really significant in cancer research. The not so extensively known fractional reaction-diffusion model is presented in this research as a plausible model for the invasion of tumour and its consequences in the human organ in which it resides. In this model we make use of fractional derivatives as a replacement for the normal derivatives to express the diffusion of the tumour cells, normal cells and hydrogen ion concentration. The nonlinear analysis of this model predicts that a death situation always arise if some of the parameters are of a certain magnitude. In this situation, the tumour cells population increases (with an eventual bound after death) by eating up all the normal cells of their host organ. Effectively, the model predicts that the relative interaction between the tumor and normal cells population gives rise to a bifurcation parameter for which a Hopf bifurcation occurs between the period of attack and death. Also the death situation using this model comes faster than the predictions of other models for tumour invasion. This research may therefore give a useful indication of the possible timing of the drugs and dosages required for the treatment or control of tumour progression in human. Key words: Tumour, fractional reaction diffusion equation, trotter product formula, hopf bifurcation, vasculature, oxygenation, Hypoxia.

Optimum advertising policy over time for subscriber service innovations in the presence of service cost learning and customers’ disadoption

On the theoretical side, this paper characterizes qualitatively optimal advertising policy for new subscriber services. A monopolistic market is analyzed first for which customers’ disadoption, discounting of future profits streams and a service cost learning curve are allowed. After characterizing the optimal policy for a general diffusion model, the results pertaining to a specific diffusion model for which advertising affects the coefficient of innovation that incorporates the disadoption rate are reported. The results of the theoretical research show that the advertising policy of the service firm in the presence of customers’ disadoption could be very different from the same when disadoption is ignored. On the empirical side, four alternative diffusion models are estimated and their predictive powers using a one-step-ahead forecasting procedure compared. The diffusion data analyzed are related to the Canadian cable TV industry. Empirical research findings suggest that the specific diffusion model considered above is not only of theoretical appeal but also of major empirical relevance. The analytical findings of the study are documented in six theoretical propositions for which proofs are provided in a separate Appendix. The results of a related numerical experiment together with the analytical findings pertaining to the competitive role of advertising are included. Managerial implications of the study together with directions for future research are also discussed.

Complementarities in the Diffusion of Personal Computers and the Internet: Implications for the Global Digital Divide

This paper studies the cross-country diffusion of personal computers (PCs) and the Internet, and examines how the diffusive interactions across these technologies affect the evolution of the global digital divide. We adopt a generalized diffusion model that incorporates the impact of one technology's installed base on the diffusion of the other technology. We estimate the model on data from 26 developing and developed countries between 1991 and 2005. We find that the codiffusion effects between PCs and the Internet are complementary in nature and the impact of PCs on Internet diffusion is substantially stronger in developing countries as compared to developed ones. Furthermore, our results suggest that these codiffusive effects are a significant driver of the narrowing of the digital divide. We also examine the policy implications of our results, especially with respect to how complementarities in the diffusion of PC and Internet technologies might be harnessed to further accelerate the narrowing of the global digital divide.

The influence of demand factors on dynamic competitive pricing strategy: An empirical study

This empirical paper investigates the relationship between the dynamic strategic interactions among competitors in a component market and demand factors in the market for the end product. The structure of competition in the US microprocessor (MPU) industry is analyzed using data on prices and sales in both the MPU market as well as the market for personal computers. The pattern of dynamic strategic interaction between competing firms in this market on a key decision variable, price is studied. Non-nested model comparison tests based on equilibrium solutions derived for specific differential games are applied to identify the mode of competitive strategy between pairs of competing brands. The empirical fit to the longitudinal and cross-sectional data, of alternative models of competition, independent (Bertrand–Nash), Stackelberg leader–follower, and Collusion, is used to determine which dynamic model best describes actual competitive behavior over the life of each MPU. Demand for the product market which is downstream from microprocessors, that for personal computers, is estimated using a generalized diffusion model with price effects. Data from the markets for desktop and laptop computers are analyzed at the level of computer vendor and internal microprocessor. Patterns are uncovered, linking downstream demand parameters with upstream competitive strategy. There is evidence to suggest that when there are strong diffusion effects driving sales of both the competing computer brands, there is a higher likelihood of Bertrand–Nash competition among MPU firms. However, when there are higher cross-price effects (substitutability) among personal computer brands there is a greater chance of Stackelberg leader–follower price competition. When self-price effects are relatively high, the likelihood of Bertrand–Nash competition among MPU firms increases. Furthermore, when the potential demand for the computer product category is high, there is a higher likelihood of Bertrand–Nash pricing in the MPU market.

The Genetic Algorithm as a General Diffusion Model for Social Networks

Diffusion processes taking place in social networks are used to model a number of phenomena, such as the spread of human or computer viruses, and the adoption of products in viral marketing campaigns. It is generally difficult to obtain accurate information about how such spreads actually occur, so a variety of stochastic diffusion models are used to simulate spreading processes in networks instead. We show that a canonical genetic algorithm with a spatially distributed population, when paired with specific forms of Holland's synthetic hyperplane-defined objective functions, can simulate a large and rich class of diffusion models for social networks. These include standard diffusion models, such as the Independent Cascade and Competing Processes models. In addition, our Genetic Algorithm Diffusion Model (GADM) can also model complex phenomena such as information diffusion. We demonstrate an application of the GADM to modeling information flow in a large, dynamic social network derived from e-mail headers.

Simulating the Diffusion of Information

Diffusion occurs in various contexts and generally involves a network of entities and interactions between entities. Through these interactions, some property, e.g., information, ideas, etc., is spread through the network. This paper presents a general model of diffusion in dynamic networks. The authors simulate the diffusion of evacuation warnings in multiple network structures under various model settings and observe the proportion of evacuated nodes. The network dynamics occur as the result of the diffusion where nodes may leave the network after receiving the warning. The authors use the model to explore how the network structure, seeding strategy, network trust, and trust distribution affect the diffusion process. The effectiveness of the diffusion is a function of the network structure and seeding strategy used in delivering the initial broadcast. The simulation results reveal interesting observations on the effects of network trust and distribution of trust in the network.

Kinetic study of the adsorption of nitroimidazole antibiotics on activated carbons in aqueous phase

The adsorption kinetics of four nitroimidazoles, Dimetridazole (DMZ), Metronidazole (MNZ), Ronidazole (RNZ) and Tinidazole (TNZ), were studied on three activated carbons: two commercial carbons from Sorbo-Norit (S) and Merck (M) and a third prepared by chemical activation of petroleum coke (C). Experimental data of the corresponding adsorption kinetics were analyzed by applying pseudo-first and pseudo-second-order models and a general diffusion model. Application of pseudo-first and pseudo-second-order kinetic models verified the following: (i) The kinetic model used that better predicts the adsorption rates depends of both the adsorbent and adsorbate studied. (ii) Nitroimidazole adsorption rate decreases in the order MNZ > DMZ > RNZ > TNZ; therefore, in the case of MNZ, molecular size does not appear to be a determining factor in the process. (iii) Nitroimidazole adsorption rate on carbons increases in the order C < S < M, which is related to the increase in carbon hydrophobicity. Hence, in general, hydrophobic interactions appear to govern the kinetics of the adsorption process. Finally, a general diffusion model was applied that combines external mass transport and intraparticle diffusion, achieving an adequate fit to the experimental data. There are notable differences among the diffusivity values for the different nitroimidazoles that do not appear to be exclusively related to carbon textural parameters or adsorbate size. Therefore, adsorbent and adsorbate chemical characteristics are highly important to establish the adsorption mechanism of nitroimidazoles on activated carbons.

Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups

This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size a . As a consequence, we obtain the probabilities that a drawdown of size a precedes a drawup of size b and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein–Uhlenbeck process, and Cox–Ingersoll–Ross process.

Regime-Switching Univariate Diffusion Models of the Short-Term Interest Rate

This article proposes a general regime-switching univariate diffusion model to describe the dynamics of the short-term interest rate. The maximum likelihood estimates are obtained using the weekly series of U.S. three-month treasury bill rates. The estimation results reveal that there are strong evidences for the existence of high and low volatility regimes, for the time varying transition probability of the regime variable, and for the high persistence of both regimes. In both regimes, the volatility, but not the drift, is estimated accurately and plays a key role in explaining the dynamics of the interest rates. High persistence's and different volatilities of two regimes can well explain volatility clustering observed in the data. Based on the inferred probability of the process being in each regime, most of the high volatility periods correspond to some historic events. The likelihood-based test shows that misspecification can result in misleading outcomes particularly regarding the volatility and transition probabilities of the regime index.