Random Diffusion Coefficient Research Articles

Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D⊂ℝd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y i (ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x∈D and the in general countably many parameters y. We establish new regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to $V=H^{1}_{0}(D)$. These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called “generalized polynomial chaos” (gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable y∈U are established. These estimates are of the form N −r , where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family $\{V_{l}\}_{l=0}^{\infty}\subset V$of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to a smoothness space W⊂V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with $H^{2}(D)\cap H^{1}_{0}(D)$in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate $N_{\mathrm{dof}}^{-s}$in terms of the total number of degrees of freedom N dof can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.

Finite elements for elliptic problems with stochastic coefficients

We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen–Loève (KL) expansion. An O(NlogN) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M,1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r=(r1,…,rM) in an (M,r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r1 up to 10 in moderate time.

Random diffusion and leverage effect in financial markets.

We prove that Brownian market models with random diffusion coefficients provide an exact measure of the leverage effect [J-P. Bouchaud et al., Phys. Rev. Lett. 87, 228701 (2001)]. This empirical fact asserts that past returns are anticorrelated with future diffusion coefficient. Several models with random diffusion have been suggested but without a quantitative study of the leverage effect. Our analysis lets us to fully estimate all parameters involved and allows a deeper study of correlated random diffusion models that may have practical implications for many aspects of financial markets.

Intercalation thermodynamics and chemical diffusion of oxygen in the solid solution YBa2Cu3−xCoxO6+δ

The equilibrium oxygen content as a function of the temperature and oxygen pressure was measured for the solid solution YBa2Cu3−xCoxO6+δ, where x=0, 0.2, 0.4, 0.6, 0.8, by using coulometric titration in the temperature range 600–850°C and oxygen pressures between 10−5 and 1.0 atm. The change in the partial molar enthalpy and entropy of the intercalated oxygen was determined at different oxygen and cobalt contents. The oxygen chemical diffusion was studied by thermogravimetric relaxation in the oxygen-controlled atmosphere. The thermodynamic data were employed to determine how the chemical diffusion coefficient, the thermodynamic factor and the random-diffusion coefficient depend on oxygen content in specimens with different cobalt concentration. The oxygen intercalation thermodynamics and diffusivity results provide evidence of ordering phenomena on a microscopic scale in the basal plane of the tetragonal solid solution YBa2Cu3−xCoxO6+δ. A model for the oxygen diffusion is suggested to explain the large difference between the random and tracer diffusion coefficients in YBa2Cu3O6+δ

Effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems

We study the effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems. For this purpose, the Gierer-Meinhardt model of pattern formation is considered. The cases we study are: (i)randomness in the underlying lattice structure, (ii)the case in which there is a probablity p that at a lattice site both reaction and diffusion occur, otherwise there is only diffusion and lastly, the effect of (iii) anisotropic and (iv) random diffusion coefficients on the formation of Turing patterns. The general conclusion is that the Turing mechanism of pattern formation is fairly robust in the presence of randomness and anisotropy.

Reliability Approach to Service Life Prediction of Concrete Exposed to Marine Environments

In the life of concrete structures they are subjected to external actions or agents that in time may alter them from a safe state to failure or a damage state. One of the processes that may trigger the onset of corrosion of steel embedded in concrete is the ingress of chloride ions that eventually reach the reinforcement, causing the rupture of the passive film. In this paper, a method is proposed for interpretation of immersion test results and prediction of the service life of concrete structures exposed to chloride ions. Once the chloride profiles are determined after the immersion test, a realization of the random diffusion coefficient is obtained at each point for which there is a measurement of the chloride content. These samples are used to predict the probability density function of the diffusion coefficient. A reliability analysis is subsequently performed for 10 different lightweight high-strength concrete mixes. A reinforced concrete element is considered to have failed when corrosion initiates at the reinforcement, i.e., after a certain chloride concentration threshold is reached at the reinforcement. The probability of corrosion initiation in time is computed for some of the concrete mixes examined.

Diffusion of clusters with randomly growing masses.

Diffusion of fractal clusters of dimension ${\mathit{d}}_{\mathit{f}}$ in a three-dimensional space is investigated. The diffusion process is assumed to be modeled by a standard parabolic diffusion equation but with a random diffusion coefficient. The motivation for this assumption is provided by two pieces of evidence: (1) the cluster diffusion coefficients depend on the clusters' masses, sizes, and shapes; (2) the masses of clusters change stochastically in time due to random attachment or detachment of particles. Two models of the growing process are considered: (a) a Poisson process; (b) a simple birth-and-death process with linear rules. The mean square displacement of the cluster mass centers is analyzed and its anomalous behavior is demonstrated as a function of the fractal cluster dimension.

On the scale-invariant distribution of the diffusion coefficient for classical particles diffusing in disordered media

The scaling form of the whole distribution P(D) of the random diffusion coefficient D(x) in a model of classically diffusing particles is investigated. The renormalization group approach above the lower critical dimension d=0 is applied to the distribution P(D) using the n-replica approach. In the annealed approximation (n=1), the inverse gaussian distribution is found to be the stable one under rescaling. This identification is made based on symmetry arguments and subtle relations between this model and that of fluc- tuating interfaces studied by Wallace and Zia. The renormalization-group flow for the ratios between consecutive cumulants shows a regime of pure diffusion for small disorder, in which P(D) goes to delta(D-<D>), and a regime of strong disorder where the cumulants grow infinitely large and the diffusion process is ill defined. The boundary between these two regimes is associated with an unstable fixed-point and a subdiffusive behavior: <x**2>=Ct**(1-d/2). For the quenched case (n goes to 0) we find that unphysical operators are generated raisng doubts on the renormalizability of this model. Implications to other random systems near their lower critical dimension are discussed.

Diffusion of protein complexes in prokaryotic membranes: fast, free, random or directed?

The random two-dimensional diffusion coefficient of many membrane proteins estimated from a variety of biophysical measurements is much greater than must be inferred from certain more biochemically based experiments in prokaryotes. One solution to this paradox suggests that such diffusion is neither fast nor free.