Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients

Abstract

In this work, we study a one-dimensional elliptic equation with a random coefficient and derive an explicit analytical approximation. We model the random coefficient with a spatially varying random field, K(x, ω) with known covariance function. We derive the relation between the standard deviation of the solution T (x, ω) and the correlation length, η of K(x, ω). We observe that, the standard deviation, σ T of the solution, T (x, ω), initially increases with the correlation length η up to a maximum value, σ T,max at η max ∼ x(1 − x)/3 and decreases beyond η max. We observe a scaling law between σ T and η, that is, σ T ∝ η 1/2 for η → 0 and σ T ∝ η −1/2 for η → ∞. We show that, for a small value of coefficient of variation (ε K = σ K /µ K) of the random coefficient, the solution T (x, ω) can be approximated with a Gaussian random field regardless of the underlying probability distribution of K(x, ω). This approximation is valid for large value of ε K , if the correlation length, η of input random field K(x, ω) is small. We compare the analytical results with numerical ones obtained from Monte-Carlo method and polynomial chaos based stochastic collocation method. Under aforementioned conditions, we observe a good agreement between the numerical simulations and the analytical results. For a given random coefficient K(x, ω) with known mean and variance we can quickly estimate the variance of the solution at any location for a given correlation length. If the correlation length is not available which is the case in most practical situations, we can still use this analytical solution to estimate the maximum variance of the solution at any location.

Solving random boundary heat model using the finite difference method under mean square convergence

“…This provides a consistent connection between the deterministic and the random setting. It must be pointed out that this hypothesis is not new at all since this kind of condition has been imposed by other authors in dealing with the study of both random ordinary differential equations and PDEs to extend important deterministic results to the random scenario …”

Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

“…We already know that, if v N (y,t)(ω), A(ω), and B(ω) are absolutely continuous and independent random variables, then u N (x,t)(ω) has a density function f u N ðx;tÞ ðuÞ given by (15). On the other hand, if A and B are deterministic, assuming that v N (y,t)(ω) is absolutely continuous, one has that u N (x,t)(ω) has a density function f u N ðx;tÞ ðuÞ expressed by (17).…”

“…In Figure 6, we approximate the probability density function of the solution stochastic process u(x,t)(ω) at x = 5 and t = 0.2, using (15), for N = 1,2,3,4. Compare the plots with those of Example 6.1, where the boundary conditions were deterministic with constant value the mode of A and B.…”

“…In B∼Normal (2,1). Notice that E½A and E½B are the deterministic boundary conditions of Example 6.2, so the approximated density functions may resemble those from Example 6.2.In Figure 7, three plots of the path described by ϕ(x) for three different outcomes ω are presented.In Figure 8, we approximate the probability density function of the solution stochastic process u(x,t)(ω) at x = 1 and t = 0.1, using (15), for N = 1,2,3,4. These plots are very similar to those from Example 6.2.…”

“…Approximations of E½uð1; 0:1Þ and V½uð1; 0:1Þ for N = 1,2,3,4 constructed by (18) and (19), respectively, being f uN ðx;tÞ given by (15…”