Mathematical Modelling and Computer Simulation

where u(x, t) is the concentration of the species and α1 is the diffusion coefficient or diffusivity of u(x, t). Situations where 1 is space-dependent are arising in more and more modeling situations of biomedical importance from diffusion of genetically engineered organisms in heterogeneous environments to the effect of white and grey matter in the growth and spread of brain tumors. The source term or forcing term f in an ecological context, for example, could represent the birth-death process. However, the equation (1) is strictly only applicable to dilute systems, that is the diffusion is a local or short range effect. In many biological areas, such as embryological development, the densities of cells involved are not small and a local or short range diffusive flux proportional to the gradient is not sufficiently accurate. When we discuss the mechanical theory of biological pattern formation in certain circumstances, it is intuitively reasonable, perhaps necessary, to include long range effects. In 1969, Othmer derived the following formulation (1).


Introduction
Mathematical modeling and computer simulation are nowadays widely used tools to predict the behavior of biological research problems. To illustrate the idea, we consider nonlocal effects and long range diffusion mathematical biology model [1]. The classical approach to diffusion is the following form ( ) ( ) 1 ( , ) . , , , x u u f u x t t (1) where u(x, t) is the concentration of the species and α 1 is the diffusion coefficient or diffusivity of u(x, t). Situations where 1 is space-dependent are arising in more and more modeling situations of biomedical importance from diffusion of genetically engineered organisms in heterogeneous environments to the effect of white and grey matter in the growth and spread of brain tumors. The source term or forcing term f in an ecological context, for example, could represent the birth-death process. However, the equation (1) is strictly only applicable to dilute systems, that is the diffusion is a local or short range effect. In many biological areas, such as embryological development, the densities of cells involved are not small and a local or short range diffusive flux proportional to the gradient is not sufficiently accurate. When we discuss the mechanical theory of biological pattern formation in certain circumstances, it is intuitively reasonable, perhaps necessary, to include long range effects. In 1969, Othmer derived the following formulation (1).
where α 1 > 0 and α 2 are constants. α 2 is a measure of the long range effects and in general is smaller in magnitude than α 1 . The biharmonic term is stabilizing if α 2 > 0, or destabilizing if α 2 < 0. In this form, the first term represents an average of nearest neighbors and the second biharmonic term is a contribution from the average of nearest averages.
We then consider the stationary Dirichelet boundary value problem of the equation (2) ( ) ( ) ( ) ( ) 1 2 , , where D represents the species living area, which can be considered bounded and the Dirichelet boundary condition can be interpreted as the number of the species is zero on the boundary of the domain D. Yet many biological applications are affected by a relatively large amount of uncertainty in the input data, such as model coefficients, source term/forcing term, boundary conditions, and geometry. In the case, to obtain a reliable numerical prediction, one has to include uncertainty quantification due to uncertainty in the input data. In this paper we focus on problem (3) with a probabilistic description of the uncertainty in the input data. Let D be a convex bounded polygonal domain in  d , (d = 1, 2, 3) and ( ) , Ω ,P F be a complete probability space, where Ω is the set of outcomes, where D is the closure of D.
The space H k (D) is endowed with the norm associated to the inner product and the corresponding norm Denote by finally, we discuss the well-posedness of problem (5). To the end, we quote the following Poincare's inequality ( ) with C P = C P (D, n, k) > 0, [2].
We will prove a(u, v) is continuous and coercive. min The proof is now complete.

Finite-Dimensional Noise Assumption
In many problems the source of randomness can be approximated using just a small number of uncorrelated, sometimes independent, random variables, for example, the case of a truncated Karhunen-Loeve expansion [3]. This motivates us to make the following assumption.

Assumption:
The coefficients and forcing terms used in the computations have the forms where N ∈ N + and { } 1 have a joint probability density function : After making Assumption, the solution u of the stochastic fourthorder elliptic boundary value problem (5) Then, the goal is to approximate the u(y, x), where ∈ Γ N y and ∈ x D . Observe that the stochastic variational formulation (5) has a deterministic equivalent which is the following: find ( ) ( ) Since the solution of (1.6) is unique and is also a solution of (5), it follows that the solution has the form ( ) x . The stochastic boundary value problem (4) now becomes a deterministic boundary value problem (6) for a fourth-order elliptic PDE with an N-dimensional parameter. For convenience, we consider the solution u as a function ( ) : Γ → N u H D and use the notation u(y) whenever we want to highlight the dependence on the parameter y. We use similar notation for the coefficient α 1 , α 2 , and the forcing term f. Given Thus, we turn the original stochastic fourth-order elliptic equation into a deterministic parametric fourth-order elliptic equation and we will adopt finite element technique to approximate the solution of the resulting deterministic problem.

Regularity Assumption
The convergence properties of the collocation techniques that will be developed in the next section depend on the regularity that the solution u has with respect to y. Denote * 1, = ≠ Γ = Π Γ N n j j n j , and let * n y be an arbitrary element of * Γ n . Here we require the solution of problem (4) to satisfy the following assumption. To make this assumption, we introduce the functional space where v is continuous in y.
Assumption: For each ∈ Γ with λ a constant independent of n.
The following lemma will verify that this assumption is sound.
where C 0 is a constant depending on a min , a max and Poincare's constant C P Proof: For simplicity, we first study the following problem: there exists For every point ∈ Γ N y , the k th -derivative of u with respect to y n is obtained by above equation, which satisfies as follows Taking φ = ∂ n k y u , we obtain the following form Using the Poincare' inequality, to obtain Combination of (8), (9) yields and using the bounds on the derivatives of α and f, we get the recursive inequality Next, we will prove the following form , and obtain ( ) ( ) By Lemma 5.1, the following estimate holds (11) follows from (12) and (13). Using that Hence (11) and (14) imply For every ∈ Γ n n y , we get the final estimates on the growth of the derivatives of u.
For every : Then, we will prove the uniform convergence of the power series (16) with norm in We get the following formula [4] ( Similarly, for the solution ( ) * , , n n u y y x of problem (7), the conclusion which is above drawn is correct. This finishes the proof.

Example 1:
Let us consider the case where the coefficient α(ω,x) is expanded in a linear truncated Karhunen-Loeve expansion provided that such an expansion guarantees α(ω) ≥ a min for almost every ω∈Ω and x∈D [5], in the case we have

Collocation techniques
We seek a numerical approximation to the solution of (7)   , , In order to prove error estimates for stochastic partial differential equation, we need estimates for deterministic fourth order elliptic problem. Let us consider the stationary deterministic problem we make the following assumptions: (AA 1 ) there exist a min ; a max > 0 such that The variational form of problem (2.2) is to where <⋅;⋅ > represents the duality pairing.
, , , , , Then, we will estimate the error between u and uh. In order to get the estimate, we need the following two lemmas. Lemma 6.1: Suppose the conditions (1) (H; (⋅;⋅)) is a Hilbert space, and V is (closed) subspace of H, (2) a(⋅;⋅) is a bilinear form on V , which is continuous and coercive on V , and that u solves, Given F ∈ V′ [6].
For the finite element variational problem: Given a finite- Then, the following inequality holds where C, a are the continuity constant and the coercivity constant of a(⋅,⋅) on V, respectively.

Smolyak approximation
Here we follow closely the work [7] and describe the Somlyak isotropic ( , ) w N A . The Smolyak formulas are just linear combinations of product formula (23) with the following key properties: only products with a relatively a small number of points are used. With Given an integer � + ∈ w , hereafter called the level, we define the sets ( ) ( ) 1 , : , 1: ( ) ( ) 1 , : , 1: and for i  + ∈ N we set (28) Equivalently, formula (28) can be written as [8] ( Obviously, A (w; 1) = U w. To compute A (w;N)(u), one only needs to know function values on the "spare grid" denotes the set of abscissas used by i U .

Choice of collocation nodes
In this section, we will determine how to select the collocation nodes. To the end, we introduce a conclusion.  ; inf where the constant C 1 is independent of p.

Error Analysis
In this section we show error estimates that will help us understand the sparse grid stochastic collocation method in this situation is efficient. Collocation methods can be used to approximate the solution using Thus, we will only concern ourselves with the convergence results when implementing the Smolyak approximation formula, namely, our primary concern will be to analyze the Smolyak approximation error ; , τ τ Γ = ∈ Γ < ∑ z ,dist z C of the complex plane, for some τ > 0, in this case, τ is smaller than the distance between 1 Γ ⊂  and the nearest singularity of u(Z) in the complex plane. For the caseN=1, we quote the following results.  : In what follows we will use shorthand notions Obviously when d = 1, we have