Background Material

4.3.1 Random PDE Theory

Given some probability space (Ω,F,P) where Ω is our sample space, F is a sigma-

algebra and P :F → [0,1] is a probability measure where we define samples from

our sample space asϑ∈Ω. We assume we have a domainD⊂_Rd_{ford <∞ where} D is a Lipschitz domain with boundary ∂D. Given a random field κ(x;ϑ) we are interested in finding the solutionp(x;ϑ) to the random PDE

−∇ ·(κ(x;ϑ)∇p(x;ϑ)) =f, ∈D, (4.3.1a)

p(x;ϑ) = 0, ∈∂D, (4.3.1b) where we have imposed Dirichlet boundary conditions. Before discussing a well- posedness theorem of (4.3.1) we first review a number of assumptions regarding both the source termf and the random coefficientκ(x;ϑ).

Assumption 4.3.1. There exist constants 0< κmin< κmax <∞ such that

P(κmin≤κ(x;ϑ)≤κmax, ∀x∈D) = 1. (4.3.2) where κ(x;ϑ)∈L∞(D;R) andesssupκ(x;ϑ) =κmax(ϑ)>0,P−a.s.

Assumption 4.3.2.LetΓ⊂_RK _{be called the parameter space. Letκ}

0, κ1, . . . , κK ∈ L∞(D), and let u1, . . . , uK be independent random variables from Ω taking values in R such that u = (u1, u2, . . . , uK) ∈ Γ. Then our random coefficient can be be expressed through the following series

κ(x;ϑ) =κ0(x) + K X

k=1

κk(x)uk(ϑ). (4.3.3)

That is,κ(x;ϑ) has a linear dependence on finitely many random variables. For ease of computation and analysis, we will assume that the uk’s are chosen

uniformly at random from the uniform distribution U[−1,1]. Given these two as- sumptions we present the uniqueness and existence theorem of random PDEs which is an application of the Lax-Millgram Theorem:

Theorem 4.3.1. Assume thatf ∈V∗_,

P−a.s, and

essinfκ(x;ϑ) =κmin(ϑ)>0, P−a.s. ThenP−a.s (4.3.1) has a unique solution which satisfies

kp(·;ϑ)k_V ≤ kfkV∗

κmin(ϑ) . Assume further that either:

(i) f ∈V∗ is deterministic and κ is distributed according to a uniform prior. (ii) κ=eu_{, u∈L}∞_(D;

R) deterministic andf ∈L2_P(Ω;V∗). Then (4.3.1) has a solutionP−a.s and in L2_P(Ω;V).

Proof. The proof can be found in [99] which is based on the Lax-Milgram Lemma.

Remark 4.3.1. We note that the Lax-Milgram Theorem in Theorem 4.3.1 holds also for a random source term f(x;ϑ), but for the purposes of this work we keep f deterministic as stated above.

4.3.2 Finite Element Method

To solve numerically a realisation of (4.3.1), we use a finite element method (FEM), which is based on the Galerkin projection. In particular, letD= [0,1] andh∈(0,1)

xh

i −xhi−1=h, for all i∈ {1, . . . , Nh}. Define the basis functions {φj}Nj=1h such that φj(xhi) = δij, for all i ∈ {0,1, . . . , Nh}, and interpolate linearly between any two

points of the partition. Then{φj}Nj=1h is a basis of a finite dimensional subspaceVh

ofH1

0(D) which contains all the functions q∈C0(D) such that q|[xh

i−1,xhi]is a linear

polynomial, for alli∈ {1, . . . , Nh}.

We now consider the family {Vh}h∈(0,1) of the finite dimensional subspaces

of H1

0(D), generated by the discretisation parameters h ∈ (0,1), and the finite

dimensional equations

A(ph(ϑ), qh;ϑ) =l(qh), ∀qh∈Vh. (4.3.4)

We know that there exists a unique solutionph(ϑ) of (4.3.4) and it is called

the Galerkin projection of the solution p(ϑ) onto Vh. As Vh = span{φ1, . . . , φNh},

whereNh = dimVh and{φi}N_i=1h is the basis ofVh, we can express the solutionph(ϑ)

in terms of the basis functions{φi}Ni=1h

(ph(ϑ))(x) = Nh X

i=1

Pi(ϑ)φi(x), (4.3.5)

where {Pi(ϑ)}N_i=1h are real numbers still to be calculated. In fact such a

calculation can be done by solving the following system of linear equations

Nh X

i=1

A(φi, φj;ϑ)Pi(y) =l(φj), j= 1, . . . , Nh. (4.3.6)

It is more convenient to think of the above linear system as

Aϑ_h·P_hϑ=lh, (4.3.7)

whereAϑ h ∈RNh

×Nh_{is called the stiffness matrix given byA}ϑ

h(i, j) =A(φi, φj;ϑ), for

all i, j ∈ {1, . . . , Nh}, P_hy = (P1(ϑ), . . . , PNh(ϑ)) and lh = (l(φ1), . . . , l(φNh)). Note

that the stiffness matrix is tri-diagonal because each basis function only overlaps with the two neighbouring basis functions. This implies that for large enoughNh

the matrix is sparse.

A good approximation result of the Galerkin projections {ph(ϑ)}h∈(0,1) is

given by C´eas Lemma which in general states the following.

Lemma 4.3.1. Let H be a Hilbert space, A be a bilinear form on H, which is coercive with constantκmin>0 and bounded with constantκmax >0, and l a linear

functional onH. Let p∈H such that A(p, q) =l(q), for all q ∈H, and consider a finite dimensional subspaceV of H andpV such that A(pV, q) =l(v), for all q∈V. Then kp−pVkH ≤ κmax κmin inf q∈Vkp−qkH. (4.3.8)

In the case of the FEM the lemma above takes the following form

kp(ϑ)−ph(ϑ)kH1 0(D) ≤C_qinf h∈V kp(ϑ)−qhkH1 0(D), (4.3.9) forC= κmax κmin >0.

Finally, we have the following convergence result in terms ofh, provided that

p(ϑ)∈H2_{(D), the coefficientκ(·, ϑ)∈C}1_{(D) andDis a convex, bounded, Lipschitz}

boundary

kp(ϑ)−ph(ϑ)kH1

0(D)≤Chkp(ϑ)kH2(D). (4.3.10)

Note that the classical theory of PDEs implies that p(ϑ) ∈ H2_{(D) since} f ∈L2_(D).