Numerical Results

4.7.1 Uniform Prior

From our experiments in subsection 4.5.5 our random coefficient was based on the expansion (4.5.8). For the RB-EKI our random coefficient will be based on a prior which is that of a geometric type. These priors will have the general form of

u=φ0+ ∞ X

j=1

ujφj, (4.7.1)

whereu≡κ. (4.7.1) consists of random functionsuj =γjξj forj= 1, . . . ,∞, where ξ={ξi}∞j=1 is an i.i.d. sequence with ξ1 ∼ U[−1,1] and{γ}∞j=1 ∈`1. We note that

{φj}∞j=1 ∈L∞(D) is an infinite sequence. Note the form of the random coefficient

in (4.7.1) is a general case, for our experiments we will use a truncated expansion.

4.7.2 Single Phase 2D Prior

Recalling the RB experiments conducted for the forward problem had a random coefficient that was based on a form given in (4.7.1). Now that we are working towards implementing the RBM within the inverse solver we need to define the form of the random coefficient as a prior µ0. In order to do so we have to remain

consistent with the assumptions of the random coefficient; that it has an affine form which is independent of the parametersµ= (µ1, . . . , µk).

In the RBM literature it is common to choose the random coefficient either uniformly or log-normally. Modifying the coefficient based on this can change the setup of the RBM, due to this we continue to use to assume a uniformly distributed random coefficient. In the context of groundwater flow recent priors that have been developed by Iglesias et al. [80] have showcased to perform well which are based on channelized flow. This is based on some non-linearities. For our prior will we use a modification of the channel flow prior defined in [80] neglecting the non-linearities. In order to define our channelized flow we have two equations which govern the channel which are defined as

t1= x d3 +d1, (4.7.2) t2= x d3 +d1+d2. (4.7.3)

From equations (4.7.2) and (4.7.3) we have three main parameters within our prior. The first beingd ∈_R+_{, which can be though of as determining the steepness of the}

channel,d1 ∈R+ which defines the initial point andd2 ∈R+ defining the height of

the channel. As well as the channel parameters we also have the the values ofκ1, κ2

which are the values of the permeability in and out the channel. All the parameters within the model are distributed accordingly to a uniform which is provided in Table 4.2. By relating this to the assumptions on the random coefficient, we design our prior such thatκmin = 1 ad κmax =5.5.

Parameter Prior distribution

d1 U[0,0.5]

d2 U[0,1]

d3 U[1,20]

κ1 U[1,1.5]

κ2 U[5,5.5]

Table 4.2: Prior associated with single phase flow.

The unknown parameter for this model isu= (d1, . . . , d3, κ1, κ2)∈R5.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Figure 4.7: Random draws from single phase prior.

4.7.3 RB-EKI Numerics

We now look to test the RB-EKI method for the model problem (4.5.7). For the inclusion of the inverse solver we need to define our truthu† which we stated would

be piecewise constant. Our truth is given below in Figure 4.8.

Our aim for choosing this particular prior form is in hope for a good recon- struction, hence why it is of a similar form to the truth. We are interested in an underdetermined system where we have 64 observational points uniformly spread. For the regularisation we choose ρ = 0.8 and τ ≈ 1.25. The number of ensem- ble members chosen is J = 150, with an iteration count of n = 20. Our noise

η∼ N(0, γ2_{I) will be chosen such thatγ = 0.4.}

For the forward solver we initialise the numerics similarly as before where we have a mesh size of h= 1/40 where our training set Ξtrain will be based on the

Lebesgue optimal points for Γ = [−1,1]9 _{with a tolerance level of T OL= 10}−9_{. As}

before we generate 50 points in our parameter space. Apart from the mean of the outputuEKI we are also interested in two other quantities of interest:

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Figure 4.8: Channelized geometric truth u†.

• Relative error - ku †_−u EKIkL2(D) ku†_k L2(D) .

• Data misfit -y−G¯(u_EKI) 2 Γ.

Method Dimension Time (seconds) FEM-EKI 9 3773

RB-EKI 9 2287

Table 4.3: 2D RB-EKI numerics. Performance of the different iterative methods. From the numerics conducted we gain an indication of the performance of RB-EKI in terms of both the mean reconstruction of the truth i.e. uEKI, the dif-

0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Figure 4.9: 2D RB-EKI numerics. Left: True permeability. Centre: Finite element reconstruction. Right: Reduced basis reconstruction.

Iteration 5 10 15 20 Relative Error 0.4 0.5 0.6 0.7 0.8 0.9 1 Iteration 5 10 15 20 25 Log Data-Misfit 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 4.10: 2D RB-EKI numerics. Left: RB-relative error. Right: RB-log-data misfit.

and the relative error. Firstly regarding Figure 4.9 we see the performance of the RB-EKI which shows a relatively good recovery of the true permeability shown on the left hand side. We also see the reconstruction of the iterative method with a FEM where we see a similar performance to that of the RB-EKI. Despite the per- meability levels being slightly off as well as the width of the channel the overall structure is recovered. This is aided by Figure 4.10 which demonstrates the effec- tiveness of the regularized properties within the iterative method. As the number of iterations increase we see a decline in both the data misfit and the relative error which terminates after the 20th iteration.

In terms of the error and the computational time when comparing both methods, we see an decrease in computational time needed with the inclusion of the RBM while showing similar errors with the FEM. This speed is up is significantly better than the results we were obtaining within the 1D elliptic problem.