Optimal-control methods

Optimal control-based methods were applied to the four synthetic datasets as well as to the two real datasets. The following priors were used: penalisation of the first derivative, penalisation of the second derivative, and penalisation based on the maximum entropy prior. Penalisation of the derivatives was also applied to the logarithm of the input function. The maximum entropy prior was not applied to the logarithm of the function, as this prior inherently enforces non-negativity. Applying a maximum entropy prior in the log domain would therefore constrain the input function to be greater than1 at all time points, which is clearly undesirable. For the cases where the derivative of the input function was penalised directly, non-negativity constraints were added to the optimisation problem in order to avoid non-physical solutions.

For the choice of function parameterisation, only piecewise constant basis functions were used. The rationale behind this decision is that, as long as the parameterisation is sufficiently fine-grained, it is for all practical purposes able to represent any function of interest. For this dataset, the input function was discretised in100intervals, uniformly distributed from time0to the time of the final measurement. Optimal-control methods can in many cases handle such fine parameterisation, as will be shown in the results section. Furthermore, this parameterisation makes the multiple shooting and collocation schemes maximally sparse, since each function coefficient only participates in the continuity and collocation constraints of a single interval.

As the piecewise constant functions are not differentiable at the jump points, the derivative-based priors were computed using finite differences, and integration was approximated by summing over all intervals. Denoting the input function value in thekth interval byuk, the approximation of the first-derivative regularisation term

is given by ER= Z t_f ti du(t) dt 2 dt≈ NB−2 X k=0 (uk+1−uk)2 ∆t . (4.3)

where∆tis the length of each discretisation interval. Similarly, the second derivative regularisation term is given by

ER= Z t_f ti d2_u(t) dt2 2 dt≈ NB−2 X k=1 (uk+1−2uk+uk−1)2 (∆t)3 . (4.4)

function and does not require any further discretisation.

All three direct optimal-controls methods presented in Section 3.2 were used: single shooting, multiple shooting, and collocation. For simplicity, the shooting and collocation intervals were chosen to coincide with the intervals chosen for the input parameterisation. The collocation method used Lagrange polynomials of degree 3

with Radau collocation points, which are given by the roots to the polynomial

dd−1 dtd−1(t

(d−1)_(t−1)d_{) (4.5)}

wheredis the order of the polynomial (Hairer and Wanner 1999). Ford= 3, the roots are located at the positions

h 2 5 − √ 6 10, 25 + √ 6 10, 1 i

. Although other collocation schemes could have been investigated, these settings were assumed to be sensible defaults.

For the most part, the discrepancy criterion was used to determine the regularisation parameterτ, despite the fact that this has been shown to overestimate τ to some extent (Twomey 1965). The reason for this choice is that this criterion makes it possible to determine τ as well as the optimal solution using a single optimisation procedure. The discrepancy criterion suggests that χ2, the sum of squared prediction residuals, scaled by the measurement standard deviation, should not exceed the number of measurements:

χ2 = n−1 X j=0 y(_jpred)−yj σj !2 ≤n, (4.6)

wherenis the number of measurements,y_j(pred)is the predicted measurement at time j, yj is the measurement at time j, and σj is the assumed measurement standard

deviation at timej. This makes it possible to pose the optimal-control problem, for the case of single shooting, as:

minimise a ER(a) (4.7a) such that χ2 ≤n (4.7b) x(ti) =x(0) (4.7c) and h(a,x(t))≤0 (4.7d) where x(t) = Φ(ti, t,x(0),a).

The termER(a)is the regularisation term. The inequality constraintsh(·)are

in this case the nonnegativity constraints for the input function. For multiple shooting and collocation, the problem is modified analogously: the objective is changed from the penalised log-likelihood to the regulariser, and theχ2 value is introduced as an extra inequality constraint. When negative entropy is used as the regulariser ER,

and the system is linear, this algorithm becomes identical to the maximum entropy algorithm presented by Charter and Gull (1987) and Hattersley et al. (2008).

In addition to the discrepancy criterion, the L-curve approach was applied to the real datasets, using penalisation of the second derivative. Estimation was performed for 80 values of τ, logarithmically spaced between 103 and 1011. This range was considered to cover any reasonable setting ofτ. Collocation was used as the optimisation method, in order to keep the total running time low. The resulting L-curves were plotted, and suitable values ofτ were manually selected. To assess the sensitivity of the estimate toτ, three values were selected for each dataset: one at the “knee” of the curve, and one on either side of the knee.

For the single shooting method, there areNB = 100decision variables, each

representing the input function for each discretisation interval. This method has one inequality constraint representing the discrepancy criterion, and an additionalNB

inequality constraints enforcing nonnegativity of the input function, for the cases where such constraints are desired. The multiple shooting method adds an additional NB·dxdecision variables, representing the states at the beginning of each discretisation

interval, and an additionalNB·dx equality constraints enforcing the initial conditions

and the continuity constraints. The collocation methods require, in addition to the variables and constraints of single shooting, an additional NB·(d+ 1)·dx decision

variables, as the trajectory of each state in each interval is represented by a polynomial of degreed. They also add an additionalNB·(d+1)·dxequality constraints, accounting

for the initial conditions, continuity constraints, and collocation constraints. Table 4.3 reports the number of variables and constraints per method.

In summary each of the four test datasets and the two real datasets were ana- lysed using15combinations of priors and optimisation methods. This is summarised in Table 4.4.