Jq

Fig. 7.18. Plots of the cost-functional J = J(pi,p2) as a function of the centre (pi,p2) of the infarction for various levels of noise in the observation data. The size of the parameters pi and p2 are represented by the 'x' and 'y' axes, respectively. (For the color version, see Figure A.18 on page 296).

Fig. 7.18. Plots of the cost-functional J = J(pi,p2) as a function of the centre (pi,p2) of the infarction for various levels of noise in the observation data. The size of the parameters pi and p2 are represented by the 'x' and 'y' axes, respectively. (For the color version, see Figure A.18 on page 296).

- In the cases of no and 5% noise in the data, J is fairly convex and has a unique minimum. Moreover, the minimum is located at the right position « (3,3)

- Even with a noise level of 100%, our cost-functional seems to capture the basic properties of the problem at hand. However, in such cases, it seems to be extremely difficult to actually compute the minimum of J.

Considering these results, one might ask the question; is the problem (7.133) well-posed? As explained in Section 7.1.5, discretizing an ill-posed system of equations often leads to a regularized approximation of the underlying problem. Based on physical considerations, it seems to be reasonable to conclude that the continuous counterpart, using infinitely many infarction parameters, of (7.133) is ill-posed. Thus, the number n of infarction parameters used in (7.133) defines a regulariza-tion parameter. In the present example n = 2, and consequently we obtain a well-behaved minimization problem (7.154).

We have seen that if the number n of infarction parameters is low, then the objective function J is well-behaved. Thus, it seems to be important to investigate the number of parameters needed to accurately characterize an infarction. In addition, efficient minimization algorithms, using the adjoint problem approach described in Section 7.4.3, for solving (7.133) should be implemented and tested. Such methods will typically require the solution of (7.121)-(7.129) and (7.152)-(7.153) for many different choices of the infarction parameters pl,...,pn. Both of these problems are extremely CPU-demanding. Hence, the methods for solving the direct problem modelling the electrical activity in the human body must be further improved. We hope to return to these issues in our future work.

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