Biomedical Engineering Reference
In-Depth Information
A categorization of common optimization algorithms following the consider-
ations introduced in the previous section is shown in Table 3.2 (deterministic
algorithms) and Table 3.3 (probabilistic algorithms).
Probabilistic methods, as shown in Table 3.3 , are preferably employed if the
dimensionality of the problem space is large, comprising a large number of
variables in the parameter vector or if a global optimum is sought in the presence
of several relative (local) optima. In addition, probabilistic methods are preferable,
along with zero-order methods, if the objective function is discontinuous, difficult
to differentiate or even non-differentiable at some point in the problem space. Also,
probabilistic methods are preferable if the functional relation between the vari-
ables and the objective function is not easily accessible. These methods do,
however, generally exhibit slow convergence rates. Due to economical aspects,
they are advantageously employed if objective function evaluation is not overly
time consuming.
A pure random choice of parameters, as well as a random initial set of start
parameters is not practicable in the present optimization process since parameters
employed in model functions, as introduced in this Chapter, Sect. 3.2.6.1 , must
fulfil certain restrictions and thus can not take an arbitrary value. If random ele-
ments are still to be included in the algorithm, a heuristic scheme must keep the
parameter choice in the prescribed valid bounds.
A global solution, however, even if desirable, is usually difficult to locate and to
identify. It is even challenging to determine whether the current solution represents
a local or a global optimum. Especially nonlinear problems may exhibit various
local solutions (multimodality) that are not global solutions. Here, at least some of
the objectives or the constraints are nonlinear functions of the variables x i , together
with a high dimensionality of the search space, where the objective function
depends on multiple variables. Thus, the search for a global solution may not be
feasible. This is often the case in practical engineering problems, since search time
may become exhaustive or the particular algorithm cannot prevent premature
convergence at local optima, as well as increase the probability of finding a global
optimum. In case of time limitation, a feasible approach therefore incorporates the
initial use of probabilistic methods to explore the region where the global optimum
is expected. Subsequently, a more efficient method can then be used for refined
optimum location.
3.4.3 Downhill Simplex Strategy
The issues discussed in Sects. 4.3 and 5.3 involve mechanical characterisation of
foam and tissue material. The underlying constitutive equations employed to
describe long- and short-term tissue and support material behaviour comprise
constants (material parameters) which need to be determined with respect to the
particular experimental scenarios. In this case, parameter optimization, as
described in more detail in Sect. 3.4.4 , is performed where the objective function
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