The Haunted Swamps of Heuristics: Uncertainty in Problem Solving - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 53

Fig. 5.5 Close-up of a Julia set ( a ) and the Mandelbrot set as parameter map of all Julia sets ( b )
(courtesy of Falconer [ 8 ] © John Wiley and Sons)

same time. Dynamical systems are an illustrative example where parameter-space
analysis has already been applied for a long time. Julia sets (Fig. 5.5 a) are the result
of iterating a simple quadratic polynomial in the complex plane. Each polynomial is
characterized by a parameter p . Different parameters lead to greatly varying results
where the outcome may for example be a connected or disconnected Julia set. Doing
an analysis of all possible parameters leads to a parameter space display where the
beautiful and immensely intricate Mandelbrot set appears (Fig. 5.5 b). The parameter
p of the Julia sets turns into a variable in the parameter space where the Mandelbrot
set resides. As a side note: the Mandelbrot sets comprises all those values p whose
corresponding Julia sets are connected. With dynamical systems a parameter-space
analysis may be local or global. A local investigation looks at small perturbations
of a parameter to identify for example stability properties which could be direction
dependent. A global investigation looks at larger structures in parameter space, e.g.,
asymptotic behavior, basins of attraction, bifurcations or topological items like sep-
aratrices. In the visualization domain parameter-space analyses become feasible as
well for data ensembles and parameterized simulation runs. Parameter-space inves-
tigations from other fields might act as guiding examples, though the peculiarities of
our applications have to be taken into account. While for dynamical systems parame-
ters often change continuously, in our applications parameters may be for example
discontinuous, discrete, or categorical in nature. Certain regions of parameter space
may be uninteresting or even meaningless because of the physical properties of the
underlying phenomenon. With the holistic view on large ensembles or simulation
runs interesting questions arise:

￿ What is the local stability of a (visualization) parameter setting?

￿ How do different parameters influence each other?

￿ What are permissible (visualization) parameter ranges?

￿ How can we automatically define parameter settings that optimize certain proper-

ties?

￿ How sensitive is the visualization outcome on parameter perturbations?

￿ How to efficiently sample the high dimensional parameter spaces?

￿

How to do reconstruction in these parameter spaces?