This greatly complicates the task of fitting models to data. Indeed, the problems of fitting model parameters inhibit the creation of high-fidelity models. As described earlier, increasing model resolution to include smaller scales in a problem may increase the number of parameters that must be determined to the point that the computations to do so are not feasible. Bifurcation theory provides a framework for direct determination of information about how system behavior changes qualitatively with parameter variations. From my perspective as a dynamicist, limitations of computer simulation technology are evident.
Similar shortcomings occur in many other areas of numerical computation.
Let me cite an example that illustrates how even algorithms for classical problems need improvements. The mathematical problem is to relate the spectrum of eigenvalues for the Laplace operator with Dirichlet boundary conditions to the geometry of a planar domain. Classical results give formulas for geometric quantities such as the area of the domain and the length of its boundary in terms of the spectrum. Gordon, Webb, and Wolpert gave the first examples of noncongruent planar domains with precisely the same spectrum. Examples include polygonal domains with fewer than a dozen edges.