Broadly speaking, science progresses steadily with the study of single length scales. Complex phenomena are reduced to mathematical equations, in a tradition dating back to Issac Newton, and after several centuries with progress on linear problems, the modern computer opened the door in a systematic manner to nonlinear problems. This is the insight of von Neumann. However, the methodology breaks when multiple length scales interact within a single problem. If the scales are widely separated, then they can often be treated separately, as isolated effects, with for example the conclusions of the study of the smaller length scales as generating parameters and coefficients which enter into the solution of the equations at the larger scales.
This insight was the basis of an article [ GliSha97], which introduced the notion of multiscale science into the culture and discourse of science. As a conceptual framework, the idea was wildly successful, to the extent that entire journals and conference series refer to this idea in their titles. The article also spawned a large research program. In spite of of this large effort, the frontier has not advanced dramatically. Perhaps most fundamentally, the slowness of the progress is related to the difficulty of the problem. The tools which have been most useful for single scale science, mathematical formalisms, linear theories and numerical methods tend to fit multiscale problems rather poorly.
The more intractable problems are strongly coupled, making theory difficult and linear analysis ill adapted. But also the computer is not naturally adapted to this problem. Each increased power of 10 in numerical resolution requires as a minimum an increase of effort of 10,000 for space time resolution, and usually more. According to Moore's law, with computer power doubling every 1.5 years, it takes a minimum of 45 years to achieve this additional factor of 10 in resolution. While the end of Moore's law has been falsely predicted many times, we should note that the fraction of the national budget needed for such leading computers is not a constant, and increases as well. So it seems plausible that a combination of technical and economic factors will intervene, and even waiting 45 years will not give the answers in a straightforward manner. Moreover, it is often the case that the missing length scales are not one factor of ten but require multiple such factors of improved resolution.
Of course, there remains the possibility of human cleverness, which has always been the wellspring of scientific progress.
The Institute for Multiscale Science is dedicated to solutions to multiscale problems. We prefer to find the solutions "in place", that is in the context of real problems, rather than as a broad and systematic theory. We believe this route to be more promising, with the general theories to follow after the special cases have been resolved. All present methods (mathematical analysis, linear and nonlinear theoretical and numerical methods) are welcome, but may not be sufficient, absent new developments.