"Dreams are highly visual and often illogical in nature, which makes them ripe for the type of out-of-the-box thinking that some problem-solving requires," said Deirdre Barrett, a psychologist at Harvard University.
Maybe if academia built a sort of way to train people (as with the movie Inception) to problem solve while dreaming we may have a way for people to approach problems 24 hours a day, 7 days a week; part in the conscious world part in the subconscious world. If the logical structure and understanding of the tangible world were built within the conscious world perhaps the dreams would be more practical to the problem. Only recently have neuroscientists considered consiousness a realistic research topic. The greatest brainteaser in this field has been to explain how processes in the brain give rise to subjective experiences. This leads me to think about the useability of the conscious world in problem solving. What if we could create the context for which problem solving would be fully experienced? This context is also relative to the person solving the problem.
An "architect" as in the movie Inception would be able to build the world in which the dreamer would be able to analyze the problem. Maybe it is all relative to the problem, meaning that some problems will relate to the tangible world while other problems would relate to a general perception or even a perception within a paradigm. The subjectivity and fluidity of these perceptions would be the scope in which the problem would be solved. The depth needed and the amount of research required for each problem solving event would be immense, but the returns could solve some of the world's most complicating problems.