We know that the most of VAT problems published and studied up to these days have been concerned with the Bottom-Up predicaments. That means the Lower Scale physical and mathematical statement being formulated as for Homogeneous matter physics problem and then thought the Upper Scale VAT statement and its solution. Though, there is the another direction of scale communications.
The problems mostly are, at least for technology related physical tasks, of the opposite nature - when the Upper Scale is evident and easy to formulate, but as a Homogeneous medium physical problem. At the same time it is known that the medium actually is the Heterogeneous one. And the Upper Scale should be formulated as the VAT statement. Then, we have for the Upper Scale the two statements - the legitimate VAT Upper Scale mathematical statement, and the Homogeneous matter statement. The latter one is the fictitious statement, the statement for which we know or need to know its physical characteristics to make it justifiable to some extent. Or to use these characteristics for another task.
The second main goal is to find out the Lower Scale characteristics for other or the same reasons. This constitutes an inverse physical and mathematical problem. In these days inverse problems mostly known and stated, and studied for the homogeneous media. The VAT (heterogeneous) inverse transport problems are the subject of some level of speculations right now. There are known attempts to approach this area with the methods developed for homogeneous media inverse transport theories.
The closure task for the Top-Down VAT statement unequivocally demands the exact mathematical dependence of the lower scale properties to the Upper scale Would be Terms and properties in the VAT governing equations. So, the mission becomes more strict in its determination.
Anyway, in this mathematical arrangement we have the Top-Down VAT statement problem. So far, it is known the only published study of this kind -
Travkin, V.S., Catton, I., Ponomarenko, A.T., and Kalinin, Yu.E., ''Bottom Up and Top Down, from Nano-Scale to Micro-Scale, Hierarchical Descriptions of Electrodynamic, Thermal and Magnetic Fields in Ferromagnets and HTSCs'', in Proc. DOE 20th Symposium on Energy Engineering Sciences, Argonne National Laboratory, pp. 296-304, (2002).
We would stop short here to further describe and investigate in this subsection the issues related to the Closure and the Top-Down VAT heterogeneous physics statements and techniques. For those, who found that their application belongs in the heterogeneous class of problems, we invite to Contact us