VSGC Intracranial Pressure Modeling Project is an interdisciplinary research group of applied mathematicians and neurosurgeons which is coordinated by Professor William D. Lakin, Director of the VSGC and Chairman of the Department of Mathematics and Statistics at the University of Vermont. Prof. Lakin and colleagues at IBM are currently exploring the application of hybrid solution techniques to systems of differential equations in models which simulate semiconductor device performance. The new hybrid methods, which involve a consistent blending of multi scale asymptotic and numerical methods, were originally developed in the context of solving mathematical models for human intracranial pressure dynamics.  A related project ( Effects of Cerebrovascular Autoregulation in Models of Intracranial Pressure Dynamics) involves consistent introduction of autoregulation into compartmental models of the human intracranial system.

The intracranial pressure models contain terms capable of adjustment on highly disparate time scales, leading to severely stiff differential equations which are impractical to solve using the commonly applied numerical techniques. The new hybrid method resolves difficulties inherent in solving such equations by producing a dynamics decoupling of processes on the various time scales. The hybrid technique is then able to consistently replace the original stiff system by a reduced non stiff system complemented by conservation laws, and evolution equations for the longtime behavior of the solution can then be derived. The associated numerics involve an asymptotic-numerical matching procedure which smoothly connects solutions valid on the various time scales. Mathematical models which simulate semiconductor devices have many features in common with intracranial pressure models, particularly the cascade of disparate scales for significant changes in the evolving solution. Further, the semiconductor problems, when discretized, involve very large systems with very sparse matrices. It would therefore appear that successfully transferring the hybrid asymptotic-numerical method into the semiconductor modeling context could produce a tremendous reduction in the required computational effort and make such modeling a true design tool for future device development.

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