The proposed research emphasizes that dramatic advances in computation must be driven by a fundamental understanding of the theory behind numerical methods. One of the main innovations in this proposal is the development of central schemes. While central schemes are considered to be among the most powerful numerical methods for time dependent PDEs, in particular for solving transport equations, there are still several fundamental issues to be resolved if they are to reach their full potential. This proposal also concentrates on the investigation of the power of anisotropy of level set methods, the implementation of implicit methods, and the power of multi-scale techniques in data assimilation.
Most real world problems are solved by computers. Many such problems are so complicated that the existing computational methods are insufficient to provide the accuracy needed. This situation cannot be solved by simply building faster computers. Indeed, the quest for finer resolution and more accurate models far outstrips gains in computational power. Moreover, as has been demonstrated often in the past, innovations in computer algorithms and software pay larger dividends than advances in computational power. The research in this project seeks to make innovative advances in the mathematics behind computer programs (so-called numerical algorithms) which will have the effect of significantly speeding up computation and thereby solving many scientific and engineering problems facing this country such as tracking pollutants in coastal areas or developing fast and accurate sensors for medical imaging.
