Collaborative Research: Numerical Methods for Fully and Implicitly Nonlinear Equations

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The main goal of this project is to further investigate the numerical solution of fully nonlinear elliptic equations such as Monge-Ampère?s, in order to extend previous work done with the support of NSF Grant DMS-0412267. The main findings of these previous investigations are that, after regularization of the data (which is not always necessary), well-chosen least-squares formulations in appropriate Hilbert spaces lead to robust solution methods able to compute classical solutions, or generalized ones if classical solutions do not exist. The objectives of the present project are: (i) To improve the performances of the iterative methods used to solve the least-squares problems. This will require the development of novel algorithms to solve the many (one per grid point) low dimensional nonlinear eigenvalue problems obtained from the decomposition of the least squares problems, since this results in a number of small but intricate constrained eigenvalue problems. (ii) To demonstrate the effectiveness of these new algorithms on a variety of test problems (Monge-Ampère, Pucci, Gaussian curvature, sigma-2, in dimension 2, 3, and even 4 for the Pucci problem, using parallelization). (iii) To determine whether the remarkable homogenization properties observed in two-dimensions for some of these fully nonlinear elliptic equations when a coefficient in the operator varies periodically or randomly in space persist in higher dimensions. (iv) To apply, ultimately, the above methodology (or close variant of it) to the solution of some implicitly nonlinear partial differential equations from non-smooth differential geometry that model folding phenomena. What motivates these investigations is the fact that fully nonlinear elliptic equations play an important role in areas as diverse as material sciences, nonlinear elasticity, fluid mechanics, atmospheric sciences, nonlinear elasticity, shape design in electrical and structural engineering (antennas, car shape,?), finance, applied and theoretical physics, differential geometry and others. The related mathematical problems have generated a large literature. In contrast these problems have the reputation to be difficult from a computational standpoint explaining why the computational and applied mathematicians have not made significant progress on their numerical solution. One of the goals of this project is to close the gap between the various communities concerned with fully nonlinear elliptic equations so that each of them will learn from the others, setting an example of interdisciplinary science. Such an effort will also benefit science and engineering professionals and students, via publications, dedicated web sites and post-graduate courses, lectures at conferences, and of course direct involvement for some graduate students. It will also stimulate the contributions of other scientists to these important areas. Since computational methods developed previously by the Principal Investigators and their associates are currently used in many areas of Science and Engineering, Academia and Industry, one can expect a similar endeavor for the results and products originating from this collaborative project.