Contributions to the solution of large nonlinear systems via model-order reduction and interval constraint solving techniques
Many engineering problems boil down to solving partial differential equations (PDEs) that describe real-life phenomena. Nevertheless, efficiently and reliably solving such problems constitutes a major challenge in computational sciences and in engineering in general. PDE-based systems can reach sizes so large after they are discretized. The large size in these problems generate several issues, among them we can mention: large space of storing, computing time, and the most important, lost of accuracy. A popular approach to solving such problems is assume that the PDE's solution is in a subspace, and the solution is sought there. This assumption and later searching is named Model-Order Reduction (MOR). As we have mentioned before, MOR aims at reducing the size of the original large problem by projecting it onto a subspace. The quality of MOR is highly dependent on the right choice of the projection. Assuming that the projection is relevant, i.e. the behavior of the projected system reproduces that of the original system, the projected smaller system is solved much more easily and its solution is ``uncompressed'' into the solution of the original system. Identifying effective projections is still an open problem. Among existing popular approaches are Krylov methods, Proper Orthogonal Decomposition (POD), and Wavelets. None of these methods is perfect, each with has pros and cons. Once the original problem is reduced, challenges still exist: nonlinear problems, solving it require going back and forth between compressed and original problems in order to solve them, which is time and space consuming. So there is a need for clever approaches to optimization that allow avoiding these pitfalls. In any case, we can consider the following added challenges: (1) No current popular technique is global, which means that there is no evidence that the solutions we obtain are the best. (2) None of these techniques take into account additional constraints, such as contact constraints, that should be satisfied by resulting solutions. (3) Uncertainty is not taken into account and thus the solution that are obtained may not be robust enough to possible variations in the input or the model. The task is to address each of the above-mentioned challenges, which all tend to achieving the same goal: efficiently and reliably solving large nonlinear systems. To do so, we propose (i) to further study using wavelets for projection / compression as well as to explore the design of cost-efficient snapshot-based approaches that require minimal expert knowledge, (ii) to employ intervals and constraint-solving techniques for directly address the nonlinear systems, and (iii) to analize regularization-based approaches for optimization to ensure faster convergence and less back-and-forth computations between smaller and larger systems.
Valera, Leobardo, "Contributions to the solution of large nonlinear systems via model-order reduction and interval constraint solving techniques" (2015). ETD Collection for University of Texas, El Paso. AAI10000776.