Introduction:
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The dimension scale of problems arising in our modern information society became very large. A new area of science and engineering is now urgently needed in order to extract and interpret significant information from the universe of data collected from a variety of modern sources (Internet, physics experiments, medical diagnostics, etc.). Numerical simulations at the required scale will be one of the great challenges of the 21st century. In short, we need to become capable of organizing and understanding complexity. The most notable recent advances in data analysis and numerical simulation are based on the observation that in several situations, even for very complex phenomena, only a few governing components/degrees of freedom are required to describe the whole dynamics; a dimensionality reduction can be achieved by demanding that the solution be "sparse" or "compressible". Since the relevant degrees of freedom are not prescribed, and may depend on the particular solution, we need efficient optimization methods for solving the hard combinatorial problem of identifying them. Within this Special Session we are first addressing results in designing efficient algorithms which allow us to achieve sparse optimization in high-dimension. Secondly, such tools developed for achieving adaptive dimensionality reductions are used as building blocks for solving large-scale optimal control problems in dynamical systems, partial differential equations, and variational problems arising in various contexts. The novelty of the results presented in our session is precisely the combination of sparsity promoting optimizations, adaptive discretizations, and their use and application to model parsimonius control of complex dynamics. The search for the minimal amount of degrees of freedom to allow the control of systems opens as well interesting connections with information based complexity and information theory. The interest in insisting in combining optimal control of dynamical systems and PDEs is due to their intimate relationship, e.g., by means of discretization processes and finite dimensional approximations. One relevant scope of this session is to further emphasize and explore the relationship between finite and infinite dimensional optimal control problems, with special emphasis on sparse control.
Finally, we will be interested in innovative applications in image processing, social-dynamics control, such as guided consensus models and controlled pattern formation, automatic learning and observability of dynamical systems and parameter identification in PDEs via sparse control. Moreover, sparsity based formulations allow for elegant approaches to attack problems of optimal sensor/actuator placement in the context of optimal control of PDEs as well as of point source reconstruction and related inverse problems with PDEs. |
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