PDEs with Compressed Solutions

Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an $L^1$ norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by minimization (to obtain an elliptic PDE) or by gradient flow (to obtain a parabolic PDE). Also, we show that some PDEs can be rewritten in an $L^1$ form, such as the divisible sandpile problem and signum-Gordon. Addition of an $L^1$ term in the variational principle leads to a modified PDE where a subgradient term appears. It is known that modified PDEs of this form will often have solutions with compact support, which corresponds to the discrete solution being sparse. We show that this is advantageous numerically through the use of efficient algorithms for solving $L^1$ based problems.Comment: 21 pages, 15 figure

Robotic load balancing for mobility-on-demand systems

In this paper we develop methods for maximizing the throughput of a mobility-on-demand urban transportation system. We consider a finite group of shared vehicles, located at a set of stations. Users arrive at the stations, pickup vehicles, and drive (or are driven) to their destination station where they drop-off the vehicle. When some origins and destinations are more popular than others, the system will inevitably become out of balance: vehicles will build up at some stations, and become depleted at others. We propose a robotic solution to this rebalancing problem that involves empty robotic vehicles autonomously driving between stations. Specifically, we utilize a fluid model for the customers and vehicles in the system. Then, we develop a rebalancing policy that lets every station reach an equilibrium in which there are excess vehicles and no waiting customers and that minimizes the number of robotic vehicles performing rebalancing trips. We show that the optimal rebalancing policy can be found as the solution to a linear program. We use this solution to develop a real-time rebalancing policy which can operate in highly variable environments. Finally, we verify policy performance in a simulated mobility-on-demand environment and in hardware experiments.Singapore-MIT Alliance for Research and Technology CenterUnited States. Office of Naval Research (Grant N000140911051)National Science Foundation (U.S.) (Grant EFRI0735953

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is changing in time, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, material science, or biological systems.The emerging mathematical models account for various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. The focus in this workshop was on novel techniques and ideas in the mathematical modelling, analysis, the numerical discretization and the upscaling of problems as described above

Design, Modeling, and Measurement of a Metamaterial Electromagnetic Field Concentrator

This document addresses the need to improve the design process for creating an optimized metamaterial. In particular, two challenges are addressed: creating an electromagnetic concentrator and optimizing the design of metamaterial used to create the electromagnetic concentrator. The first challenge is addressed by developing an electromagnetic field concentrator from a design of concentric geometric shapes. The material forming the concentrator is derived from the application of transformation optics. The resulting anisotropic, spatially variant constitutive parameter tensors are then approximated with metamatieral inclusions using the combination of an AFIT rapid metamaterial design process and a design process created for rapid metamaterial production. The second challenge of optimizing the design of the metamaterial is addressed by considering factors such as circuit board selection, various sets of metamaterial cell geometry combinations, and optimization of the ratio of the widths for the concentric geometric shapes. The resulting optimized design is simulated and shown to compress and concentrate the vertical electric field component of incident plane waves. A physical device is constructed based on the simulations and tested to confirm the entire design process. Experimental data do not definitely show concentration however an optimized design process has been proven

Reactive Flow and Transport Through Complex Systems

The meeting focused on mathematical aspects of reactive flow, diffusion and transport through complex systems. The research interest of the participants varied from physical modeling using PDEs, mathematical modeling using upscaling and homogenization, numerical analysis of PDEs describing reactive transport, PDEs from fluid mechanics, computational methods for random media and computational multiscale methods

Dynamical Systems; Proceedings of an IIASA Workshop, Sopron, Hungary, September 9-13, 1985

The investigation of special topics in systems dynamics -- uncertain dynamic processes, viability theory, nonlinear dynamics in models for biomathematics, inverse problems in control systems theory -- has become a major issue at the System and Decision Sciences Research Program of IIASA. The above topics actually reflect two different perspectives in the investigation of dynamic processes. The first, motivated by control theory, is concerned with the properties of dynamic systems that are stable under variations in the systems' parameters. This allows us to specify classes of dynamic systems for which it is possible to construct and control a whole "tube" of trajectories assigned to a system with uncertain parameters and to resolve some inverse problems of control theory within numerically stable solution schemes. The second perspective is to investigate generic properties of dynamic systems that are due to nonlinearity (as bifurcations theory, chaotic behavior, stability properties, and related problems in the qualitative theory of differential systems). Special stress is given to the applications of nonlinear dynamic systems theory to biomathematics and ecology. The proceedings of a workshop on the "Mathematics of Dynamic Processes", dealing with these topics is presented in this volume

Modélisation et analyse de variations paramétriques d'un système mécatronique par l'inclusion différentielle

Pour étudier les incertitudes paramétriques liées au tolérancement des pièces de production on a utilisé l'approche des inclusions différentielles. Cette méthode donne en résultat toutes les valeurs possibles que peut atteindre le système en variant le paramètre dans son intervalle de tolérancement et en évoluant au cour du temps, le résultat est « le domaine atteignable ». La mise en uvre de la méthode nécessite une application qui montre ses avantages et ses limites. Pour cela on a utilisé un système mécatronique et on a évalué les résultats par simulation sous Mathematica

Conservation based uncertainty propagation in dynamic systems

Uncertainty is present in our everyday decision making process as well as our understanding of the structure of the universe. As a result an intense and mathematically rigorous study of how uncertainty propagates in the dynamic systems present in our lives is warranted and arguably necessary. In this thesis we examine existing methods for uncertainty propagation in dynamic systems and present the results of a literature survey that justifies the development of a conservation based method of uncertainty propagation. Conservation methods are physics based and physics drives our understanding of the physical world. Thus, it makes perfect sense to formulate an understanding of uncertainty propagation in terms of one of the fundamental concepts in physics: conservation. We develop that theory for a small group of dynamic systems which are fundamental. They include ordinary differential equations, finite difference equations, differential inclusions and inequalities, stochastic differential equations, and Markov chains. The study presented considers uncertainty propagation from the initial condition where the initial condition is given as a prior distribution defined within a probability structure. This probability structure is preserved in the sense of measure. The results of this study are the first steps into a generalized theory for uncertainty propagation using conservation laws. In addition, it is hoped that the results can be used in applications such as robust control design for everything from transportation systems to financial markets