Proyectos

Transferencia y adopción de Tecnologías para la Gestión de Riesgo en el Proceso Productivo de la Cereza: hacia una agricultura de precisión para la Región de O’Higgins

Objetivo general: sesarrollar un modelo de balance hídrico en acuífero de roca fracturada. Objetivos específicos: (i) identificar un sitio piloto de roca fracturada; (ii) instrumentar el piloto para monitorear los flujos hídricos; (iii) diseñar e implementar la metodología de balance hídrico en acuíferos de roca fracturada.

Stochasticity in optimization and game theory is a very important aspect to model more accurately real-world problems in many different areas (see for instance [6]). In optimization problems as well as in one-level games, namely, Nash equilibrium problems, stochasticity aspects have received quite a lot of attention for a while and have also been well studied [22]. However, for the branch of bilevel games quite few studies have included in their analysis stochastic aspects in their models. A bilevel game is basically to split a finite set of players into two levels: the leaders or upper-level players, and the followers or lower-level players. In the model, the followers react in a passive way to the leaders' actions, while the leaders compete in the upper level trying to actively anticipate the followers' reaction. Moreover, in each level, the interaction is non-cooperative as in Nash equilibrium problems. Bilevel games have been recognized as one of the most complex and at the same time very useful models in the literature [17]. Bilevel games, and more precisely the problem of the leaders in a bilevel game, face an ambiguity/uncertainty whenever the followers' reaction is not necessarily uniquely determined for each leaders' decision. To deal with this ambiguity two main approaches are well-known the optimistic and the pessimistic. The weakness of these two approaches is that both are quite extreme and the optimistic one lacks of real modeling foundations, putting the leaders in a quite naive position. Recently, in [9] a general stochastic approach has been proposed to solve this ambiguity, which is seen as an uncertainty of the problem, providing also a specific approach that seems to be more reasonable than the optimistic one, from a modeling point of view. The stochasticity in the stochastic approach is an endogenous one since it corresponds to a decision-dependent uncertainty [1, 23]. But, of course, stochasticity might also come from an exogenous side, that is, when some of the parameters defining the game, such as future demand and prices, 1 forecasts of winds and clouds, are uncertain and possibly follow some probability distribution. This has been considered in [34, 12, 13, 16]. In the second part of the project, which is the applied part, we are interested in using the developed theoretical framework of bilevel games with stochastic aspects to a problem of contaminated water resource management, which has high levels of stochasticity. The scarcity of water resource and its efficient use has been recognized as an extremely important problem in Chile and the whole world, for agricultural, industrial, and human use. Moreover, after any use, there is an outflow of water which has generally more contaminants than the inflow. Depending on the type and quantity of contaminants the outflow of water could be reused, but sometimes giving less profit to the entity. The general situation is full of uncertainties, since the entities do not share their information. Moreover, the main source of information for us will be measurements on the quality of water at different strategic points and punctual events of contamination registered by inspection, which is simply a qualitative data. Therefore, we propose first to apply predictive models and machine learning to do inverse engineering in order to understand the game played by the different actors. Then we want to study the underlying one-level game and study the design of mechanisms (bilevel game) so that we can move the equilibrium to a desired goal. In a somehow similar spirit, in [30, 31, 10] game theory techniques have been used to analyze the behavior of companies sharing contaminated water in the context of eco-industrial parks, while in [36] also a bilevel model is used for a water resource optimal allocation problem.

Transferencia y adopción de Tecnologías para la Gestión de Riesgo en el Proceso Productivo de la Cereza: hacia una agricultura de precisión para la Región de O’Higgins

Transferencia y adopción de Tecnologías para la Gestión de Riesgo en el Proceso Productivo de la Cereza: hacia una agricultura de precisión para la Región de O’Higgins

Las ondas y las estructuras coherentes están presentes como entidades individuales en varios contextos físicos, astronómicos y geofísicos, y particularmente en los fluidos. Las situaciones realistas generalmente involucran a ambos, lo que lleva a procesos de interacción complejos que son difíciles de separar y desenredar. En esta propuesta nos enfocamos en estudiar experimentalmente cómo la presencia de estructuras coherentes, tales como vórtices o singularidades, afectan las propiedades de las ondas superficiales en un régimen turbulento. Para ello, construiremos dos montajes experimentales para poder estudiar de forma sistemática las propiedades estadísticas de las ondas cuando interactúan con las estructuras mencionadas. Ambos sistemas tienen la ventaja de que, ajustando los parámetros de forzamiento, podemos controlar la aparición e intensidad de las estructuras. Por lo tanto, un estudio sistemático de su influencia en turbulencia de ondas (WT) es sencillo. La naturaleza intermitente del campo de ondas, así como los mecanismos detrás de la ruptura del espectro de WT en presencia de estas estructuras son algunas de las preguntas que pretendemos responder. Para abordar estas preguntas, proponemos realizar mediciones espaciotemporales, como la fotografía de luz difusa (DLP) y la velocimetría de imagen de partículas (PIV). Los resultados que surjan de esta investigación podrían ser de gran importancia para una teoría que, si bien es válida en muchos sistemas, aún está incompleta. Las aplicaciones de los resultados a otros sistemas, como los flujos geofísicos, también podrían ser posibles y bastante relevantes para una amplia comunidad.

Large systems which can be described in terms of mean-field equations provide a general framework for modeling complex systems where the understanding of few collective modes cannot explain long term behavior. They frequently arise in biological applications, in particular in theoretical neuroscience. The study of mean field equations in theoretical neuroscience started long ago with integrate-and-fire systems and has been a driving force in the development of new tools and techniques in functional analysis, numerics, stochastic systems and partial differential equations. This project is placed in these mathematical fields and aims to study, for a particular kinetic equation modeling a multi-population network of FitzHugh- Nagumo interconnected neurons with chemical synapses three fundamental aspects: (1) Consistency between finite particle system and the respective limit mean field equation; (2)Qualitative properties such as existence, stability, and exponential convergence under different parameters regime; and (3) Convergence of the solutions to Dirac masses for strongly interconnectivity parameters. The particle system corresponds to a fully connected FHN neurons with synapses given by a Destexhe- Sejnowski gating model. Moreover, the channels are noisy and modeled as reflected processes. The mean field model consists of two coupled non-linear partial differential equations with unbounded coefficients, where the right-hand side includes the first moment of the unknowns. They correspond to the law of finding a neuron of each subpopulation in a particular state. The equation is obtained via the Fokker-Planck formulation, and therefore is related to a mean-field stochastic process. In this proposal, we describe our plans to get new results on consistency, existence of solutions and their stability for the aforementioned class of models. From a transdiscliplinary viewpoint, the main novelty is that in certain parameter regimes, numerical simulations on the finite system indicates that our model converges towards Dirac masses with support on a manifold where the inhibition and excitation balance condition hold true. This approach provides a novel interpretation of the balance in brain and requires the development and adaptation of new mathematical tools. Moreover, under external excitatory input, the system comes back - in a faster time scale - towards the E/I balance condition, showing high excitation activity followed by inhibition. This results is in line with important functional properties of inhibition in the brain, particularly in auditory and somatosensory cortex. This question raises a number of new mathematical problems that this proposal aims at addressing. The methods that we propose combine techniques from several fields such as stochastic processes, partial differential equations, functional analysis, viscosity solutions and numerics.