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Model order reduction methods for chemical kinetics in combustion and biology

In thermodynamics, gas dynamics, chemical kinetics, ecology, biology and other sciences, it is increasingly important to develop dynamical models able to describe the time evolution of the non-equilibrium states of the system of interest. Often the detailed kinetic model (DKM) of the system consists of a large number of differential equations for all state variables, coupled to an even greater number of non-linear equations that link the kinetic constants of each of the different mechanisms in place to the state variables. When the DKM must be coupled to the equations of the transport of mass, momentum and energy, as well as models of turbulence or capillarity or interface, the computational costs often become formidable, also for the intrinsic presence of a wide range of characteristic times and lengths . In chemical kinetics, these difficulties have motivated the development of several techniques of model order reduction. Over the years we contributed to the advancement of the method known as Rate Controlled Constrained Equilibrium in collaboration with the pioneer of the method, J.C. Keck, and with colleagues of the Northeastern University. We are currently working on the development of a mathematical algorithm for automatic constraint identification. More illustrative details available here. The use of the method for kinetic schemes in biology has not yet been addressed, but it is the natural application in the field of health technologies that constitute a strategic line of research in the Department.

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Formulation and application of Steepest-Entropy-Ascent models for non-equilibrium systems

In most of the existing theories of non-equilibrium, the component of the time evolution of the mathematical object representing the local state of the system that accounts for its dissipative behavior pushes the state to evolve along the direction of Steepest Entropy Ascent (SEA) in state space with respect to a suitable metric that physically represents the coupled generalized conductivities of the system characterizing the strength of the system’s spontaneous tendency towards equilibrium. This is true in the near-equilibrium limit, where in most non-equilibrium frameworks it is possible to show that the traditional assumption of linear relaxation coincides with the SEA principle. The SEA principle, however, is valid also in the far-non-equilibrium regime. The family of SEA dynamical models has the fundamental feature of a strong built-in consistency with the second law of thermodynamics which follows from the non-negativity of the local entropy production density and holds regardless of the details of the underlying metric tensor. In a variety of fields of application, the SEA principle provides a unifying modeling approach that we hope to show will provide a new basis for effective numerical and theoretical models of irreversible, conservative relaxation towards equilibrium from far non-equilibrium states, as well as for a systematic approach to model order reduction techniques.

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