Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes. The aim is to approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold. We present a novel method for computing approximations of such manifolds using trajectory-based optimization. We discuss Riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics. The optimization criteria correspond to a suitable mathematical formulation of "minimal relaxation" of chemical forces along reaction trajectories under given constraints. We present various geometrically motivated criteria and the results of their application to three test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.Comment: 22 pages, 18 figure

On the Application of Trajectory-based Optimization for Nonlinear Kinetic Model Reduction

The use of increasingly detailed reaction mechanisms for the chemistry description in computational fluid dynamics simulations plays a major role in combustion research. Model reduction adresses the discrepancy between the need to develop detailed high-dimensional multi-scale models (e.g. in chemical kinetics) and the ineffciency of their use in computationally demanding numerical simulations. Many modern model reduction approaches are based on the approximation of slow invariant manifolds of lower dimension than the original system. Pursuing the work of Lebiedz on Minimum Entropy Production Trajectories (MEPT), this work presents a way to approximate slow invariant manifolds based on the optimization of trajectories. This work particularly covers the generalization of the MEPT concept for the construction of invariant manifolds of arbitrary dimension and guides the way for the construction of general relaxation criteria. Especially the minimization of curvature of trajectories - used for model reduction purposes for the first time within the scope of this work - leads to highly encouraging results, where the optimized trajectories are close to the slow invariant manifold. For the derivation of a reliable method for the construction of low-dimensional manifolds to be used in chemical kinetics, this work deals with advanced numerical methods, ideas from thermodynamics and differential geometry

Approximation of slow attracting manifolds in chemical kinetics by trajectory-based optimization approaches

Many common kinetic model reduction approaches are explicitly based on inherent multiple time scales and often assume and directly exploit a clear time scale separation into fast and slow reaction processes. They approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones. The corresponding restrictive assumption of full relaxation of fast modes often renders the resulting approximation of slow attracting manifolds inaccurate as a representation of the reduced model and makes the numerical solution of the nonlinear “reduction equations” particularly difficult in many cases where the gap in intrinsic time scales is not large enough. We demonstrate that trajectory optimization approaches can avoid such severe restrictions by computing numerical solutions that correspond to “maximally relaxed” dynamical modes in a suitable sense. We present a framework of trajectory-based optimization for model reduction in chemical kinetics and a general class of reduction criteria characterizing the relaxation of chemical forces along reaction trajectories. These criteria can be motivated geometrically exploiting ideas from differential geometry and fundamental physics and turn out to be highly successful in example applications. Within this framework, we provide results for the computational approximation of slow attracting low-dimensional manifolds in terms of families of optimal trajectories for a six-component hydrogen combustion mechanism

NaCl salinity affects lateral root development in Plantago maritima

Contains fulltext : 60375.pdf (publisher's version ) (Closed access

Tumors diagnosed as cerebellar glioblastoma comprise distinct molecular entities.

In this multi-institutional study we compiled a retrospective cohort of 86 posterior fossa tumors having received the diagnosis of cerebellar glioblastoma (cGBM). All tumors were reviewed histologically and subjected to array-based methylation analysis followed by algorithm-based classification into distinct methylation classes (MCs). The single MC containing the largest proportion of 25 tumors diagnosed as cGBM was MC anaplastic astrocytoma with piloid features representing a recently-described molecular tumor entity not yet included in the WHO Classification of Tumours of the Central Nervous System (WHO classification). Twenty-nine tumors molecularly corresponded to either of 6 methylation subclasses subsumed in the MC family GBM IDH wildtype. Further we identified 6 tumors belonging to the MC diffuse midline glioma H3 K27 M mutant and 6 tumors allotted to the MC IDH mutant glioma subclass astrocytoma. Two tumors were classified as MC pilocytic astrocytoma of the posterior fossa, one as MC CNS high grade neuroepithelial tumor with BCOR alteration and one as MC control tissue, inflammatory tumor microenvironment. The methylation profiles of 16 tumors could not clearly be assigned to one distinct MC. In comparison to supratentorial localization, the MC GBM IDH wildtype subclass midline was overrepresented, whereas the MCs GBM IDH wildtype subclass mesenchymal and subclass RTK II were underrepresented in the cerebellum. Based on the integration of molecular and histological findings all tumors received an integrated diagnosis in line with the WHO classification 2016. In conclusion, cGBM does not represent a molecularly uniform tumor entity, but rather comprises different brain tumor entities with diverse prognosis and therapeutic options. Distinction of these molecular tumor classes requires molecular analysis. More than 30% of tumors diagnosed as cGBM belong to the recently described molecular entity of anaplastic astrocytoma with piloid features