17,380 research outputs found
Local strong maximal monotonicity and full stability for parametric variational systems
The paper introduces and characterizes new notions of Lipschitzian and
H\"olderian full stability of solutions to general parametric variational
systems described via partial subdifferential and normal cone mappings acting
in Hilbert spaces. These notions, postulated certain quantitative properties of
single-valued localizations of solution maps, are closely related to local
strong maximal monotonicity of associated set-valued mappings. Based on
advanced tools of variational analysis and generalized differentiation, we
derive verifiable characterizations of the local strong maximal monotonicity
and full stability notions under consideration via some positive-definiteness
conditions involving second-order constructions of variational analysis. The
general results obtained are specified for important classes of variational
inequalities and variational conditions in both finite and infinite dimensions
A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks
Approximate solutions of the chemical master equation and the chemical
Fokker-Planck equation are an important tool in the analysis of biomolecular
reaction networks. Previous studies have highlighted a number of problems with
the moment-closure approach used to obtain such approximations, calling it an
ad-hoc method. In this article, we give a new variational derivation of
moment-closure equations which provides us with an intuitive understanding of
their properties and failure modes and allows us to correct some of these
problems. We use mixtures of product-Poisson distributions to obtain a flexible
parametric family which solves the commonly observed problem of divergences at
low system sizes. We also extend the recently introduced entropic matching
approach to arbitrary ansatz distributions and Markov processes, demonstrating
that it is a special case of variational moment closure. This provides us with
a particularly principled approximation method. Finally, we extend the above
approaches to cover the approximation of multi-time joint distributions,
resulting in a viable alternative to process-level approximations which are
often intractable.Comment: Minor changes and clarifications; corrected some typo
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