4,315 research outputs found
Quasi regular Dirichlet forms and the stochastic quantization problem
After recalling basic features of the theory of symmetric quasi regular
Dirichlet forms we show how by applying it to the stochastic quantization
equation, with Gaussian space-time noise, one obtains weak solutions in a large
invariant set. Subsequently, we discuss non symmetric quasi regular Dirichlet
forms and show in particular by two simple examples in infinite dimensions that
infinitesimal invariance, does not imply global invariance. We also present a
simple example of non-Markov uniqueness in infinite dimensions
A partitioned model order reduction approach to rationalise computational expenses in multiscale fracture mechanics
We propose in this paper an adaptive reduced order modelling technique based
on domain partitioning for parametric problems of fracture. We show that
coupling domain decomposition and projection-based model order reduction
permits to focus the numerical effort where it is most needed: around the zones
where damage propagates. No \textit{a priori} knowledge of the damage pattern
is required, the extraction of the corresponding spatial regions being based
solely on algebra. The efficiency of the proposed approach is demonstrated
numerically with an example relevant to engineering fracture.Comment: Submitted for publication in CMAM
An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals: Continuum and Discrete Analysis
This paper outlines an energy-minimization finite-element approach to the
computational modeling of equilibrium configurations for nematic liquid
crystals under free elastic effects. The method targets minimization of the
system free energy based on the Frank-Oseen free-energy model. Solutions to the
intermediate discretized free elastic linearizations are shown to exist
generally and are unique under certain assumptions. This requires proving
continuity, coercivity, and weak coercivity for the accompanying appropriate
bilinear forms within a mixed finite-element framework. Error analysis
demonstrates that the method constitutes a convergent scheme. Numerical
experiments are performed for problems with a range of physical parameters as
well as simple and patterned boundary conditions. The resulting algorithm
accurately handles heterogeneous constant coefficients and effectively resolves
configurations resulting from complicated boundary conditions relevant in
ongoing research.Comment: 31 pages, 3 figures, 3 table
Bayesian Inference Application
In this chapter, we were introduced the concept of Bayesian inference and application to the real world problems such as game theory (Bayesian Game) etc. This chapter was organized as follows. In Sections 2 and 3, we present Model-based Bayesian inference and the components of Bayesian inference, respectively. The last section contains some applications of Bayesian inference
Scaling limit of stochastic dynamics in classical continuous systems
We investigate a scaling limit of gradient stochastic dynamics associated to
Gibbs states in classical continuous systems on . The
aim is to derive macroscopic quantities from a given micro- or mesoscopic
system. The scaling we consider has been investigated in \cite{Br80},
\cite{Ro81}, \cite{Sp86}, and \cite{GP86}, under the assumption that the
underlying potential is in and positive. We prove that the Dirichlet
forms of the scaled stochastic dynamics converge on a core of functions to the
Dirichlet form of a generalized Ornstein--Uhlenbeck process. The proof is based
on the analysis and geometry on the configuration space which was developed in
\cite{AKR98a}, \cite{AKR98b}, and works for general Gibbs measures of Ruelle
type. Hence, the underlying potential may have a singularity at the origin,
only has to be bounded from below, and may not be compactly supported.
Therefore, singular interactions of physical interest are covered, as e.g. the
one given by the Lennard--Jones potential, which is studied in the theory of
fluids. Furthermore, using the Lyons--Zheng decomposition we give a simple
proof for the tightness of the scaled processes. We also prove that the
corresponding generators, however, do not converge in the -sense. This
settles a conjecture formulated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}
Correlation functions from a unified variational principle: trial Lie groups
Time-dependent expectation values and correlation functions for many-body
quantum systems are evaluated by means of a unified variational principle. It
optimizes a generating functional depending on sources associated with the
observables of interest Comment: 42 page
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