88,726 research outputs found
Algorithmic Statistics
While Kolmogorov complexity is the accepted absolute measure of information
content of an individual finite object, a similarly absolute notion is needed
for the relation between an individual data sample and an individual model
summarizing the information in the data, for example, a finite set (or
probability distribution) where the data sample typically came from. The
statistical theory based on such relations between individual objects can be
called algorithmic statistics, in contrast to classical statistical theory that
deals with relations between probabilistic ensembles. We develop the
algorithmic theory of statistic, sufficient statistic, and minimal sufficient
statistic. This theory is based on two-part codes consisting of the code for
the statistic (the model summarizing the regularity, the meaningful
information, in the data) and the model-to-data code. In contrast to the
situation in probabilistic statistical theory, the algorithmic relation of
(minimal) sufficiency is an absolute relation between the individual model and
the individual data sample. We distinguish implicit and explicit descriptions
of the models. We give characterizations of algorithmic (Kolmogorov) minimal
sufficient statistic for all data samples for both description modes--in the
explicit mode under some constraints. We also strengthen and elaborate earlier
results on the ``Kolmogorov structure function'' and ``absolutely
non-stochastic objects''--those rare objects for which the simplest models that
summarize their relevant information (minimal sufficient statistics) are at
least as complex as the objects themselves. We demonstrate a close relation
between the probabilistic notions and the algorithmic ones.Comment: LaTeX, 22 pages, 1 figure, with correction to the published journal
versio
Algorithmic statistics: forty years later
Algorithmic statistics has two different (and almost orthogonal) motivations.
From the philosophical point of view, it tries to formalize how the statistics
works and why some statistical models are better than others. After this notion
of a "good model" is introduced, a natural question arises: it is possible that
for some piece of data there is no good model? If yes, how often these bad
("non-stochastic") data appear "in real life"?
Another, more technical motivation comes from algorithmic information theory.
In this theory a notion of complexity of a finite object (=amount of
information in this object) is introduced; it assigns to every object some
number, called its algorithmic complexity (or Kolmogorov complexity).
Algorithmic statistic provides a more fine-grained classification: for each
finite object some curve is defined that characterizes its behavior. It turns
out that several different definitions give (approximately) the same curve.
In this survey we try to provide an exposition of the main results in the
field (including full proofs for the most important ones), as well as some
historical comments. We assume that the reader is familiar with the main
notions of algorithmic information (Kolmogorov complexity) theory.Comment: Missing proofs adde
Thermo-micro-mechanical simulation of bulk metal forming processes
The newly proposed microstructural constitutive model for polycrystal
viscoplasticity in cold and warm regimes (Motaman and Prahl, 2019), is
implemented as a microstructural solver via user-defined material subroutine in
a finite element (FE) software. Addition of the microstructural solver to the
default thermal and mechanical solvers of a standard FE package enabled coupled
thermo-micro-mechanical or thermal-microstructural-mechanical (TMM) simulation
of cold and warm bulk metal forming processes. The microstructural solver,
which incrementally calculates the evolution of microstructural state variables
(MSVs) and their correlation to the thermal and mechanical variables, is
implemented based on the constitutive theory of isotropic
hypoelasto-viscoplastic (HEVP) finite (large) strain/deformation. The numerical
integration and algorithmic procedure of the FE implementation are explained in
detail. Then, the viability of this approach is shown for (TMM-) FE simulation
of an industrial multistep warm forging
Reversibility Violation in the Hybrid Monte Carlo Algorithm
We investigate reversibility violations in the Hybrid Monte Carlo algorithm.
Those violations are inevitable when computers with finite numerical precision
are being used. In SU(2) gauge theory, we study the dependence of observables
on the size of the reversibility violations. While we cannot find any
statistically significant deviation in observables related to the simulated
physical model, algorithmic specific observables signal an upper bound for
reversibility violations below which simulations appear unproblematic. This
empirically derived condition is independent of problem size and parameter
values, at least in the range of parameters studied here.Comment: 17 pages, 5 figures, typos corrected, comment added, matches
published versio
Elastoplastic consolidation at finite strain. Part 2: finite element implementation and numerical examples
A mathematical model for finite strain elastoplastic consolidation of fully saturated soil media is implemented into a finite element program. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. A two-field mixed finite element formulation is employed in which the nodal solid displacements and the nodal pore water pressures are coupled via the linear momentum and mass balance equations. The constitutive model for the solid phase is represented by modified Cam—Clay theory formulated in the Kirchhoff principal stress space, and return mapping is carried out in the strain space defined by the invariants of the elastic logarithmic principal stretches. The constitutive model for fluid flow is represented by a generalized Darcy's law formulated with respect to the current configuration. The finite element model is fully amenable to exact linearization. Numerical examples with and without finite deformation effects are presented to demonstrate the impact of geometric nonlinearity on the predicted responses. The paper concludes with an assessment of the performance of the finite element consolidation model with respect to accuracy and numerical stability
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