7,733 research outputs found
A Class of Free Boundary Problems with Onset of a new Phase
A class of diffusion driven Free Boundary Problems is considered which is
characterized by the initial onset of a phase and by an explicit kinematic
condition for the evolution of the free boundary. By a domain fixing change of
variables it naturally leads to coupled systems comprised of a singular
parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even
though the one dimensional case has been thoroughly investigated, results as
basic as well-posedness and regularity have so far not been obtained for its
higher dimensional counterpart. In this paper a recently developed regularity
theory for abstract singular parabolic Cauchy problems is utilized to obtain
the first well-posedness results for the Free Boundary Problems under
consideration. The derivation of elliptic regularity results for the underlying
static singular problems will play an important role
Optimal Regularity for a Class of Singular Abstract Parabolic Equations
A general class of singular abstract Cauchy problems is considered which
naturally arises in applications to certain Free Boundary Problems. Existence
of an associated evolution operator characterizing its solutions is established
and is subsequently used to derive optimal regularity results. The latter are
well known to be important basic tools needed to deal with corresponding
nonlinear Cauchy Problems such as those associated to Free Boundary Problems
Wellposedness of a nonlocal nonlinear diffusion equation of image processing
Existence and uniqueness are established for a degenerate regularization of
the well-known Perona-Malik equation proposed by the first author for
non-smooth initial data. The results heavily rely on the choice of appropriate
functional setting inspired by a recent approach to degenerate parabolic
equations via so-called singular Riemannian manifolds introduced by Herbert
Amann
On The Stability of Interpretable Models
Interpretable classification models are built with the purpose of providing a
comprehensible description of the decision logic to an external oversight
agent. When considered in isolation, a decision tree, a set of classification
rules, or a linear model, are widely recognized as human-interpretable.
However, such models are generated as part of a larger analytical process. Bias
in data collection and preparation, or in model's construction may severely
affect the accountability of the design process. We conduct an experimental
study of the stability of interpretable models with respect to feature
selection, instance selection, and model selection. Our conclusions should
raise awareness and attention of the scientific community on the need of a
stability impact assessment of interpretable models
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