3,152 research outputs found
Weighted projections into closed subspaces
In this paper we study -projections, i.e. operators of a Hilbert space
\HH which act as projections when a seminorm is considered in \HH.
-projections were introduced by Mitra and Rao \cite{[MitRao74]} for finite
dimensional spaces. We relate this concept to the theory of compatibility
between positive operators and closed subspaces of \HH. We also study the
relationship between weighted least squares problems and compatibility
Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations
This paper addresses the estimation of uncertain distributed diffusion
coefficients in elliptic systems based on noisy measurements of the model
output. We formulate the parameter identification problem as an infinite
dimensional constrained optimization problem for which we establish existence
of minimizers as well as first order necessary conditions. A spectral
approximation of the uncertain observations allows us to estimate the infinite
dimensional problem by a smooth, albeit high dimensional, deterministic
optimization problem, the so-called finite noise problem in the space of
functions with bounded mixed derivatives. We prove convergence of finite noise
minimizers to the appropriate infinite dimensional ones, and devise a
stochastic augmented Lagrangian method for locating these numerically. Lastly,
we illustrate our method with three numerical examples
A representer theorem for deep kernel learning
In this paper we provide a finite-sample and an infinite-sample representer
theorem for the concatenation of (linear combinations of) kernel functions of
reproducing kernel Hilbert spaces. These results serve as mathematical
foundation for the analysis of machine learning algorithms based on
compositions of functions. As a direct consequence in the finite-sample case,
the corresponding infinite-dimensional minimization problems can be recast into
(nonlinear) finite-dimensional minimization problems, which can be tackled with
nonlinear optimization algorithms. Moreover, we show how concatenated machine
learning problems can be reformulated as neural networks and how our
representer theorem applies to a broad class of state-of-the-art deep learning
methods
Total least squares problems on infinite dimensional spaces
We study weighted total least squares problems on infinite dimensional spaces. We present some necessary and sufficient conditions for the regularized problem to have a solution. The existence of solution can also be assured for the regularized minimization problem with a constraint to special subsets. Furthermore, we show that a regularization in infinite dimensional total least squares problems is necessary, since in most cases the problem without regularization does not admit a solution.Fil: Contino, Maximiliano. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Fongi, Guillermina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; ArgentinaFil: Maestripieri, Alejandra Laura. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Muro, Luis Santiago Miguel. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentin
Shape deformation analysis from the optimal control viewpoint
A crucial problem in shape deformation analysis is to determine a deformation
of a given shape into another one, which is optimal for a certain cost. It has
a number of applications in particular in medical imaging. In this article we
provide a new general approach to shape deformation analysis, within the
framework of optimal control theory, in which a deformation is represented as
the flow of diffeomorphisms generated by time-dependent vector fields. Using
reproducing kernel Hilbert spaces of vector fields, the general shape
deformation analysis problem is specified as an infinite-dimensional optimal
control problem with state and control constraints. In this problem, the states
are diffeomorphisms and the controls are vector fields, both of them being
subject to some constraints. The functional to be minimized is the sum of a
first term defined as geometric norm of the control (kinetic energy of the
deformation) and of a data attachment term providing a geometric distance to
the target shape. This point of view has several advantages. First, it allows
one to model general constrained shape analysis problems, which opens new
issues in this field. Second, using an extension of the Pontryagin maximum
principle, one can characterize the optimal solutions of the shape deformation
problem in a very general way as the solutions of constrained geodesic
equations. Finally, recasting general algorithms of optimal control into shape
analysis yields new efficient numerical methods in shape deformation analysis.
Overall, the optimal control point of view unifies and generalizes different
theoretical and numerical approaches to shape deformation problems, and also
allows us to design new approaches. The optimal control problems that result
from this construction are infinite dimensional and involve some constraints,
and thus are nonstandard. In this article we also provide a rigorous and
complete analysis of the infinite-dimensional shape space problem with
constraints and of its finite-dimensional approximations
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