71,320 research outputs found
Nonlinearizing linear equations to integrable systems including new hierarchies with nonholonomic deformations
We propose a scheme for nonlinearizing linear equations to generate
integrable nonlinear systems of both the AKNS and the KN classes, based on the
simple idea of dimensional analysis and detecting the building blocks of the
Lax pair. Along with the well known equations we discover a novel integrable
hierarchy of higher order nonholonomic deformations for the AKNS family, e.g.
for the KdV, the mKdV, the NLS and the SG equation, showing thus a two-fold
universality of the recently found deformation for the KdV equation.Comment: 17 pages, 5 figures, Latex, Final version to be published in J. Math.
Phy
C-HiLasso: A Collaborative Hierarchical Sparse Modeling Framework
Sparse modeling is a powerful framework for data analysis and processing.
Traditionally, encoding in this framework is performed by solving an
L1-regularized linear regression problem, commonly referred to as Lasso or
Basis Pursuit. In this work we combine the sparsity-inducing property of the
Lasso model at the individual feature level, with the block-sparsity property
of the Group Lasso model, where sparse groups of features are jointly encoded,
obtaining a sparsity pattern hierarchically structured. This results in the
Hierarchical Lasso (HiLasso), which shows important practical modeling
advantages. We then extend this approach to the collaborative case, where a set
of simultaneously coded signals share the same sparsity pattern at the higher
(group) level, but not necessarily at the lower (inside the group) level,
obtaining the collaborative HiLasso model (C-HiLasso). Such signals then share
the same active groups, or classes, but not necessarily the same active set.
This model is very well suited for applications such as source identification
and separation. An efficient optimization procedure, which guarantees
convergence to the global optimum, is developed for these new models. The
underlying presentation of the new framework and optimization approach is
complemented with experimental examples and theoretical results regarding
recovery guarantees for the proposed models
A new approach to nonlinear constrained Tikhonov regularization
We present a novel approach to nonlinear constrained Tikhonov regularization
from the viewpoint of optimization theory. A second-order sufficient optimality
condition is suggested as a nonlinearity condition to handle the nonlinearity
of the forward operator. The approach is exploited to derive convergence rates
results for a priori as well as a posteriori choice rules, e.g., discrepancy
principle and balancing principle, for selecting the regularization parameter.
The idea is further illustrated on a general class of parameter identification
problems, for which (new) source and nonlinearity conditions are derived and
the structural property of the nonlinearity term is revealed. A number of
examples including identifying distributed parameters in elliptic differential
equations are presented.Comment: 21 pages, to appear in Inverse Problem
Modeling active electrolocation in weakly electric fish
In this paper, we provide a mathematical model for the electrolocation in
weakly electric fishes. We first investigate the forward complex conductivity
problem and derive the approximate boundary conditions on the skin of the fish.
Then we provide a dipole approximation for small targets away from the fish.
Based on this approximation, we obtain a non-iterative location search
algorithm using multi-frequency measurements. We present numerical experiments
to illustrate the performance and the stability of the proposed multi-frequency
location search algorithm. Finally, in the case of disk- and ellipse-shaped
targets, we provide a method to reconstruct separately the conductivity, the
permittivity, and the size of the targets from multi-frequency measurements.Comment: 37 pages, 11 figure
Inverse problems in the design, modeling and testing of engineering systems
Formulations, classification, areas of application, and approaches to solving different inverse problems are considered for the design of structures, modeling, and experimental data processing. Problems in the practical implementation of theoretical-experimental methods based on solving inverse problems are analyzed in order to identify mathematical models of physical processes, aid in input data preparation for design parameter optimization, help in design parameter optimization itself, and to model experiments, large-scale tests, and real tests of engineering systems
On Multiple Frequency Power Density Measurements
We shall give a priori conditions on the illuminations such that the
solutions to the Helmholtz equation in \Omega,
on , and their gradients satisfy certain non-zero
and linear independence properties inside the domain \Omega, provided that a
finite number of frequencies k are chosen in a fixed range. These conditions
are independent of the coefficients, in contrast to the illuminations
classically constructed by means of complex geometric optics solutions. This
theory finds applications in several hybrid problems, where unknown parameters
have to be imaged from internal power density measurements. As an example, we
discuss the microwave imaging by ultrasound deformation technique, for which we
prove new reconstruction formulae.Comment: 26 pages, 4 figure
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