1,171 research outputs found
On the choice of the regularization parameter for iterated Tikhonov regularization of III-posed problems
AbstractWe propose a method for choosing the regularization parameter in iterated Tikhonov regularization of ill-posed linear equations that is based on quantities that arise during the calculations and leads to optimal convergence rates
Ions in Fluctuating Channels: Transistors Alive
Ion channels are proteins with a hole down the middle embedded in cell
membranes. Membranes form insulating structures and the channels through them
allow and control the movement of charged particles, spherical ions, mostly
Na+, K+, Ca++, and Cl-. Membranes contain hundreds or thousands of types of
channels, fluctuating between open conducting, and closed insulating states.
Channels control an enormous range of biological function by opening and
closing in response to specific stimuli using mechanisms that are not yet
understood in physical language. Open channels conduct current of charged
particles following laws of Brownian movement of charged spheres rather like
the laws of electrodiffusion of quasi-particles in semiconductors. Open
channels select between similar ions using a combination of electrostatic and
'crowded charge' (Lennard-Jones) forces. The specific location of atoms and the
exact atomic structure of the channel protein seems much less important than
certain properties of the structure, namely the volume accessible to ions and
the effective density of fixed and polarization charge. There is no sign of
other chemical effects like delocalization of electron orbitals between ions
and the channel protein. Channels play a role in biology as important as
transistors in computers, and they use rather similar physics to perform part
of that role. Understanding their fluctuations awaits physical insight into the
source of the variance and mathematical analysis of the coupling of the
fluctuations to the other components and forces of the system.Comment: Revised version of earlier submission, as invited, refereed, and
published by journa
Elastic-Net Regularization: Error estimates and Active Set Methods
This paper investigates theoretical properties and efficient numerical
algorithms for the so-called elastic-net regularization originating from
statistics, which enforces simultaneously l^1 and l^2 regularization. The
stability of the minimizer and its consistency are studied, and convergence
rates for both a priori and a posteriori parameter choice rules are
established. Two iterative numerical algorithms of active set type are
proposed, and their convergence properties are discussed. Numerical results are
presented to illustrate the features of the functional and algorithms
The equivalence of fluctuation scale dependence and autocorrelations
We define optimal per-particle fluctuation and correlation measures, relate
fluctuations and correlations through an integral equation and show how to
invert that equation to obtain precise autocorrelations from fluctuation scale
dependence. We test the precision of the inversion with Monte Carlo data and
compare autocorrelations to conditional distributions conventionally used to
study high- jet structure.Comment: 10 pages, 9 figures, proceedings, MIT workshop on correlations and
fluctuations in relativistic nuclear collision
Parameter identification in a semilinear hyperbolic system
We consider the identification of a nonlinear friction law in a
one-dimensional damped wave equation from additional boundary measurements.
Well-posedness of the governing semilinear hyperbolic system is established via
semigroup theory and contraction arguments. We then investigte the inverse
problem of recovering the unknown nonlinear damping law from additional
boundary measurements of the pressure drop along the pipe. This coefficient
inverse problem is shown to be ill-posed and a variational regularization
method is considered for its stable solution. We prove existence of minimizers
for the Tikhonov functional and discuss the convergence of the regularized
solutions under an approximate source condition. The meaning of this condition
and some arguments for its validity are discussed in detail and numerical
results are presented for illustration of the theoretical findings
Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide
We consider an inverse problem of reconstructing the conductivity function in
a hyperbolic equation using single space-time domain noisy observations of the
solution on the backscattering boundary of the computational domain. We
formulate our inverse problem as an optimization problem and use Lagrangian
approach to minimize the corresponding Tikhonov functional. We present a
theorem of a local strong convexity of our functional and derive error
estimates between computed and regularized as well as exact solutions of this
functional, correspondingly. In numerical simulations we apply domain
decomposition finite element-finite difference method for minimization of the
Lagrangian. Our computational study shows efficiency of the proposed method in
the reconstruction of the conductivity function in three dimensions
Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints
The Tikhonov regularization of linear ill-posed problems with an
penalty is considered. We recall results for linear convergence rates and
results on exact recovery of the support. Moreover, we derive conditions for
exact support recovery which are especially applicable in the case of ill-posed
problems, where other conditions, e.g. based on the so-called coherence or the
restricted isometry property are usually not applicable. The obtained results
also show that the regularized solutions do not only converge in the
-norm but also in the vector space (when considered as the
strict inductive limit of the spaces as tends to infinity).
Additionally, the relations between different conditions for exact support
recovery and linear convergence rates are investigated.
With an imaging example from digital holography the applicability of the
obtained results is illustrated, i.e. that one may check a priori if the
experimental setup guarantees exact recovery with Tikhonov regularization with
sparsity constraints
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