121 research outputs found
Convergence of the gradient method for ill-posed problems
We study the convergence of the gradient descent method for solving ill-posed
problems where the solution is characterized as a global minimum of a
differentiable functional in a Hilbert space. The classical least-squares
functional for nonlinear operator equations is a special instance of this
framework and the gradient method then reduces to Landweber iteration. The main
result of this article is a proof of weak and strong convergence under new
nonlinearity conditions that generalize the classical tangential cone
conditions
Inversion formulas for the linearized impedance tomography problem
We consider the linearized electrical impedance tomography problem in two
dimensions on the unit disk. By a linearization around constant coefficients
and using a trigonometric basis, we calculate the linearized
Dirichlet-to-Neumann operator in terms of moments of the conduction coefficient
of the problem. By expanding this coefficient into angular trigonometric
functions and Legendre-M\"untz polynomials in radial coordinates, we can find a
lower-triangular representation of the parameter to data mapping. As a
consequence, we find an explicit solution formula for the corresponding inverse
problem. Furthermore, we also consider the problem with boundary data given
only on parts of the boundary while setting homogeneous Dirichlet values on the
rest. We show that the conduction coefficient is uniquely determined from
incomplete data of the linearized Dirichlet-to-Neumann operator with an
explicit solution formula provided
Optimization of the shape (and topology) of the initial conditions for diffusion parameter identification
The design of an experiment, e.g., the setting of initial conditions,
strongly influences the accuracy of the whole process of determining model
parameters from data. We impose a sensitivity-based approach for choosing
optimal design variables and study the optimization of the shape (and topology)
of the initial conditions for an inverse problem of a diffusion parameter
identification. Our approach, although case independent, is illustrated at the
FRAP (Fluorescence Recovery After Photobleaching) experimental technique. The
core idea resides in the maximization of a sensitivity measure, which depends
on a specific experimental setting of initial conditions. By a numerical
optimization, we find an interesting pattern of increasingly complicated (with
respect to connectivity) optimal initial shapes. The proposed modification of
the FRAP experimental protocol is rather surprising but entirely realistic and
the resulting enhancement of the parameter estimate accuracy is significant
The Kurdyka-\L{}ojasiewicz inequality as regularity condition
We show that a Kurdyka-\L{}ojasiewicz (KL) inequality can be used as
regularity condition for Tikhonov regularization with linear operators in
Banach spaces. In fact, we prove the equivalence of a KL inequality and various
known regularity conditions (variational inequality, rate conditions, and
others) that are utilized for postulating smoothness conditions to obtain
convergence rates. Case examples of rate estimates for Tikhonov regularization
with source conditions or with conditional stability estimate illustrate the
theoretical result
Analysis and Approximation of the Canonical Polyadic Tensor Decomposition
We study the least-squares (LS) functional of the canonical polyadic (CP)
tensor decomposition. Our approach is based on the elimination of one factor
matrix which results in a reduced functional. The reduced functional is
reformulated into a projection framework and into a Rayleigh quotient. An
analysis of this functional leads to several conclusions: new sufficient
conditions for the existence of minimizers of the LS functional, the existence
of a critical point in the rank-one case, a heuristic explanation of "swamping"
and computable bounds on the minimal value of the LS functional. The latter
result leads to a simple algorithm -- the Centroid Projection algorithm -- to
compute suboptimal solutions of tensor decompositions. These suboptimal
solutions are applied to iterative CP algorithms as initial guesses, yielding a
method called centroid projection for canonical polyadic (CPCP) decomposition
which provides a significant speedup in our numerical experiments compared to
the standard methods
Heuristic Parameter Choice Rules for Tikhonov Regularisation with Weakly Bounded Noise
We study the choice of the regularisation parameter for linear ill-posed
problems in the presence of noise that is possibly unbounded but only finite in
a weaker norm, and when the noise-level is unknown. For this task, we analyse
several heuristic parameter choice rules, such as the quasi-optimality,
heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the
weakly bounded noise case. We prove convergence and convergence rates under
certain noise conditions. Moreover, we analyse and provide conditions for the
convergence of the parameter choice by the generalised cross-validation and
predictive mean-square error rules.Comment: 18 page
Convergence of Heuristic Parameter Choice Rules for Convex Tikhonov Regularisation
We investigate the convergence theory of several known as well as new
heuristic parameter choice rules for convex Tikhonov regularisation. The
success of such methods is dependent on whether certain restrictions on the
noise are satisfied. In the linear theory, such conditions are well understood
and hold for typically irregular noise. In this paper, we extend the
convergence analysis of heuristic rules using noise restrictions to the convex
setting and prove convergence of the aforementioned methods therewith. The
convergence theory is exemplified for the case of an ill-posed problem with a
diagonal forward operator in spaces. Numerical examples also provide
further insight.Comment: 32 pages, 5 figure
Towards analytical model optimization in atmospheric tomography
Modern ground-based telescopes rely on a technology called adaptive optics
(AO) in order to compensate for the loss of image quality caused by atmospheric
turbulence. Next-generation AO systems designed for a wide field of view
require a stable and high-resolution reconstruction of the refractive index
fluctuations in the atmosphere. By introducing a novel Bayesian method, we
address the problem of estimating an atmospheric turbulence strength profile
and reconstructing the refractive index fluctuations simultaneously, where we
only use wavefront measurements of incoming light from guide stars. Most
importantly, we demonstrate how this method can be used for model optimization
as well. We propose two different algorithms for solving the maximum a
posteriori estimate: the first approach is based on alternating minimization
and has the advantage of integrability into existing atmospheric tomography
methods. In the second approach, we formulate a convex non-differentiable
optimization problem, which is solved by an iterative thresholding method. This
approach clearly illustrates the underlying sparsity-enforcing mechanism for
the strength profile. By introducing a tuning/regularization parameter, an
automated model reduction of the layer structure of the atmosphere is achieved.
Using numerical simulations, we demonstrate the performance of our method in
practice
On Accelerating the Regularized Alternating Least Square Algorithm for Tensors
In this paper, we discuss the acceleration of the regularized alternating
least square (RALS) algorithm for tensor approximation. We propose a fast
iterative method using a Aitken-Stefensen like updates for the regularized
algorithm. Through numerical experiments, the fast algorithm demonstrate a
faster convergence rate for the accelerated version in comparison to both the
standard and regularized alternating least squares algorithms. In addition, we
analyze the global convergence based on the Kurdyka- Lojasiewicz inequality as
well as show that the RALS algorithm has a linear local convergence rate
Some Convergence Results on the Regularized Alternating Least-Squares Method for Tensor Decomposition
We study the convergence of the Regularized Alternating Least-Squares
algorithm for tensor decompositions. As a main result, we have shown that given
the existence of critical points of the Alternating Least-Squares method, the
limit points of the converging subsequences of the RALS are the critical points
of the least squares cost functional. Some numerical examples indicate a faster
convergence rate for the RALS in comparison to the usual alternating least
squares method
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