1,272 research outputs found
A Nonsmooth Augmented Lagrangian Method and its Application to Poisson Denoising and Sparse Control
In this paper, fully nonsmooth optimization problems in Banach spaces with
finitely many inequality constraints, an equality constraint within a Hilbert
space framework, and an additional abstract constraint are considered. First,
we suggest a (safeguarded) augmented Lagrangian method for the numerical
solution of such problems and provide a derivative-free global convergence
theory which applies in situations where the appearing subproblems can be
solved to approximate global minimality. Exemplary, the latter is possible in a
fully convex setting. As we do not rely on any tool of generalized
differentiation, the results are obtained under minimal continuity assumptions
on the data functions. We then consider two prominent and difficult
applications from image denoising and sparse optimal control where these
findings can be applied in a beneficial way. These two applications are
discussed and investigated in some detail. Due to the different nature of the
two applications, their numerical solution by the (safeguarded) augmented
Lagrangian approach requires problem-tailored techniques to compute approximate
minima of the resulting subproblems. The corresponding methods are discussed,
and numerical results visualize our theoretical findings.Comment: 36 pages, 4 figures, 1 tabl
Lagrange optimality system for a class of nonsmooth convex optimization
In this paper, we revisit the augmented Lagrangian method for a class of
nonsmooth convex optimization. We present the Lagrange optimality system of the
augmented Lagrangian associated with the problems, and establish its
connections with the standard optimality condition and the saddle point
condition of the augmented Lagrangian, which provides a powerful tool for
developing numerical algorithms. We apply a linear Newton method to the
Lagrange optimality system to obtain a novel algorithm applicable to a variety
of nonsmooth convex optimization problems arising in practical applications.
Under suitable conditions, we prove the nonsingularity of the Newton system and
the local convergence of the algorithm.Comment: 19 page
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
On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification
We present a new approach to convexification of the Tikhonov regularization
using a continuation method strategy. We embed the original minimization
problem into a one-parameter family of minimization problems. Both the penalty
term and the minimizer of the Tikhonov functional become dependent on a
continuation parameter.
In this way we can independently treat two main roles of the regularization
term, which are stabilization of the ill-posed problem and introduction of the
a priori knowledge. For zero continuation parameter we solve a relaxed
regularization problem, which stabilizes the ill-posed problem in a weaker
sense. The problem is recast to the original minimization by the continuation
method and so the a priori knowledge is enforced.
We apply this approach in the context of topology-to-shape geometry
identification, where it allows to avoid the convergence of gradient-based
methods to a local minima. We present illustrative results for magnetic
induction tomography which is an example of PDE constrained inverse problem
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