721 research outputs found
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
A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks
In this paper we propose a distributed implementation of the relaxed
Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization
of a separable convex cost function, whose terms are stored by a set of
interacting agents, one for each agent. Specifically the local cost stored by
each node is in general a function of both the state of the node and the states
of its neighbors, a framework that we refer to as `partition-based'
optimization. This framework presents a great flexibility and can be adapted to
a large number of different applications. We show that the partition-based
R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford
Splitting (R-PRS) operator which, historically, has been introduced in the
literature to find the zeros of sum of functions. Interestingly, making use of
non expansive operator theory, the proposed algorithm is shown to be provably
robust against random packet losses that might occur in the communication
between neighboring nodes. Finally, the effectiveness of the proposed algorithm
is confirmed by a set of compelling numerical simulations run over random
geometric graphs subject to i.i.d. random packet losses.Comment: Full version of the paper to be presented at Conference on Decision
and Control (CDC) 201
Multiobjective Optimization of Non-Smooth PDE-Constrained Problems
Multiobjective optimization plays an increasingly important role in modern
applications, where several criteria are often of equal importance. The task in
multiobjective optimization and multiobjective optimal control is therefore to
compute the set of optimal compromises (the Pareto set) between the conflicting
objectives. The advances in algorithms and the increasing interest in
Pareto-optimal solutions have led to a wide range of new applications related
to optimal and feedback control - potentially with non-smoothness both on the
level of the objectives or in the system dynamics. This results in new
challenges such as dealing with expensive models (e.g., governed by partial
differential equations (PDEs)) and developing dedicated algorithms handling the
non-smoothness. Since in contrast to single-objective optimization, the Pareto
set generally consists of an infinite number of solutions, the computational
effort can quickly become challenging, which is particularly problematic when
the objectives are costly to evaluate or when a solution has to be presented
very quickly. This article gives an overview of recent developments in the
field of multiobjective optimization of non-smooth PDE-constrained problems. In
particular we report on the advances achieved within Project 2 "Multiobjective
Optimization of Non-Smooth PDE-Constrained Problems - Switches, State
Constraints and Model Order Reduction" of the DFG Priority Programm 1962
"Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation
and Hierarchical Optimization"
Research in orbit determination optimization for space trajectories
Research data covering orbit determination, optimization techniques, and trajectory design for manned space flights are summarized
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