Lagrange duality for the Morozov principle

Considering a general linear ill-posed equation, we explore the duality arising from the requirement that the discrepancy should take a given value based on the estimation of the noise level, as is notably the case when using the Morozov principle. We show that, under reasonable assumptions, the dual function is smooth, and that its maximization points out the appropriate value of Tikhonov's regularization parameter.Comment: 7 pages, 1 figur

A proximal approach to the inversion of ill-conditioned matrices

We propose a general proximal algorithm for the inversion of ill-conditioned matrices. This algorithm is based on a variational characterization of pseudo-inverses. We show that a particular instance of it (with constant regularization parameter) belongs to the class of {\sl fixed point} methods. Convergence of the algorithm is also discussed

What Allotment and Subcontracting in Procurement Bidding

Allotment and subcontracting are the two alternative mechanisms enabling the participation of SMEs in procurement. We compare these two alternatives in the context of a procurement contract awarded by a first-price sealed-bid auction. When the winning large firm is constrained with respect to the degree of subcontracting, we show that only a reduction of the chosen SME's profit can reduce the expected cost of the contract. However, when the large firm is allowed to choose the subcontracting level, subcontracting can be a Pareto dominating mechanism, i.e. simultaneously increasing both firms' profits and reducing the expected total cost of the contract.allotment; subcontracting; procurement; bidding

Optimizing condition numbers

In this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite $n\times n$ matrices. We show that the condition number is a Clarke regular strongly pseudoconvex function. We prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program. This asymptotic analysis provides a valuable tool for designing an implementable algorithm for solving the problem of minimizing condition numbers

Theoretical relation between water flow rate in a vertical fracture and rock temperature in the surrounding massif

A steady-state analytical solution is given describing the temperature distribution in a homogeneous massif perturbed by cold water flow through a discrete vertical fracture. A relation is derived to express the flow rate in the fracture as a function of the temperature measured in the surrounding rock. These mathematical results can be useful for tunnel drilling as it approaches a vertical cold water bearing structure that induces a thermal anomaly in the surrounding massif. During the tunnel drilling, by monitoring this anomaly along the tunnel axis one can quantify the flow rate in the discontinuity ahead before intersecting the fracture. The cases of the Simplon, Mont Blanc and Gotthard tunnels (Alps) are handled with this approach which shows very good agreement between observed temperatures and the theoretical trend. The flow rates before drilling of the tunnel predicted with the theoretical solution are similar in the Mont Blanc and Simplon cases, as well as the flow rates observed during the drilling. However, the absence of information on hydraulic gradients (before and during drilling) and on fracture specific storage prevents direct predictions of discharge rates in the tunnel

A Framework of Point Merge-based Autonomous System for Optimizing Aircraft Scheduling in Busy TMA

International audienceIn this article we present recent work towards the development of an autonomous system with point merge (PM) that performs sequencing, merging and spacing for arrival aircraft in the busy terminal area. This autonomous arrival management system aims to safely solve the major arrival flight scheduling problems currently handled by human controllers. With PM, it has the potential to handle higher traffic demands without more workload on controllers, consequently increasing capacity and reducing delay. The main objective of this paper is to introduce the framework of this autonomous system with PM. Based on analysis of classic PM route structure, a novel PM-based route network is firstly designed for Beijing Capital International Airport. Vertically, this PM system consists of multi-layers on the sequencing legs for different categories of aircraft with Heavy and Medium, horizontally, it is shaped as a lazy “8”. Then, a multiple-objectives function is discussed for this aircraft scheduling problem, operational constraints and conflict detection and resolution are analysed in detail, a modelling strategy with sliding time window and simulated annealing algorithm is proposed for solving this real-time dynamic problem. Experimental results verify our algorithm is well adapting the high-density traffic optimisation, and finally a conclusion is made and future work is pointed ou

Optimal control techniques for thermo-acoustic tomography

Thermo-acoustic and photo-acoustic tomography are imaging techniques that combine high electromagnetic absorption contrast between two media with ultrasound high resolution. Both techniques lead to an ill-posed inverse problem of the same form which entails inversion (in the wide sense) of the spherical Radon transform. One currently has a choice between three main types of reconstruction procedures namely the filtered backprojection formulae, eigenfunction expansion methods and time reversal method. In this paper we propose to investigate this inverse problem with an alternative control formulation : in our model the function to be recovered is the control function while the (acoustic) pressure is the state function which satisfies a wave equation. We stress that our objective is to give modelling hints, and not to provide new results. Otherwise expressed, the originality of this paper lies essentially in the way the problem is stated

Potential Operational Benefits of Multi-layer Point Merge System on Dense TMA Operation Hybrid arrival trajectory optimization applied to Beijing Capital International Airport

International audience4D Trajectory optimization in dense terminal control area is one of the most challenging problems in air traffic management research. In order to efficiently and robustly land more aircraft at Beijing Capital International Airport (BCIA), one of the busiest airport in the world, a novel trajectory operation model is proposed, i.e. Multi-layer Point Merge (ML-PM) based Autonomous Arrival Management System. This paper aims at the evaluation of its potential operational benefits in terms of flight efficiency and runway throughput. Horizontal and Vertical profiles of ML-PM route network are introduced, the objective and constraints of this optimizing mathematical model are analyzed, especially the speed change profile and the conflict detection mode for merging zone. Then a case study is made by simulating arrival flows under three different operational modes: baseline, traditional point merge, and the ML-PM. Finally, the results show that rational arrival sequence and conflict-free trajectories are generated in ML-PM system, the benefits gained are very positive. Comparing with baseline and the traditional point merge system, ML-PM system shows good performance on flight time, fuel consumption, CO2 emission. The saving of fuel with ML-PM system is expected around 26838 Yuan per hour at BCIA compared with baseline scenario by numerical simulation. Furthermore, more flexible sequence position shift and continuous descent are possible in ML-PM system, and it is capable to handle the high-density operation environment