Image recovery via total variation minimization and related problems

Summary. We study here a classical image denoising technique introduced by L. Rudin and S. Osher a few years ago, namely the constrained minimization of the total variation (TV) of the image. First, we give results of existence and uniqueness and prove the link between the constrained minimization problem and the minimization of an associated Lagrangian functional. Then we describe a relaxation method for computing the solution, and give a proof of convergence. After this, we explain why the TV-based model is well suited to the recovery of some images and not of others. We eventually propose an alternative approach whose purpose is to handle the minimization of the minimum of several convex functionals. We propose for instance a variant of the original TV minimization problem that handles correctly some situations where TV fails. Received December 21, 1995 / Revised version February 26, 1996     

Modelling structural recovery in amorphous polymers

Constitutive equations are derived for linear viscoelastic response and enthalpy relaxation in amorphous polymers in the vicinity of the glass transition temperature. According to the concept of cooperative relaxation, a glassy polymer is treated as an ensemble of weakly-connected relaxing regions. Mechanical relaxation in a region occurs when the thermally activated flow unit reaches some liquid-like state. Structural recovery is modeled as a sequence of hops in which rearranging regions change their traps. A constitutive model for the linear viscoelastic behavior and enthalpy relaxation in a glassy polymer is validated using experimental data in mechanical and calorimetric tests on polycarbonate and poly (methyl methacrylate). Fair agreement is demonstrated between observations and results of numerical simulation.     

Basket problems in margin calculation: Modelling and algorithms

This paper considers combinatorial models and algorithms for the problems of constructing a basket of stock positions representing a capitalization-weighted market index for the purpose to be offset in the margin calculation by a position in other index products as index options, futures or participation units. We show how following the regulatory definition of the basket a 0–1 programming model of knapsack type can be obtained and how to develop the related efficient exact or approximation grab-the-basket algorithms. All approximations are supplied by the performance guarantee evaluations.     

Modelling total tail dependence along diagonals

An approach to modelling total tail dependence beyond the main diagonals is proposed. The concept introduced combines the principal and minor diagonals to describe total extreme dependence. A framework is introduced for the measurement of total tail dependence under model mixture. Illustrations are presented using empirical data on stock market indices and exchange rates. An extension is provided to the multivariate case and total tail dependence is considered for model mixtures.     

Modelling and optimization of carbon-in-pulp gole recovery processs

The paper presents a mixed-integer nonlinear programming (MINLP)approach to construction and operational optimization of carbon-in-pulp(CIP) processes related to hydrometallurgical gold recoveryplants. In this plant section, gold in the form of aurocyanidecomplex ions is adsorbed on the active sites of carbon particles.This takes place in a series of agitated tanks where carbonis countercurrently transferred with respect to the pulp. Threebasic process operating modes are reported and analysed: namelycarousel, continuous, and sequential. The mathematical modelof the CIP process is developed for each one of the operatingmodes examined. Economic evaluation is carried out proposingthe sum of annual gold loss to the tailings plus the total annualcost as a minimization object subject to constraints dictatedby the process model. The original problem is suitably decomposedinto a series of nonlinear programming problems and solved withsuccessive quadratic programming techniques. A characteristiccase study is presented to demonstrate the effectiveness ofthe proposed method.     

Modelling of topological derivatives for contact problems

The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization. Partially supported by the grant 4 T11A 01524 of the State Committee for the Scientific Research of the Republic of Poland     

Modelling and analysis of multistage stochastic programming problems: A software environment

In many decision problems, time and uncertainty play important roles and stochastic programming models are well suited for capturing both these aspects. In this paper we focus our attention on the difficulties which arise in developing multi-period stochastic models and we outline those challenging aspects where the use of classical modelling techniques for deterministic problems prove to be inadequate. In particular, we discuss the development of a modelling and analysis environment which combines multidimensional databases (MDDB), declarative modelling languages and procedural languages. Our aim is to develop a versatile tool which generates multi-period stochastic models and supports the modeller in browsing of data and solutions across different time stages and among different scenarios.     

Modelling and mathematical results arising from ferromagnetic problems

In this article,we investigate the equations of magnetostatics for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum.Furthermore,the ferromagnetic law takes the form B=μ0μr(|H|)H,i.e.,the magnetizing field H and the magnetic induction B are collinear,but the relative permeability μr is allowed to depend on the modulus of H.We prove the well-posedness of the magnetostatic problem under suitable convexity assumptions,and the convergence of several iterative methods,both for the original problem set in the Beppo-Levi space W1(R3),and for a finite-dimensional approximation.The theoretical results are illustrated by numerical examples,which capture the known physical phenomena.     

Single machine total tardiness maximization problems: complexity and algorithms

In this paper, we consider some scheduling problems on a single machine, where weighted or unweighted total tardiness has to be maximized in contrast to usual minimization problems. These problems are theoretically important and have also practical interpretations. For the total weighted tardiness maximization problem, we present an NP-hardness proof and a pseudo-polynomial solution algorithm. For the unweighted total tardiness maximization problem with release dates, NP-hardness is proven. Complexity results for some other classical objective functions (e.g., the number of tardy jobs, total completion time) and various additional constraints (e.g., deadlines, weights and/or release dates of jobs may be given) are presented as well.     

Modelling and long-term simulation of a heat recovery steam generator

Developing accurate non-linear dynamical models for heat recovery steam generator (HRSG) units is presented in this article. The common non-linear autoregressive with exogenous input (NARX) system topology was employed to develop the neuro-fuzzy models based on the experimental data taken during field experiments. In this structure, the non-linear behaviours of the HRSG unit can be characterized through interpolation of local linear models associated with different operating regions via fuzzy inference mechanism. The operating regimes were recognized by applying a genetic algorithm-based fuzzy clustering technique to the prepared data sets. The structures of the fuzzy models are defined with respect to the obtained optimal cluster centres and the corresponding membership functions. The parameters of fuzzy rules were adjusted by recursive least-squares estimation method to fit the model responses to real data. The performances of developed models were evaluated by performing a comparison between the model responses and the responses of the real plant. In addition, the stability of the developed models was assessed by perturbing the model inputs from the nominal values. This guarantees the long-term simulation capabilities of the developed models. A comparison between the responses of the corresponding models and the models obtained from some recent modelling approaches was performed to show the advantages of the developed models. The results show the accuracy and reliability of the developed models at transient and steady-state conditions.     

Modelling of containerized air cargo forwarding problems under uncertainty

This paper examines air container renting and cargo loading problems experienced by freight forwarding companies. Containers have to be booked in advance, in order to obtain discounted rental rates from airlines; renting or returning containers on the day of shipping will incur a heavy penalty. We first propose a mixed-integer model for the certain problem, in which shipment information is known with certainty, when booking. We then present a two-stage recourse model to handle the uncertainty problem, in which accurate shipment information cannot be obtained when booking, and all cargoes have to be shipped without delay. The first-stage decision is made at the booking stage, to book specific numbers of different types of containers. The second-stage decision is made on the day of shipping, depending on the extent to which the uncertainty has been realized. The decisions include number of additional containers of different types that are required to be rented, or the number of containers to be returned, under the scenario that might occur on the day of shipping. We then extend the recourse model into a robust model for dealing with the situation in which cargoes are allowed to be shipped later. The robust model provides a quantitative method to measure the trade-off between risk and cost. A series of experiments demonstrate the effectiveness of the robust model in dealing with risk and uncertainty.     

Randomized algorithms for total least squares problems

Motivated by the recently popular probabilistic methods for low‐rank approximations and randomized algorithms for the least squares problems, we develop randomized algorithms for the total least squares problem with a single right‐hand side. We present the Nyström method for the medium‐sized problems. For the large‐scale and ill‐conditioned cases, we introduce the randomized truncated total least squares with the known or estimated rank as the regularization parameter. We analyze the accuracy of the algorithm randomized truncated total least squares and perform numerical experiments to demonstrate the efficiency of our randomized algorithms. The randomized algorithms can greatly reduce the computational time and still maintain good accuracy with very high probability.     

Modelling and information systems for senior management in local government: problems and prospects

The paper considers the role of modelling systems at senior levels in democratically accountable multi-service public sector organisations such as local government.The main basis for the arguments proposed is three years of project work terminating in mid-1977. During this period the writer acted as a consultant to the IBM (UK) Scientific Centre and was responsible for the day-to-day co-ordination of a major project undertaken in the first tier Welsh local authority, Clwyd County Council.The project, which involved substantial resource inputs from both IBM and the local authority, led to definitive conclusions as to the nature and potential value of the modelling system developed.The main purpose of this paper is to consider the extent to which multi-service modelling systems in general are of relevance to local government. The system developed in the Welsh local authority may be described as affording means oriented resource planning facilities. Further it was designed to generate short and medium term (up to five years) information.It is concluded that although at first sight the implementation of an extensive modelling system may seem to involve intervention on a narrow front the reality is that it impinges upon all the main organisational variables. Because of this the prospect for realising the potential of the system will vary considerably from organisation to organisation. It is important that this should be appreciated by senior management before they approve the commitment of resources to developments of this nature.     

On constructing total orders and solving vector optimization problems with total orders

In this paper, we introduce a construction method of total ordering cone on \mathbbRⁿ{\mathbb{R}^n} . It is shown that any total ordering cone on \mathbbRⁿ{\mathbb{R}^n} is isomorphic to the cone \mathbbRⁿ_lex{\mathbb{R}^n_{lex}} . Existence of a total ordering cone that contain given cone with a compact base is shown. By using this cone, a solving method of vector and set valued optimization problems is presented.     

Solving regularized total least squares problems via a sequence of eigenvalue problems

A computational approach for solving regularized total least squares problems via a sequence of eigenvalue problems has recently been introduced by Renaut and Guo. Combining this approach with thick starts using the nonlinear Arnoldi method lead to a very efficient method for large RTLS problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling and solving optimal placement problems in wireless sensor networks

In this work, the optimal sensor displacement problem in wireless sensor networks is addressed. It is assumed that a network, consisting of independent, collaborative and mobile nodes, is available. Starting from an initial configuration, the aim is to define a specific sensors displacement, which allows the network to achieve high performance, in terms of energy consumption and travelled distance. To mathematically represent the problem under study, different innovative optimization models are proposed and defined, by taking into account different performance objectives. An extensive computational phase is carried out in order to assess the behaviour of the developed models in terms of solution quality and computational effort. A comparison with distributed approaches is also given, by considering different scenarios.     

Maximin investment problems for discounted and total wealth

Email: nikolaidokuchaev{at}trentu.ca Received on 27 March 2006. Accepted on 14 September 2007. We study an optimal investment problem for a continuous-timeincomplete market model such that the risk-free rate, the appreciationrates and the volatility of the stocks are all random; theyare not necessarily adapted to the driving Brownian motion,and their distributions are unknown, but they are supposed tobe currently observable. The optimal investment problem is statedin ‘maximin’ setting which leads to maximizationof the minimum of expected utility over all distributions ofparameters. We found that the presence of the non-discountedwealth in the performance criterion (in addition to the discountedwealth) implies an additional condition for the saddle pointof the maximin problem: the saddle point must include the minimumof the possible risk-free return. This is different from thecase when the utility depends on the discounted wealth only.Using this result, the maximin problem is reduced to a linearparabolic equation and minimization over two scalar parameters.It is an important development of the results obtained in Dokuchaev(2002, Dynamic Portfolio Strategies: Quantitative Methods andEmpirical Rules for Incomplete Information. Boston: Kluwer;2006, IMA J. Manage. Math., 17, 257–276).     

Dominance-based heuristics for one-machine total cost scheduling problems

We study the one-machine scheduling problem with release dates and we look at several objective functions including total (weighted) tardiness and total (weighted) completion time. We describe dominance rules for these criteria, as well as techniques for using these dominance rules to build heuristic solutions. We use them to improve certain well-known greedy heuristic algorithms from the literature. Finally, we introduce a Tabu Search method with a neighborhood based on our dominance rules. Experiments show the effectiveness of our techniques in obtaining very good solutions for all studied criteria.     

Tikhonov regularization for weighted total least squares problems

In this work, we study and analyze the regularized weighted total least squares (RWTLS) formulation. Our regularization of the weighted total least squares problem is based on the Tikhonov regularization. Numerical examples are presented to demonstrate the effectiveness of the RWTLS method.