Solving the anticipatory route guidance generation problem using a generalization of secant methods

Route guidance refers to information provided to travelers in an attempt to facilitate their decisions relative to departure time, travel mode and route. We are specifically interested in consistent anticipatory route guidance, in which real-time traffic measurements are used to make short-term predictions, involving complex simulation tools, of future traffic conditions. These predictions are the basis of the guidance information that is provided to users. By consistent, we mean that the anticipated traffic conditions used to generate the guidance must be similar to the traffic conditions that the travelers are going to experience on the network. The problem is tricky because, contrarily to weather forecast where the real system under consideration is not affected by information provision, the very fact of providing travel information may modify the future traffic conditions and, therefore, invalidate the prediction that has been used to generate it. Bottom (2000) has proposed a general fixed point formulation of this problem with the following characteristics. First, as guidance generation involves considerable amounts of computation, this fixed point problem must be solved quickly and accurately enough for the results to be timely delivered to drivers. Secondly the unavailability of analytical forms for the objective function and the presence of noise due to the use of simulation tools prevent from using classical algorithms. We propose in this paper an adaptation of the generalized secant method (cf. the related presentation “A generalization of secant methods for solving nonlinear systems of equations”) in order to handle the intrinsic characteristics of the consistent anticipatory route guidance generation, especially the very high dimension associated with real problems. We present then a number of simulation experiments based on two simulation tools in order to compare the performances of the diverse algorithms. The first is a simple simulator implementing the framework of the route guidance generation on a small network, which is used to illustrate the properties of this problem and the behavior of the algorithms. Then, we present a large-scale case study of size 124575 using DynaMIT, a simulation-based real-time Dynamic Traffic Assignment system designed to compute and disseminate anticipatory route guidance. These results point out the real-time potential of the method as its ability to handle large scale problem.

New algorithmic methods for real-time transportation problems

Frank Crittin

Two of the most basic problems encountered in numerical optimization are least-squares problems and systems of nonlinear equations. The use of more and more complex simulation tools on high performance computers requires solving problems involving an increasingly large number of variables. The main thrust of this thesis the design of new algorithmic methods for solving large-scale instances of these two problems. Although they are relevant in many different applications, we concentrate specifically on real applications encountered in the context of Intelligent Transportation Systems to illustrate their performances. First we propose a new approach for the estimation and prediction of OriginDestination tables. This problem is usually solved using a Kalman filter approach, which refers to both formulation and resolution algorithm. We prefer to consider a explicit least-squares formulation. It offers convenient and flexible algorithms especially designed to solve largescale problems. Numerical results provide evidence that this approach requires significantly less computation effort than the Kalman filter algorithm. Moreover it allows to consider larger problems, likely to occur in real applications. Second a new class of quasi-Newton methods for solving systems of nonlinear equations is presented. The main idea is to generalize classical methods by building a model using more than two previous iterates. We use a least-squares approach to calibrate this model, as exact interpolation requires a fixed number of iterates, and may be numerically problematic. Based on classical assumptions we give a proof of local convergence of this class of methods. Computational comparisons with standard quasi-Newton methods highlight substantial improvements in terms of robustness and number of function evaluations. We derive from this class of methods a matrix-free algorithm designed to solve large-scale systems of nonlinear equations without assuming any particular structure on the problems. We have successfully tried out the method on problems with up to one million variables. Computational experiments on standard problems show that this algorithm outperforms classical large-scale quasi-Newton methods in terms of efficiency and robustness. Moreover, its numerical performances are similar to Newton-Krylov methods, currently considered as the best to solve large-scale systems of equations. In addition, we provide numerical evidence of the superiority of our method for solving noisy systems of nonlinear equations. This method is then applied to the consistent anticipatory route guidance generation. Route guidance refers to information provided to travelers in an attempt to facilitate their decisions relative to departure time, travel mode and route. We are specifically interested in consistent anticipatory route guidance, in which real-time traffic measurements are used to make short-term predictions, involving complex simulation tools, of future traffic conditions. These predictions are the basis of the guidance information that is provided to users. By consistent, we mean that the anticipated traffic conditions used to generate the guidance must be similar to the traffic conditions that the travelers are going to experience on the network. The problem is tricky because, contrarily to weather forecast where the real system under consideration is not affected by information provision, the very fact of providing travel information may modify the future traffic conditions and, therefore, invalidate the prediction that has been used to generate it. Bottom (2000) has proposed a general fixed point formulation of this problem with the following characteristics. First, as guidance generation involves considerable amounts of computation, this fixed point problem must be solved quickly and accurately enough for the results to be timely and of use to drivers. Secondly the unavailability of a closed-form objective function and the presence of noise due to the use of simulation tools prevent from using classical algorithms. A number of simulation experiments based on two system software including DynaMIT a state-of-the-art, real-time computer system for traffic estimation and prediction, developed at the Intelligent Transportation Systems Program of the Massachusetts Institute of Technology (MIT), have been run. These numerical results underline the good behavior of our large-scale method compared to classical fixed point methods for solving the consistent anticipatory route guidance problem. We close with some comments about future promising directions of research.

EPFL2004

An optimal prediction problem in financial modelling

Violetta Bernyk

The subject of the present thesis is an optimal prediction problem concerning the ultimate maximum of a stable Lévy process over a finite interval of time. Such "optimal prediction" problems are of both theoretical and practical interest, in particular they have applications in finance. For instance, suppose that an investor has a long position in one financial asset, whose price is modelled by some stochastic process. The investor's objective is to determine a "best moment" at which to close out the position and to sell the asset at the highest possible price. This optimal decision must be based on continuous observations of the asset price performance and only on the information accumulated to date. Hence, the investor should use a prediction (forecasting) of the future evolution of the price of the financial security. We examine this problem in the case where the asset price is modelled by a Lévy process. Indeed, during the last several years, the application of Lévy processes in the modelling financial asset returns has become one of the active research directions in quantitative finance. Thus, this thesis contains suitable new results concerning Lévy processes. We derive the law of the supremum process associated with a strictly stable Lévy process with no negative jumps which is not a subordinator. We note that the latter problem dates back to 1973. In particular, we show that the probability density function of the supremum process can be expressed using an explicit power series representation or via an integral representation. We also derive the infinitesimal generator of the reflected process associated with a general strictly stable Lévy process. Throughout this thesis, we apply the theory of optimal stopping, the methods of fractional differential calculus, and some results from fluctuation theory. Implementing these theories in the context of Lévy processes requires the development of specific analytical results. In the case where the asset price is modelled by a spectrally positive stable Lévy process, we describe the optimal strategy under certain conditions on the model parameters. The optimal strategy is of the following form: the investor must stop the observation of the price process and sell the asset as soon as the associated reflected process crosses for the first time a particular stopping boundary. We also provide numerical estimates and simulation examples of the results obtained by using this strategy.

EPFL2007

Rate allocation and optimized decoding in inter-session network coding

Eirina Bourtsoulatze

Recent advances in data processing and communication systems have led to a continuous increase in the amount of data communicated over today’s networks. These large volumes of data pose new challenges on the current networking infrastructure that only offers a best effort mechanism for data delivery. The emergence of new distributed network architectures, such as peer-to-peer networks and wireless mesh networks, and the need for efficient data delivery mechanisms have motivated researchers to reconsider the way that information is communicated and processed in the networks. This has given rise to a new research field called network coding. The network coding paradigm departs from the traditional routing principle where information is simply relayed by the network nodes towards the destination, and introduces some intelligence in the network through coding at the intermediate nodes. This in-network data processing has been proved to substantially improve the performance of data delivery systems in terms of throughput and error resilience in networks with high path diversity. Motivated by the promising results in the network coding research, we focus in this thesis on the design of network coding algorithms for simultaneous transmission of multiple data sources in overlay networks. We investigate several problems that arise in the context of inter-session network coding, namely (i) decoding delay minimization in inter-session network coding, (ii) distributed rate allocation for inter-session network coding and (iii) correlation-aware decoding of incomplete network coded data. We start by proposing a novel framework for data delivery from multiple sources to multiple clients in an overlay wireline network, where intermediate nodes employ randomized inter-session network coding. We consider networks with high resource diversity, which creates network coding opportunities with possibly large gains in terms of throughput, delay and error robustness. However, the coding operations in the intermediate nodes must be carefully designed in order to enable efficient data delivery. We look at the problem from the decoding delay perspective and design solutions that lead to a small decoding delay at clients through proper coding and rate allocation. We cast the optimization problem as a rate allocation problem, which seeks for the coding operations that minimize the average decoding delay in the client population. We demonstrate the validity of our algorithm through simulations in representative network topologies. The results show that an effective combination of intra- and inter-session network coding based on randomized linear coding permits to reach small decoding delays and to better exploit the available network resources even in challenging network settings. Next, we design a distributed rate allocation algorithm where the users decide locally how many intra- and inter-session network coded packets should be requested from the parent nodes in order to get minimal decoding delay. The capability to take coding decisions locally with only a partial knowledge of the network statistics is of crucial importance for applications where users are organized in dynamic overlay networks. We propose a receiver-driven communication protocol that operates in two rounds. First, the users request and obtain information regarding the network conditions and packet availability in their local neighborhood. Then, every user independently optimizes the rate allocation among different possible intra- and inter-session packet combinations to be requested from its parents. We also introduce the novel concept of equivalent flows, which permits to efficiently estimate the expected number of packets that are necessary for decoding and hence to simplify the rate allocation process. Experimental results indicate that our algorithm is capable of eliminating the bottlenecks and reducing the decoding delay of users with limited resources. We further investigate the application of the proposed distributed rate allocation algorithm to the transmission of video sequences and validate the performance of our system using the NS-3 simulator. The simulation results show that the proposed rate allocation algorithm is successful in improving the quality of the delivered video compared to intra-session network coding based solutions. Finally, we investigate the problem of decoding the source information from an incomplete set of network coded data with the help of source priors in a finite algebraic field. The inability to form a complete decoding system can be often caused by transmission losses or timing constraints imposed by the application. In this case, exact reconstruction of the source data by conventional algorithms such as Gaussian elimination is not feasible; however, partial recovery of the source data may still be possible, which can be useful in applications where approximate reconstruction is informative. We use the statistical characteristics of the source data in order to perform approximate decoding. We first analyze the performance of a hypothetical maximum a posteriori decoder, which recovers the source data from an incomplete set of network coded data given the joint statistics of the sources. We derive an upper bound on the probability of erroneous source sequence decoding as a function of the system parameters. We then propose a constructive solution to the approximate decoding problem and design an iterative decoding algorithm based on message passing, which jointly considers the network coding and the correlation constraints. We illustrate the performance of our decoding algorithm through extensive simulations on synthetic and real data sets. The results demonstrate that, even by using a simple correlation model expressed as a correlation noise between pairs of sources, the original source data can be partially decoded in practice from an incomplete set of network coded symbols. Overall, this thesis addresses several important issues related to the design of efficient data delivery methods with inter-session network coding. Our novel framework for decoding delay minimization can impact the development of practical inter-session network coding algorithms that are appropriate for applications with low delay requirements. Our rate allocation algorithms are able to exploit the high resource diversity of modern networking systems and represent an effective alternative in the development of distributed communication systems. Finally, our algorithm for data recovery from incomplete network coded data using correlation priors can contribute significantly to the improvement of the delivered data quality and provide new insights towards the design of joint source and network coding algorithms.

EPFL2013