Optimal Aircraft Trajectory Planning in the Presence of Stochastic Convective Weather Cells

The Air Traffic Management system is heavily influenced by meteorological uncertainty, and convective weather cells represent one of the most relevant uncertain meteorological phenomena. They are weather hazards that must be avoided through tactical trajectory modifications. As a consequence of the existence in uncertainty in meteorological forecasts and nowcasts, it is important to consider the convective weather cells to be avoided as a stochastic, time-dependent process. In this paper we present a comparative analysis of two methodologies for handling stochastic storms in trajectory planning: one based on stochas- tic reachability and a second one, based on robust optimal control. In the former, the thunderstorm avoidance problem is modelled as a stochastic reach-avoid problem, consid- ering the motion of the aircraft as a discrete-time stochastic system and the weather haz- ards as random set-valued obstacles. Dynamic programming is used to compute a Markov feedback policy that maximizes the probability of reaching the target before entering the unsafe set, i.e., the hazardous weather zones. For the latter, the stochastic dynamics of the storms are modeled in continuous time. We implement an optimal control formulation that allows different possible realizations of the stochastic process to be considered. The resulting problem is then transcribed to a nonlinear programming (NLP) problem through the use of direct numerical methods. A benchmark case study is presented, in which the effectiveness of the two proposed approaches are analyzed.

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

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

Dimensionality Reduction in Dynamic Optimization under Uncertainty

Napat Rujeerapaiboon

Dynamic optimization problems affected by uncertainty are ubiquitous in many application domains. Decision makers typically model the uncertainty through random variables governed by a probability distribution. If the distribution is precisely known, then the emerging optimization problems constitute stochastic programs or chance constrained programs. On the other hand, if the distribution is at least partially unknown, then the emanating optimization problems represent robust or distributionally robust optimization problems. In this thesis, we leverage techniques from stochastic and distributionally robust optimization to address complex problems in finance, energy systems management and, more abstractly, applied probability. In particular, we seek to solve uncertain optimization problems where the prior distributional information includes only the first and the second moments (and, sometimes, the support). The main objective of the thesis is to solve large instances of practical optimization problems. For this purpose, we develop complexity reduction and decomposition schemes, which exploit structural symmetries or multiscale properties of the problems at hand in order to break them down into smaller and more tractable components. In the first part of the thesis we study the growth-optimal portfolio, which maximizes the expected log-utility over a single investment period. In a classical stochastic setting, this portfolio is known to outperform any other portfolio with probability 1 in the long run. In the short run, however, it is notoriously volatile. Moreover, its performance suffers in the presence of distributional ambiguity. We design fixed-mix strategies that offer similar performance guarantees as the classical growth-optimal portfolio but for a finite investment horizon. Moreover, the proposed performance guarantee remains valid for any asset return distribution with the same mean and covariance matrix. These results rely on a Taylor approximation of the terminal logarithmic wealth that becomes more accurate as the rebalancing frequency is increased. In the second part of the thesis, we demonstrate that such a Taylor approximation is in fact not necessary. Specifically, we derive sharp probability bounds on the tails of a product of non-negative random variables. These generalized Chebyshev bounds can be computed numerically using semidefinite programming--in some cases even analytically. Similar techniques can also be used to derive multivariate Chebyshev bounds for sums, maxima, and minima of random variables. In the final part of the thesis, we consider a multi-market reservoir management problem. The eroding peak/off-peak spreads on European electricity spot markets imply reduced profitability for the hydropower producers and force them to participate in the balancing markets. This motivates us to propose a two-layer stochastic programming model for the optimal operation of a cascade of hydropower plants selling energy on both spot and balancing markets. The planning problem optimizes the reservoir management over a yearly horizon with weekly granularity, and the trading subproblems optimize the market transactions over a weekly horizon with hourly granularity. We solve both the planning and trading problems in linear decision rules, and we exploit the inherent parallelizability of the trading subproblems to achieve computational tractability.

EPFL2016

An optimal prediction problem in financial modelling

Violetta Bernyk

EPFL2007

Dimensionality Reduction in Dynamic Optimization under Uncertainty

Napat Rujeerapaiboon

EPFL2016

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

EPFL2016