Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: application to urban drainage simulation

International audienceThis paper presents an efficient surrogate modeling strategy for the uncertainty quantification and Bayesian calibration of a hydrological model. In particular, a process-based dynamical urban drainage simulator that predicts the discharge from a catchment area during a precipitation event is considered. The goal is to perform a global sensitivity analysis and to identify the unknown model parameters as well as the measurement and prediction errors. These objectives can only be achieved by cheapening the incurred computational costs, that is, lowering the number of necessary model runs. With this in mind, a regularity-exploiting metamodeling technique is proposed that enables fast uncertainty quantification. Principal component analysis is used for output dimensionality reduction and sparse polynomial chaos expansions are used for the emulation of the reduced outputs. Sensitivity measures such as the Sobol indices are obtained directly from the expansion coefficients. Bayesian inference via Markov chain Monte Carlo posterior sampling is drastically accelerated

Equiconvergence in summation associated with elliptic polinomial

We compare the Fourier integral with the Fourier series in summation associated with elliptic polinomial

Aircraft Trajectory Planning Considering Ensemble Forecasting of Thunderstorms

Mención Internacional en el título de doctorConvective weather poses a major threat that compromises the safe operation of flights while inducing delay and cost. The aircraft trajectory planning problem under thunderstorm evolution is addressed in this thesis, proposing two novel heuristic approaches that incorporate uncertainties in the evolution of convective cells. In this context, two additional challenges are faced. On the one hand, studies have demonstrated that given the computational power available nowadays, the best way to characterize weather uncertainties is through ensemble forecasting products, hence compatibility with them is crucial. On the other hand, for the algorithms to be used during a flight, they must be fast and deliver results in a few seconds. As a first methodology, three variants of the Scenario-Based Rapidly-Exploring Random Trees (SB-RRTs) are proposed. Each of them builds a tree to explore the free airspace during an iterative and random process. The so-called SB-RRT, the SB-RRT∗ and the Informed SB-RRT∗ find point-to-point safe trajectories by meeting a user-defined safety threshold. Additionally, the last two techniques converge to solutions of minimum flight length. In a second instance, the Augmented Random Search (ARS) algorithm is used to sample trajectories from a directed graph and deform them iteratively in the search for an optimal path. The aim of such deformations is to adapt the initial graph to the unsafe set and its possible changes. In the end, the ARS determines the population of trajectories that, on average, minimizes a combination of flight time, time in storms, and fuel consumption Both methodologies are tested considering a dynamic model of an aircraft flying between two waypoints at a constant flight level. Test scenarios consist of realistic weather forecasts described by an ensemble of equiprobable members. Moreover, the influence of relevant parameters, such as the maximum number of iterations, safety margin (in SB-RRTs) or relative weights between objectives (in ARS) is analyzed. Since both algorithms and their convergence processes are random, sensitivity analyses are conducted to show that after enough iterations the results match. Finally, through parallelization on graphical processing units, the required computational times are reduced substantially to become compatible with near real-time operation. In either case, results show that the suggested approaches are able to avoid dangerous and uncertain stormy regions, minimize objectives such as time of flight, flown distance or fuel consumption and operate in less than 10 seconds.Los fenómenos convectivos representan una gran amenaza que compromete la seguridad de los vuelos, a la vez que incrementa los retrasos y costes. En esta tesis se aborda el problema de la planificación de vuelos bajo la influencia de tormentas, proponiendo dos nuevos métodos heurísticos que incorporan incertidumbre en la evolución de las células convectivas. En este contexto, se intentará dar respuesta a dos desafíos adicionales. Por un lado, hay estudios que demuestran que, con los recursos computacionales disponibles hoy en día, la mejor manera de caracterizar la incertidumbre meteorológica es mediante productos de tipo “ensemble”. Por tanto, la compatibilidad con ellos es crucial. Por otro lado, para poder emplear los algoritmos durante el vuelo, deben de ser rápidos y obtener resultados en pocos segundos. Como primera aproximación, se proponen tres variantes de los “Scenario-Based Rapidly-Exploring Random Trees” (SB-RRTs). Cada uno de ellos crea un árbol que explora el espacio seguro durante un proceso iterativo y aleatorio. Los denominados SB-RRT, SB-RRT∗ e Informed SB-RRT∗ calculan trayectorias entre dos puntos respetando un margen de seguridad impuesto por el usuario. Además, los dos últimos métodos convergen en soluciones de mínima distancia de vuelo. En segundo lugar, el algoritmo “Augmented Random Search” (ARS) se utiliza para muestrear trajectorias de un grafo dirigido y deformarlas iterativamente en busca del camino óptimo. El fin de tales deformaciones es adaptar el grafo inicial a las zonas peligrosas y a los cambios que puedan sufrir. Finalmente, el ARS calcula aquella población de trayectorias que, de media, minimiza una combinación del tiempo de vuelo, el tiempo en zonas tormentosas y el consumo de combustible. Ambas metodologías se testean considerando un modelo de avión volando punto a punto a altitud constante. Los casos de prueba se basan en datos meteorológicos realistas formados por un grupo de predicciones equiprobables. Además, se analiza la influencia de los parámetros más importantes como el máximo número de iteraciones, el margen de seguridad (en SB-RRTs) o los pesos relativos de cada objetivo (en ARS). Como ambos algoritmos y sus procesos de convergencia son aleatorios, se realizan análisis de sensibilidad para mostrar que, tras suficientes iteraciones, los resultados coinciden. Por último, mediante técnicas de paralelización en procesadores gráficos, se reducen enormemente los tiempos de cálculo, siendo compatibles con una operación en tiempo casi-real. En ambos casos los resultados muestran que los algoritmos son capaces de evitar zonas inciertas de tormenta, minimizar objetivos como el tiempo de vuelo, la distancia recorrida o el consumo de combustible, en menos de 10 segundos de ejecución.Programa de Doctorado en Ingeniería Aeroespacial por la Universidad Carlos III de MadridPresidente: Ernesto Staffetti Giammaria.- Secretario: Alfonso Valenzuela Romero.- Vocal: Valentin Polishchu

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Machine Learning in Aerodynamic Shape Optimization

Machine learning (ML) has been increasingly used to aid aerodynamic shape optimization (ASO), thanks to the availability of aerodynamic data and continued developments in deep learning. We review the applications of ML in ASO to date and provide a perspective on the state-of-the-art and future directions. We first introduce conventional ASO and current challenges. Next, we introduce ML fundamentals and detail ML algorithms that have been successful in ASO. Then, we review ML applications to ASO addressing three aspects: compact geometric design space, fast aerodynamic analysis, and efficient optimization architecture. In addition to providing a comprehensive summary of the research, we comment on the practicality and effectiveness of the developed methods. We show how cutting-edge ML approaches can benefit ASO and address challenging demands, such as interactive design optimization. Practical large-scale design optimizations remain a challenge because of the high cost of ML training. Further research on coupling ML model construction with prior experience and knowledge, such as physics-informed ML, is recommended to solve large-scale ASO problems

Inference via low-dimensional couplings

We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a target measure of interest. Direct simulation from the desired measure can then be achieved by pushing forward reference samples through the map. Yet characterizing such a map---e.g., representing and evaluating it---grows challenging in high dimensions. The central contribution of this paper is to establish a link between the Markov properties of the target measure and the existence of low-dimensional couplings, induced by transport maps that are sparse and/or decomposable. Our analysis not only facilitates the construction of transformations in high-dimensional settings, but also suggests new inference methodologies for continuous non-Gaussian graphical models. For instance, in the context of nonlinear state-space models, we describe new variational algorithms for filtering, smoothing, and sequential parameter inference. These algorithms can be understood as the natural generalization---to the non-Gaussian case---of the square-root Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure

Nonlinear Gaussian Filtering : Theory, Algorithms, and Applications

By restricting to Gaussian distributions, the optimal Bayesian filtering problem can be transformed into an algebraically simple form, which allows for computationally efficient algorithms. Three problem settings are discussed in this thesis: (1) filtering with Gaussians only, (2) Gaussian mixture filtering for strong nonlinearities, (3) Gaussian process filtering for purely data-driven scenarios. For each setting, efficient algorithms are derived and applied to real-world problems