A novel method for solving high-dimensional backward stochastic differential equations using Malliavin calculus and deep learning

Backward stochastic differential equations (BSDE) are known to be a powerful tool in mathematical modeling due to their inherent connection with second-order parabolic partial differential equations (PDE) established by the non-linear Feynman-Kac relations. The fundamental power of BSDEs lies in the fact that with them one does not merely obtain the solution of the corresponding PDE but also its spatial gradient through the control process Z. Classical numerical methods tackling the system face the so-called curse of dimensionality and cannot be used to solve high-dimensional problems. In recent years, multiple approaches have been developed to overcome this computational burden, building on deep learning and showing remarkable empirical success even beyond 10 dimensions. However, such Deep BSDE methods struggle with giving accurate approximations for the Z-process throughout the whole time horizon. In this thesis we propose a novel approach aimed to give better estimations for the control problem, exploiting the natural dynamics of the Z-process given by Malliavin calculus. The proposed methods use deep learning parametrizations taking advantage of the universal approximation capability of neural networks. The Malliavin derivatives are estimated through the Malliavin chain rule. Two discrete numerical methods are developed which are called One-Step Malliavin (OSM) and Multi-Step Malliavin (MSM) schemes respectively. An error analysis is carried out proving the consistency of the algorithms and showing first-order convergence under certain assumptions. Numerical experiments are presented to demonstrate the efficiency of the Malliavin formulation compared to other Deep BSDE solvers

Custom optimization algorithms for efficient hardware implementation

The focus is on real-time optimal decision making with application in advanced control systems. These computationally intensive schemes, which involve the repeated solution of (convex) optimization problems within a sampling interval, require more efficient computational methods than currently available for extending their application to highly dynamical systems and setups with resource-constrained embedded computing platforms. A range of techniques are proposed to exploit synergies between digital hardware, numerical analysis and algorithm design. These techniques build on top of parameterisable hardware code generation tools that generate VHDL code describing custom computing architectures for interior-point methods and a range of first-order constrained optimization methods. Since memory limitations are often important in embedded implementations we develop a custom storage scheme for KKT matrices arising in interior-point methods for control, which reduces memory requirements significantly and prevents I/O bandwidth limitations from affecting the performance in our implementations. To take advantage of the trend towards parallel computing architectures and to exploit the special characteristics of our custom architectures we propose several high-level parallel optimal control schemes that can reduce computation time. A novel optimization formulation was devised for reducing the computational effort in solving certain problems independent of the computing platform used. In order to be able to solve optimization problems in fixed-point arithmetic, which is significantly more resource-efficient than floating-point, tailored linear algebra algorithms were developed for solving the linear systems that form the computational bottleneck in many optimization methods. These methods come with guarantees for reliable operation. We also provide finite-precision error analysis for fixed-point implementations of first-order methods that can be used to minimize the use of resources while meeting accuracy specifications. The suggested techniques are demonstrated on several practical examples, including a hardware-in-the-loop setup for optimization-based control of a large airliner.Open Acces

Principled and Efficient Bilevel Optimization for Machine Learning

Automatic differentiation (AD) is a core element of most modern machine learning libraries that allows to efficiently compute derivatives of a function from the corresponding program. Thanks to AD, machine learning practitioners have tackled increasingly complex learning models, such as deep neural networks with up to hundreds of billions of parameters, which are learned using the derivative (or gradient) of a loss function with respect to those parameters. While in most cases gradients can be computed exactly and relatively cheaply, in others the exact computation is either impossible or too expensive and AD must be used in combination with approximation methods. Some of these challenging scenarios arising for example in meta-learning or hyperparameter optimization, can be framed as bilevel optimization problems, where the goal is to minimize an objective function that is evaluated by first solving another optimization problem, the lower-level problem. In this work, we study efficient gradient-based bilevel optimization algorithms for machine learning problems. In particular, we establish convergence rates for some simple approaches to approximate the gradient of the bilevel objective, namely the hypergradient, when the objective is smooth and the lower-level problem consists in finding the fixed point of a contraction map. Leveraging such results, we also prove that the projected inexact hypergradient method achieves a (near) optimal rate of convergence. We establish these results for both the deterministic and stochastic settings. Additionally, we provide an efficient implementation of the methods studied and perform several numerical experiments on hyperparameter optimization, meta-learning, datapoisoning and equilibrium models, which show that our theoretical results are good indicators of the performance in practice

An integrated study of earth resources in the State of California using remote sensing techniques

The author has identified the following significant results. The supply, demand, and impact relationships of California's water resources as exemplified by the Feather River project and other aspects of the California Water Plan are discussed

Estimation and tracking of rapidly time-varying broadband acoustic communication channels

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2006This thesis develops methods for estimating wideband shallow-water acoustic communication channels. The very shallow water wideband channel has three distinct features: large dimension caused by extensive delay spread; limited number of degrees of freedom (DOF) due to resolvable paths and inter-path correlations; and rapid fluctuations induced by scattering from the moving sea surface. Traditional LS estimation techniques often fail to reconcile the rapid fluctuations with the large dimensionality. Subspace based approaches with DOF reduction are confronted with unstable subspace structure subject to significant changes over a short period of time. Based on state-space channel modeling, the first part of this thesis develops algorithms that jointly estimate the channel as well as its dynamics. Algorithms based on the Extended Kalman Filter (EKF) and the Expectation Maximization (EM) approach respectively are developed. Analysis shows conceptual parallels, including an identical second-order innovation form shared by the EKF modification and the suboptimal EM, and the shared issue of parameter identifiability due to channel structure, reflected as parameter unobservability in EKF and insufficient excitation in EM. Modifications of both algorithms, including a two-model based EKF and a subspace EM algorithm which selectively track dominant taps and reduce prediction error, are proposed to overcome the identifiability issue. The second part of the thesis develops algorithms that explicitly find the sparse estimate of the delay-Doppler spread function. The study contributes to a better understanding of the channel physical constraints on algorithm design and potential performance improvement. It may also be generalized to other applications where dimensionality and variability collide.Financial support for this thesis research was provided by the Office of Naval Research and the WHOI Academic Program Office

Development of a hybrid genetic programming technique for computationally expensive optimisation problems

The increasing computational power of modern computers has contributed to the advance of nature-inspired algorithms in the fields of optimisation and metamodelling. Genetic programming (GP) is a genetically-inspired technique that can be used for metamodelling purposes. GP main strength is in the ability to infer the mathematical structure of the best model fitting a given data set, relying exclusively on input data and on a set of mathematical functions given by the user. Model inference is based on an iterative or evolutionary process, which returns the model as a symbolic expression (text expression). As a result, model evaluation is inexpensive and the generated expressions can be easily deployed to other users. Despite genetic programming has been used in many different branches of engineering, its diffusion on industrial scale is still limited. The aims of this thesis are to investigate the intrinsic limitations of genetic programming, to provide a comprehensive review of how researchers have tackled genetic programming main weaknesses and to improve genetic programming ability to extract accurate models from data. In particular, research has followed three main directions. The first has been the development of regularisation techniques to improve the generalisation ability of a model of a given mathematical structure, based on the use of a specific tuning algorithm in case sinusoidal functions are among the functions the model is composed of. The second has been the analysis of the influence that prior knowledge regarding the function to approximate may have on genetic programming inference process. The study has led to the introduction of a strategy that allows to use prior knowledge to improve model accuracy. Thirdly, the mathematical structure of the models returned by genetic programming has been systematically analysed and has led to the conclusion that the linear combination is the structure that is mostly returned by genetic programming runs. A strategy has been formulated to reduce the evolutionary advantage of linear combinations and to protect more complex classes of individuals throughout the evolution. The possibility to use genetic programming in industrial optimisation problems has also been assessed with the help of a new genetic programming implementation developed during the research activity. Such implementation is an open source project and is freely downloadable from http://www.personal.leeds.ac.uk/~cnua/mypage.html

Stochastic modeling of DNA demethylation dynamics in ESCs

DNA methylation and demethylation are opposing processes that when in balance create stable patterns of epigenetic memory. The control of DNA methylation pattern formation in replication dependent and independent demethylation processes has been suggested to be influenced by Tet mediated oxidation of a methylated cytosine, 5mC, to a hydroxylated cytosine, 5hmC. Based only on in vitro experiments, several alternative mechanisms have been proposed on how 5hmC influences replication dependent maintenance of DNA methylation and replication independent processes of active demethylation. In this thesis we design an extended and easily generalizable hidden Markov model that uses as input hairpin (oxidative-)bisulfite sequencing data to precisely determine the over time dynamics of 5mC and 5hmC, as well as to infer the activities of the involved enzymes at a single CpG resolution. Developing the appropriate statistical and computational tools, we apply the model to discrete high-depth sequenced genomic loci, and on a whole genome scale with a much smaller sequencing depth. Performing the analysis of the model’s output on mESCs data, we show that the presence of Tet enzymes and 5hmC has a very strong impact on replication dependent demethylation by establishing a passive demethylation mechanism, implicitly impairing methylation maintenance, but also down-regulating the de novo methylation activity.DNA-Methylierung und Demethylierung sind gegenläufige Prozesse, die im Gleichgewicht stabile Muster des epigenetischen Gedächtnisses erzeugen. Es wird angenommen, dass die Kontrolle der DNA-Methylierungsmusterbildung in replikationsabhängige und unabhängige Demethylierungsprozesse durch Tet-regulierte Oxidation eines methylierten Zytosins (5mC) zu einem hydroxylierten Zytosin (5hmC) beeinflusst wird. Aufgrund von In-Vitro-Experimenten, wurden verschiedene Mechanismen vorgeschlagen wie 5hmC die replikationsabhängige Aufrechterhaltung der DNA-Methylierung und die replikationsunabhängigen Prozesse der aktiven Demethylierung beeinflusst. In dieser Arbeit entwerfen wir ein erweitertes und leicht verallgemeinertes Hidden Markov Modell, das mit Hilfe von Hairpin (oxidative-)Bisulfit Sequenzierung gewonnener Daten die Zeitdynamik von 5mC und 5hmC genau bestimmt und die Aktivitäten der beteiligten Enzyme auf der Ebene einzelner CpGs scha ̈tzt. Wir entwickeln geeignete statistische Methoden, um das Modell sowohl auf der Ebene der sequenzspezifischen Tiefensequenzierung einzelner Loci, als auch auf genomweiter Ebene mit stark verringerter Sequenzierungstiefe anzuwenden. Wir zeigen, dass die Anwesenheit von Tet-Enzymen und 5hmC einen sehr starken Einfluss auf die replikationsabhängige Demethylierung hat, indem sie einen passiven Demethylierungsmechanismus etabliert, der die Methylierungserhaltung implizit beeinträchtigt, aber auch die de novo-Methylierung herunterreguliert

Working Papers: Astronomy and Astrophysics Panel Reports

The papers of the panels appointed by the Astronomy and Astrophysics survey Committee are compiled. These papers were advisory to the survey committee and represent the opinions of the members of each panel in the context of their individual charges. The following subject areas are covered: radio astronomy, infrared astronomy, optical/IR from ground, UV-optical from space, interferometry, high energy from space, particle astrophysics, theory and laboratory astrophysics, solar astronomy, planetary astronomy, computing and data processing, policy opportunities, benefits to the nation from astronomy and astrophysics, status of the profession, and science opportunities