Stochastic gradient descent (SGD) is arguably the most important algorithm used in optimization problems for large-scale machine learning. Its behaviour has been studied extensively from the viewpoint of mathematical analysis and probability theory; it is widely held that in the limit where the learning rate in the algorithm tends to zero, a specific stochastic differential equation becomes an adequate model of the dynamics of the algorithm. This study exhibits some of the research in this field by analyzing the application of a recently proven theorem to the problem of tensor principal component analysis. The results, originally discovered in an article by Gérard Ben Arous, Reza Gheissari and Aukosh Jagannath from 2022, illustrate how the phase diagram of functions of SGD differ in the high-dimensional regime from that of the classical fixed-dimensional setting.

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<p>This dissertation focuses on the simulation efficiency of the Markov process for two scenarios: Stochastic differential equations(SDEs) and simulated weather data. </p> <p><br></p> <p>For SDEs, we propose a novel Gibbs sampling algorithm that allows sampling from a particular class of SDEs without any discretization error and shows the proposed algorithm improves the sampling efficiency by orders of magnitude against the existing popular algorithms.  </p> <p><br></p> <p>In the weather data simulation study, we investigate how representative the simulated data are for three popular stochastic weather generators. Our results suggest the need for more than a single realization when generating weather data to obtain suitable representations of climate. </p>

In this thesis, we introduce several closely related stochastic models which generalize the deterministic Hodgkin-Huxley formalism to an SDE framework. We provide analytical results on the existence and uniqueness of solutions as well as the exact formulas for the moments of a simplified model, with simplifications motivated by the experiments performed by Hodgkin and Huxley in their seminal paper.For more complicated models, we provide an approach for the approximation and simulation of solutions to the corresponding SDEs, and show several realizations of the sample paths and moments of these simulations to verify qualitative behavior in this case. All code for the project is written in the Julia language and can be obtained upon request by the reader.

Quantum communication protocols require maximally entangled state of pair of qubits (spin-1/2 states in this context) to be shared between sender and the receiver. The entangled qubits lose entanglement because of random magnetic field disturbances. The dynamics in the form of joint density matrix of random pure entangled state provide the steady (joint) state and the associated timescales (time taken by the pair to reach the steady state) providing a scope in future to quantify the effective utilization of quantum communication protocols. / The dynamics of an identical pair of entangled spin-1/2 particles, both subjected to the identical, independent, correlated random magnetic fields is studied. The dynamics of the pure joint state of the pair is derived using stochastic calculus. In case of identical fields, an ensemble of such pure states are combined using the modified spin joint density matrix and the joint relaxation time is obtained for the pair of spin-1/2 particles. These dynamics can be interpreted as special kind of correlations involving the spatial components of the Bloch polarization vectors of the constituent entangled spin-1/2 particles. In case of independent random magnetic fields, the dynamics are obtained by considering a pure joint state of entangled spin-1/2 particles. The disentanglement time defined as the time taken for the particles to become disentangled, is obtained. In case of correlated random magnetic fields, the dynamics of a maximally entangled pair of spin-1/2 particles are derived in terms of the joint density matrix of the entangled pair from which the steady state density matrix and the associated timescale for it to be reached are obtained. The asymptotic density matrix in this case represents a state of (partial) disentanglement. In other words, there is a persistent entanglement in case of correlated field disturbances. / Thesis / Doctor of Philosophy (PhD) / Maximally entangled pair of quantum bits (in the form of spin-1/2 states) lose entanglement either partially or completely depending upon the nature of random magnetic field disturbances around them (correlated/independent/identical fields). The dynamics of entangled states (in the form of density matrix of a random pure state) in the presence of random magnetic fields are obtained using the ideas of stochastic calculus to understand the steady state of the pair and the associated timescales to be reached.

We consider a stochastic differential equation on a domain D in n-dimensional real space, where the associated dynamical system is linear, and D contains a repulsive type stationary point at the origin O. We obtain an exit law for the first exit time of the solution process from a ball of arbitrary radius centered at the origin, which involves additive scaling as in Day (1995). The form of the scaling constant is worked out and shown to depend on the structure of the Jordan form of the linear drift. We then obtain an extension of this exit law to the first exit time of the solution process from the general domain D by considering the exit in two stages: first from the origin O to the boundary of the ball, for which the aforementioned exit law applies, and then from the boundary of the ball to the boundary of D. In this way we are able to determine for which Jordan forms we can obtain a limiting distribution for the first exit time to the boundary of D as the noise approaches 0. In particular, we observe there are cases for which the exit time distribution diverges as the noise approaches 0. / Ph. D.

An accelerated pace of discovery in biological sciences is made possible by a new generation of computational biology and bioinformatics tools. In this dissertation we develop novel computational, analytical, and high performance simulation techniques for biological problems, with applications to the yeast cell division cycle, and to the RNA-Sequencing of the yellow fever mosquito. Cell cycle system evolves stochastic effects when there are a small number of molecules react each other. Consequently, the stochastic effects of the cell cycle are important, and the evolution of cells is best described statistically. Stochastic simulation algorithm (SSA), the standard stochastic method for chemical kinetics, is often slow because it accounts for every individual reaction event. This work develops a stochastic version of a deterministic cell cycle model, in order to capture the stochastic aspects of the evolution of the budding yeast wild-type and mutant strain cells. In order to efficiently run large ensembles to compute statistics of cell evolution, the dissertation investigates parallel simulation strategies, and presents a new probabilistic framework to analyze the performance of dynamic load balancing algorithms. This work also proposes new accelerated stochastic simulation algorithms based on a fully implicit approach and on stochastic Taylor expansions. Next Generation RNA-Sequencing, a high-throughput technology to sequence cDNA in order to get information about a sample's RNA content, is becoming an efficient genomic approach to uncover new genes and to study gene expression and alternative splicing. This dissertation develops efficient algorithms and strategies to find new genes in Aedes aegypti, which is the most important vector of dengue fever and yellow fever. We report the discovery of a large number of new gene transcripts, and the identification and characterization of genes that showed male-biased expression profiles. This basic information may open important avenues to control mosquito borne infectious diseases. / Ph. D.

This doctoral thesis is concerned with some theoretical and practical questions related to backward stochastic differential equations (BSDEs) and more specifically their connection with some parabolic partial differential equations (PDEs). The thesis is made of three parts. In the first part, we study the probabilistic representation for a class of multidimensional PDEs with quadratic nonlinearities of a special form. We obtain a representation formula for the PDE solution in terms of the solutions to a Lipschitz BSDE. We then use this representation to obtain an estimate on the gradient of the PDE solutions by probabilistic means. In the course of our analysis, we are led to prove some results for the associated multidimensional quadratic BSDEs, namely an existence result and a partial uniqueness result. In the second part, we study the well-posedness of a very general quadratic reflected BSDE driven by a continuous martingale. We obtain the comparison theorem, the special comparison theorem for reflected BSDEs (which allows to compare the increasing processes of two solutions), the uniqueness and existence of solutions, as well as a stability result. The comparison theorem (from which uniqueness follows) and the special comparison theorem are obtained through natural techniques and minimal assumptions. The existence is based on a perturbative procedure, and holds for a driver whis is Lipschitz, or slightly-superlinear, or monotone with arbitrary growth in y. Finally, we obtain a stability result, which gives in particular a local Lipschitz estimate in BMO for the martingale part of the solution. In the third and last part, we study the time-discretization of BSDEs having nonlinearities that are monotone but with polynomial growth in the primary variable. We show that in that case, the explicit Euler scheme is likely to diverge, while the implicit scheme converges. In fact, by studying the family of θ-schemes, which are mixed explicit-implicit, θ characterizing the degree of implicitness, we find that the scheme converges when the implicit component is dominant (θ ≥ 1/2 ). We then propose a tamed explicit scheme, which converges. We show that the implicit-dominant schemes with θ > 1/2 and our tamed explicit scheme converge with order 1/2 , while the trapezoidal scheme (θ = 1/2) converges with order 7/4.

The motivation for this research is to leverage the increasing deployment of the phasor measurement unit (PMU) technology by electric utilities in order to improve situational awareness in power systems. PMUs provide unprecedentedly fast and synchronized voltage and current measurements across the system. Analyzing the big data provided by PMUs may prove helpful in reducing the risk of blackouts, such as the Northeast blackout in August 2003, which have resulted in huge costs in past decades. In order to provide deeper insight into early warning signs (EWS) of catastrophic events in power systems, this dissertation studies changes in statistical properties of high-resolution measurements as a power system approaches a critical transition. The EWS under study are increases in variance and autocorrelation of state variables, which are generic signs of a phenomenon known as critical slowing down (CSD). Critical slowing down is the result of slower recovery of a dynamical system from perturbations when the system approaches a critical transition. CSD has been observed in many stochastic nonlinear dynamical systems such as ecosystem, human body and power system. Although CSD signs can be useful as indicators of proximity to critical transitions, their characteristics vary for different systems and different variables within a system. The dissertation provides evidence for the occurrence of CSD in power systems using a comprehensive analytical and numerical study of this phenomenon in several power system test cases. Together, the results show that it is possible extract information regarding not only the proximity of a power system to critical transitions but also the location of the stress in the system from autocorrelation and variance of measurements. Also, a semi-analytical method for fast computation of expected variance and autocorrelation of state variables in large power systems is presented, which allows one to quickly identify locations and variables that are reliable indicators of proximity to instability.

In this thesis, first we study the unique classical solution of quasi-linear second order parabolic partial differential equations (PDEs). For this, we study the existence and uniqueness of the $L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{d}) \otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k})\otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k\times d})$ valued solution of forward backward stochastic differential equations (FBSDEs) with finite horizon, the regularity property of the solution of FBSDEs and the connection between the solution of FBSDEs and the solution of quasi-linear parabolic PDEs. Then we establish their connection in the Sobolev weak sense, in order to give the weak solution of the quasi-linear parabolic PDEs. Finally, we study the unique weak solution of quasi-linear second order elliptic PDEs through the stationary solution of the FBSDEs with infinite horizon.