Performing simulations of large-scale bio-molecular systems has long been one of the great challenges of molecular biophysics. Phenomena, such as the folding and conformational rearrangement of proteins, often takes place over the course milliseconds-to-seconds. The methods of traditional molecular dynamics used to simulate such systems are on the other hand typically limited to giving trajectories of nanosecond-to-microsecond duration. The failure of traditional methods has thus motivated the development of many special purpose techniques that propose to capture the essential characteristics of systems over conventionally inaccessible timescales. This dissertation first focuses on presenting a set of advances made on one such technique, Milestoning. Milestoning gives a statistical procedure for recovering long trajectories of the system based on observations of many short trajectories that start and end on hypersurfaces in the system’s phase space. Justification of the method’s validity typically relies on the assumption that trajectories of the system lose all memory between crossing successive milestones. We start by giving a modified milestoning procedure in which both the memory loss assumption is relaxed and reaction mechanisms are more easily extracted. We follow with numerical examples illustrating the success of new procedure. Then we show how milestoning may be used to compute an experimentally relevant timescale known as the transit time (also known as the reaction path time). Finally, we discuss how time reversal symmetry may be exploited to improve sampling of the trajectory fragments that connect milestones. After discussing milestoning, the dissertation shifts focus to a different way of approaching the problem of simulating long timescales. We consider two polymers models that are sufficiently simple to permit numerical integration of the desired long trajectories of the system. In some limiting cases, we see their simplicity even permits some questions about the dynamcis to be answered analytically. Using these models, we make a series of experimentally verifiable predictions about the dynamics of unfolded polymers. / text

Methods of efficient Monte-Carlo simulation when rare events are involved have been studied for several decades. Rare events are very important in the context of evaluating high quality computer/communication systems. Meanwhile, the efficient simulation of systems involving rare events poses great challenges. A simulation method is said to be efficient if the number of replicas required to get accurate estimates grows slowly, compared to the rate at which the probability of the rare event approaches zero. Despite the great success of the two mainstream methods, importance sampling (IS) and importance splitting, either of them can become inefficient under certain conditions, as reported in some recent studies. The purpose of this study is to look for possible enhancement of fast simulation methods. I focus on the ``level/phase process', a Markov process in which the level and the phase are two state variables. Furthermore, changes of level and phase are induced by events, which have rates that are independent of the level except at a boundary. For such a system, the event of reaching a high level occurs rarely, provided the system typically stays at lower levels. The states at those high levels constitute the rare event set. Though simple, this models a variety of applications involving rare events. In this setting, I have studied two efficient simulation methods, the rate tilting method and the adaptive splitting method, concerning their efficiencies. I have compared the efficiency of rate tilting with several previously used similar methods. The experiments are done by using queues in tandem, an often used test bench for the rare event simulation. The schema of adaptive splitting has not been described in literature. For this method, I have analyzed its efficiency to show its superiority over the (conventional) splitting method. The way that a system approaches a designated rare event set is called the system's large deviation behavior. Toward the end of gaining insight about the relation of system behavior and the efficiency of IS simulation, I quantify the large deviation behavior and its complexity. This work indicates that the system's large deviation behavior has a significant impact on the efficiency of a simulation method.

Methods of efficient Monte-Carlo simulation when rare events are involved have been studied for several decades. Rare events are very important in the context of evaluating high quality computer/communication systems. Meanwhile, the efficient simulation of systems involving rare events poses great challenges. A simulation method is said to be efficient if the number of replicas required to get accurate estimates grows slowly, compared to the rate at which the probability of the rare event approaches zero. Despite the great success of the two mainstream methods, importance sampling (IS) and importance splitting, either of them can become inefficient under certain conditions, as reported in some recent studies. The purpose of this study is to look for possible enhancement of fast simulation methods. I focus on the ``level/phase process', a Markov process in which the level and the phase are two state variables. Furthermore, changes of level and phase are induced by events, which have rates that are independent of the level except at a boundary. For such a system, the event of reaching a high level occurs rarely, provided the system typically stays at lower levels. The states at those high levels constitute the rare event set. Though simple, this models a variety of applications involving rare events. In this setting, I have studied two efficient simulation methods, the rate tilting method and the adaptive splitting method, concerning their efficiencies. I have compared the efficiency of rate tilting with several previously used similar methods. The experiments are done by using queues in tandem, an often used test bench for the rare event simulation. The schema of adaptive splitting has not been described in literature. For this method, I have analyzed its efficiency to show its superiority over the (conventional) splitting method. The way that a system approaches a designated rare event set is called the system's large deviation behavior. Toward the end of gaining insight about the relation of system behavior and the efficiency of IS simulation, I quantify the large deviation behavior and its complexity. This work indicates that the system's large deviation behavior has a significant impact on the efficiency of a simulation method.

In this dissertation we present theoretical and experimental investigation of the performance quality of fiber optical communication systems, and find new and inexpansive ways of increasing the rate of theinformation transmission.The first part of this work discuss the two major factors limiting the quality of information channels in the fiber optical communication systems. Using methods of large deviation theory from statisticalphysics, we carry out analytical and numerical study of error statistics in optical communication systems in the presence of the temporal noise from optical amplifiers and the structural disorder of optical fibers. In the slowly varying envelope approximation light propagation through optical fiber is described by Schr\{o}dinger's equation. Signal transmission is impeded by the additive (amplifiers) and multiplicative (birefringence) noise This results in signal distortion that may lead to erroneous interpretation of the signal. System performance is characterized by the probability of error occurrence. Fluctuation of spacial disorder due to changing external factors (temperature, vibrations, etc) leads to fluctuations of error rates. Commonly the distribution of error rates is assumed to be Gaussian. Using the optimal fluctuation method we show that this distribution is in fact lognormal. Sucha distribution has ""fat"" tails implying that the likelihood of system outages is much higher than itwould be in the Gaussian approximation. We present experimental results that provide excellent confirmation of our theoretical predictions.In the second part of this dissertation we present some published work on bisolitons in the dispersion managed systems. Modern communication systems use light pulses to transmit tremendous amounts of information. These systems can be modeled using variations of the Nonlinear Shrodinger Equation where chromatic dispersion and nonlinear effects in the glass fiber are taken into account. The best system performance to date is achieved using dispersion management. We will see how the dispersion management works and how it can be modeled. As you pack information more tightly the interaction between the pulsesbecomes increasingly important. In Fall 2005, experiments in Germany showed that bound pairs of pulses (bisolitons) could propagate significant distances. Through numerical investigation we found parametric bifurcation of bisolitonic solutions, and developed a new iterative method with polynomial correction for the calculation of these solutions. Using these solutions in the signal transmission could increase the transmission rates.

<p>I denna avhandling studerar vi nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska processer.</p><p>I introduktionen ger vi nödvändiga grundläggande definitioner och beskrivningar av modeller som betraktas i avhandlingen, samt ger några exempel på situationer i vilka metoder av första-sällan-händelsetider kan vara lämpliga att använda. Dessutom analyserar vi publicerade resultat om asymptotiska problem för stokastiska funktionaler som definieras på semi-Markovska processer.</p><p>I artikel A betraktar vi första-sällan-händelsetider för semi-Markovska processer med en ändlig mängd av lägen. Vi ger också en sammanfattning av våra resultat om nödvändiga och tillräckliga villkor för svag konvergens, samt diskuterar möjliga tillämpningar inom aktuarie-området.</p><p>I artikel B redovisar vi i detalj de resultat som annonseras i artikel A och bevisen för dem. Vi ger också nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska processer med en ändlig mängd av lägen i ett icke-triangulärt tillstånd. Dessutom beskriver vi med hjälp av Laplacetransformationen klassen av alla möjliga gränsfördelningar.</p><p>I artikel C studerar vi villkor av svag konvergens av flöden av sällan-händelser i ett icke-triangulärt tillstånd. Vi formulerar nödvändiga och tillräckliga villkor för konvergens, och beskriver klassen av alla möjliga gränsflöden. Vi tillämpar också våra resultat i asymptotisk analys av icke-ruin-sannolikheten för störda riskprocesser.</p><p>I artikel D ger vi nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska rocesser med en ändlig mängd av lägen i ett triangulärt tillstånd, samt beskriver klassen av alla möjliga gränsfördelningar. Resultaten utvidgar slutsatser från artikel B till att gälla för ett allmänt triangulärt tillstånd.</p><p>I artikel E ger vi nödvändiga och tillräckliga villkor för svag konvergens av flöden av sällan-händelser för semi-Markovska processer i ett triangulärt tillstånd. Detta generaliserar resultaten från artikel C till att beskriva ett allmänt triangulärt tillstånd. Vidare ger vi tillämpningar av våra resultat på asymptotiska problem av störda riskprocesser och till kösystemen med snabb service.</p> / <p>In this thesis we study necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes, we describe the class of all possible limit distributions, and give the applications of the results to risk theory and queueing systems.</p><p>In paper <b>A</b>, we consider first-rare-event times for semi-Markov processes with a finite set of states, and give a summary of our results concerning necessary and sufficient conditions for weak convergence of first-rare-event times and their actuarial applications.</p><p>In paper <b>B</b>, we present in detail results announced in paper <b>A</b> as well as their proofs. We give necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in non-triangular-array mode and describe the class of all possible limit distributions in terms of their Laplace transforms.</p><p>In paper <b>C</b>, we study the conditions for weak convergence for flows of rare events for semi-Markov processes with a finite set of states in non-triangular array mode. We formulate necessary and sufficient conditions of convergence and describe the class of all possible limit stochastic flows. In the second part of the paper, we apply our results to the asymptotical analysis of non-ruin probabilities for perturbed risk processes.</p><p>In paper <b>D</b>, we give necessary and sufficient conditions for the weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in triangular array mode as well as describing the class of all possible limit distributions. The results of paper <b>D</b> extend results obtained in paper <b>B</b> to a general triangular array mode.</p><p>In paper <b>E</b>, we give the necessary and sufficient conditions for weak convergence for the flows of rare events for semi-Markov processes with a finite set of states in triangular array case. This paper generalizes results obtained in paper <b>C</b> to a general triangular array mode. In the second part of the paper, we present applications of our results to asymptotical problems of perturbed risk processes and to queueing systems with quick service</p>

I denna avhandling studerar vi nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska processer. I introduktionen ger vi nödvändiga grundläggande definitioner och beskrivningar av modeller som betraktas i avhandlingen, samt ger några exempel på situationer i vilka metoder av första-sällan-händelsetider kan vara lämpliga att använda. Dessutom analyserar vi publicerade resultat om asymptotiska problem för stokastiska funktionaler som definieras på semi-Markovska processer. I artikel A betraktar vi första-sällan-händelsetider för semi-Markovska processer med en ändlig mängd av lägen. Vi ger också en sammanfattning av våra resultat om nödvändiga och tillräckliga villkor för svag konvergens, samt diskuterar möjliga tillämpningar inom aktuarie-området. I artikel B redovisar vi i detalj de resultat som annonseras i artikel A och bevisen för dem. Vi ger också nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska processer med en ändlig mängd av lägen i ett icke-triangulärt tillstånd. Dessutom beskriver vi med hjälp av Laplacetransformationen klassen av alla möjliga gränsfördelningar. I artikel C studerar vi villkor av svag konvergens av flöden av sällan-händelser i ett icke-triangulärt tillstånd. Vi formulerar nödvändiga och tillräckliga villkor för konvergens, och beskriver klassen av alla möjliga gränsflöden. Vi tillämpar också våra resultat i asymptotisk analys av icke-ruin-sannolikheten för störda riskprocesser. I artikel D ger vi nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska rocesser med en ändlig mängd av lägen i ett triangulärt tillstånd, samt beskriver klassen av alla möjliga gränsfördelningar. Resultaten utvidgar slutsatser från artikel B till att gälla för ett allmänt triangulärt tillstånd. I artikel E ger vi nödvändiga och tillräckliga villkor för svag konvergens av flöden av sällan-händelser för semi-Markovska processer i ett triangulärt tillstånd. Detta generaliserar resultaten från artikel C till att beskriva ett allmänt triangulärt tillstånd. Vidare ger vi tillämpningar av våra resultat på asymptotiska problem av störda riskprocesser och till kösystemen med snabb service. / In this thesis we study necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes, we describe the class of all possible limit distributions, and give the applications of the results to risk theory and queueing systems. In paper <b>A</b>, we consider first-rare-event times for semi-Markov processes with a finite set of states, and give a summary of our results concerning necessary and sufficient conditions for weak convergence of first-rare-event times and their actuarial applications. In paper <b>B</b>, we present in detail results announced in paper <b>A</b> as well as their proofs. We give necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in non-triangular-array mode and describe the class of all possible limit distributions in terms of their Laplace transforms. In paper <b>C</b>, we study the conditions for weak convergence for flows of rare events for semi-Markov processes with a finite set of states in non-triangular array mode. We formulate necessary and sufficient conditions of convergence and describe the class of all possible limit stochastic flows. In the second part of the paper, we apply our results to the asymptotical analysis of non-ruin probabilities for perturbed risk processes. In paper <b>D</b>, we give necessary and sufficient conditions for the weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in triangular array mode as well as describing the class of all possible limit distributions. The results of paper <b>D</b> extend results obtained in paper <b>B</b> to a general triangular array mode. In paper <b>E</b>, we give the necessary and sufficient conditions for weak convergence for the flows of rare events for semi-Markov processes with a finite set of states in triangular array case. This paper generalizes results obtained in paper <b>C</b> to a general triangular array mode. In the second part of the paper, we present applications of our results to asymptotical problems of perturbed risk processes and to queueing systems with quick service

Background and Objectives: I investigated the use of sensitivity analyses in assessing statistical results or analytical approaches in three different statistical issues: (1) accounting for within-subject correlations in analyzing discrete choice data, (2) handling both-armed zero-event studies in meta-analyses for rare event outcomes, and (3) incorporating external information using Bayesian approach to estimate rare-event rates. Methods: Project 1: I empirically compared ten statistical models in analyzing correlated data from a discrete choice survey to elicit patient preference for colorectal cancer screening. Logistic and probit models with random-effects, generalized estimating equations or robust standard errors were applied to binary, multinomial or bivariate outcomes. Project 2: I investigated the impacts of including or excluding both-armed zero-event studies on pooled odds ratios for classical meta-analyses using simulated data. Five commonly used pooling methods: Peto, Mantel-Haenszel fixed/random effects and inverse variance fixed/random effects, were compared in terms of bias and precision. Project 3: I explored the use of Bayesian approach to incorporate external information through priors to verify, enhance or modify the study evidence. Three study scenarios were derived from previous studies to estimate inhibitor rates for hemophilia A patients treated with rAHF-PFM: 1) a single cohort of previously treated patients, 2) individual patient data meta-analysis, and 3) an previously unexplored patient population with limited data. Results and Conclusion: Project 1: When within-subject correlations were substantial, the results from different statistical models were inconsistent. Project 2: Including both-armed zero-event studies in meta-analyses increased biases for pooled odd ratios when true treatment effects existed. Project 3: Through priors, Bayesian approaches effectively incorporated different types of information to strengthen or broaden research evidence. Through this thesis I demonstrated that when analyzing complicated health research data, it was important to use sensitivity analyses to assess the robustness of analysis results or proper choice of statistical models. / Dissertation / Doctor of Philosophy (PhD)

Response adaptive designs intend to allocate more patients to better treatments without undermining the validity and the integrity of the trial. The immediacy of the primary response (e.g. deaths, remission) determines the efficiency of the response adaptive design, which often requires outcomes to be quickly or immediately observed. This presents difficulties for survival studies, which may require long durations to observe the primary endpoint. Therefore, we introduce auxiliary endpoints to assist the adaptation with the primary endpoint, where an auxiliary endpoint is generally defined as any measurement that is positively associated with the primary endpoint. Our proposed design (referred to as bivariate adaptive design) is based on the classical response adaptive design framework. The connection of auxiliary and primary endpoints is established through Bayesian method. We extend parameter space from one dimension to two dimensions, say primary and auxiliary efficacies, by implementing a conditional weigh function on the loss function of the design. The allocation ratio is updated at each stage by optimization of the loss function subject to the information provided for both the auxiliary and primary outcomes. We demonstrate several methods of joint modeling the auxiliary and primary outcomes. Through simulation studies, we show that the bivariate adaptive design is more effective in assigning patients to better treatments as compared with univariate optimal and balanced designs. As hoped, this joint-approach also reduces the expected number of patient failures and preserves the comparable power as compared with other designs.

Rare-event phenomena are ubiquitous in nature. We propose a new strategy, kappa-dynamics, to model rare event dynamics. In this methodology we only assume that the important rare-event dynamics obey first-order kinetics. Exact rates are not required in the calculation and the reaction path is determined on the fly. kappa-dynamics is highly parallelizable and can be implemented on computer clusters and distributed machines. Theoretical derivations and several examples of atomic scale dynamics are presented. With single-molecule (SM) techniques, the individual molecular process can be resolved without being averaged over the ensemble. However, factors such as apparatus stability, background level, and data quality will limit the amount of information being collected. We found that the correlation function calculated from the finite-size SM rotational diffusion trajectory will deviate from its true value. Therefore, care must be taken not to interpret the difference as the evidence of new dynamics occurred in the system. We also proposed an algorithm of single fluorophore orientation reconstruction which converts three measured intensities {I₀,I₄₅,I₉₀} to the dipole orientation. Fluctuations in the detected signals caused by the shot noise result in a different prediction from the true orientation. This difference should not be interpreted as the evidence of the nonisotropic rotational motion. Catalytic reactions are also governed by the rare-events. Studying the dynamics of catalytic processes is an important subject since the more we learn, the more we can improve current catalysts. Fuel cells have become a promising energy source in the past decade. The key to make a high performance cell while keeping the price low is the choice of a suitable catalyst at the electrodes. Density functional theory calculations are carried out to study the role of geometric relaxation in the oxygen-reduction reaction for nanoparticle of various transition metals. Our calculations of Pt nanoparticles show that the structural deformation induced by atomic oxygen binding can energetically stabilize the oxidized states and thus reduces the catalytic activity. The catalytic performance can be improved by making alloys with less deformable metals. / text

L’évaluation de risques est conditionnée par les connaissances et les informations disponibles au moment où l’analyse est faite. La modélisation et la simulation sont des moyens d’explorer et de comprendre le comportement du système, d’identifier des scénarios critiques et d’éviter des surprises. Un certain nombre de simulations du modèle sont exécutées avec des conditions initiales et opérationnelles différentes pour identifier les scénarios conduisant à des conséquences critiques et pour estimer leurs probabilités d’occurrence. Pour les systèmes complexes, les modèles de simulations peuvent être : i) de haute dimension ; ii) boite noire ; iii) dynamiques ; iv) coûteux en termes de calcul, ce qu’empêche l’analyste d’exécuter toutes les simulations pour les conditions multiples qu’il faut considérer.La présente thèse introduit des cadres avancés d’évaluation des risques basée sur les simulations. Les méthodes développées au sein de ces cadres sont attentives à limiter les coûts de calcul requis par l’analyse, afin de garder une scalabilité vers des systèmes complexes. En particulier, toutes les méthodes proposées partagent l’idée prometteuse de focaliser automatiquement et de conduire d’une manière adaptive les simulations vers les conditions d’intérêt pour l’analyse, c’est-à-dire, vers des informations utiles pour l'évaluation des risques.Les avantages des méthodes proposées ont été montrés en ce qui concerne différentes applications comprenant, entre autres, un sous-réseau de transmission de gaz, un réseau électrique et l’Advanced Lead Fast Reactor European Demonstrator (ALFRED). / Risk assessment is conditioned on the knowledge and information available at the moment of the analysis. Modeling and simulation are ways to explore and understand system behavior, for identifying critical scenarios and avoiding surprises. A number of simulations of the model are run with different initial and operational conditions to identify scenarios leading to critical consequences and to estimate their probabilities of occurrence. For complex systems, the simulation models can be: i) high-dimensional; ii) black-box; iii) dynamic; and iv) computationally expensive to run, preventing the analyst from running the simulations for the multiple conditions that need to be considered.The present thesis presents advanced frameworks of simulation-based risk assessment. The methods developed within the frameworks are attentive to limit the computational cost required by the analysis, in order to keep them scalable to complex systems. In particular, all methods proposed share the powerful idea of automatically focusing and adaptively driving the simulations towards those conditions that are of interest for the analysis, i.e., for risk-oriented information.The advantages of the proposed methods have been shown with respect to different applications including, among others, a gas transmission subnetwork, a power network and the Advanced Lead Fast Reactor European Demonstrator (ALFRED).