# Browsing by Subject "Mathematics"

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Item Open Access (1,1) L-space knots(COMPOSITIO MATHEMATICA, 2018-05-01) Greene, JE; Lewallen, S; Vafaee, FWe characterize the (1, 1) knots in the three-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non- trivial L-space surgeries. We also recover a characterization of the Berge manifold amongst 1-bridge braid exteriors.Item Open Access A classical proof that the algebraic homotopy class of a rational function is the residue pairing(Linear Algebra and Its Applications, 2020-06-15) Kass, JL; Wickelgren, K© 2020 Elsevier Inc. Cazanave has identified the algebraic homotopy class of a rational function of 1 variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functions in 1 variable. Finally, we announce results answering this question for functions in an arbitrary number of variables.Item Open Access A Deep Learning Model for V50%, V60%, and V66.7% Prediction in LINAC-based Treatment Planning of Single-Iso-Multiple-Targets (SIMT) Stereotactic Radiosurgery (SRS)(2023) Khazaieli, MercedehBrain metastases are a common complication of many types of cancer, including lung, breast, and melanoma. Approximately 30-40% of patients develop brain metastases that originate from primary systemic tumors during the course of cancer treatment. One treatment method is a LINAC-based single-isocenter multiple-target (SIMT) stereotactic radiosurgery (SRS). High plan quality has been one of the important goals in radiotherapy treatment planning. Generation of a high quality SRS treatment plan, particularly a SIMT plan, usually requires planners’ extensive planning experience, multiple runs of planning and trial-and-error, and frequent communication among planners, physicians and other radiation oncology team members. In clinical practice with potentially limited resources, SIMT SRS planning could be time-consuming and may have large variations in plan dosimetric quality. Therefore, an estimation of achievable dosimetric outcome can help reduce plan quality variation and improve planning efficiency. Assuming 20Gy in a single fraction of treatment, the volume of normal brain tissue receiving 10Gy (V50%), 12Gy (V60%), and 13Gy (V66.7%) are known predictors of brain tissue toxicity, or radionecrosis. We developed deep learning networks for the prediction of V50%, V60%, and V66.7% based on each patient’s target delineation. A prediction of achievable V10Gy, V12Gy, and V13Gy (assuming 20Gy x 1fx) can assist physicians in the determination of fractionation schemes (i.e., single fx vs. multiple fx). Such predictions can be used as guidelines for planners to generate a SIMT plan more rapidly with reduced dosimetric variability. A key technical innovation of this work is the spherical projection design: by projecting target distribution on a spherical surface, the target distribution in 3D is collapsed to a polar-azimuthal angular distribution map. This transformation enables a dimensional reduction for deep learning input without losing volumetric information. Our results indicate promising potential but there is a need for further work to improve the accuracy of our predictions.

Item Open Access A Dynamical Nephrovascular Model of Renal Autoregulation(2014) Sgouralis, IoannisThe main functions of the kidney take place in the nephrons. For their proper operation, nephrons need to be supplied with a stable blood flow that remains constant despite fluctuations of arterial pressure. Such stability is provided by the afferent arterioles, which are unique vessels in the kidney capable of adjusting diameter. By doing so, afferent arterioles regulate blood delivery downstream, where the nephrons are located. The afferent arterioles respond to signals initiated by two mechanisms: the myogenic response which operates to absorb pressure perturbations within the vasculature, and tubuloglomerular feedback which operates to stabilize salt reabsorption.

In this thesis, a mathematical model of the renal nephrovasculature that represents both mechanisms in a dynamical context is developed. For this purpose, de- tailed representations of the myogenic mechanism of vascular smooth muscles and the tubular processes are developed and combined in a single comprehensive model. The resulting model is formulated with a large number of ordinary differential equations that represent the intracellular processes of arteriolar smooth muscles, coupled with a number of partial differential equations, mainly of the advection-diffusion-reaction type, that represent blood flow, glomerular filtration and the tubular processes. Due to its unique activation characteristics, the myogenic response is formulated with a set of delay differential equations.

The model is utilized to assess a verity of physiological phenomena: the conduction of vasomotor responses along the afferent arteriole, autoregulation under physiologic as well as pathophysiologic conditions, and renal oxygenation. A first attempt to model the impact of diabetes mellitus on renal hemodynamics is also made. Further, an application with clinical significance is presented. Namely, renal oxygenation is estimated under conditions that simulate those observed during cardiopulmonary surgery. Results indicate the development of renal hypoxia, which suggests an important pathway for the development of acute kidney injury.

Item Open Access A Generalized Lyapunov Construction for Proving Stabilization by Noise(2012) Kolba, Tiffany NicoleNoise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.

The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.

Item Open Access A new fully automated approach for aligning and comparing shapes.(Anatomical record (Hoboken, N.J. : 2007), 2015-01) Boyer, Doug M; Puente, Jesus; Gladman, Justin T; Glynn, Chris; Mukherjee, Sayan; Yapuncich, Gabriel S; Daubechies, IngridThree-dimensional geometric morphometric (3DGM) methods for placing landmarks on digitized bones have become increasingly sophisticated in the last 20 years, including greater degrees of automation. One aspect shared by all 3DGM methods is that the researcher must designate initial landmarks. Thus, researcher interpretations of homology and correspondence are required for and influence representations of shape. We present an algorithm allowing fully automatic placement of correspondence points on samples of 3D digital models representing bones of different individuals/species, which can then be input into standard 3DGM software and analyzed with dimension reduction techniques. We test this algorithm against several samples, primarily a dataset of 106 primate calcanei represented by 1,024 correspondence points per bone. Results of our automated analysis of these samples are compared to a published study using a traditional 3DGM approach with 27 landmarks on each bone. Data were analyzed with morphologika(2.5) and PAST. Our analyses returned strong correlations between principal component scores, similar variance partitioning among components, and similarities between the shape spaces generated by the automatic and traditional methods. While cluster analyses of both automatically generated and traditional datasets produced broadly similar patterns, there were also differences. Overall these results suggest to us that automatic quantifications can lead to shape spaces that are as meaningful as those based on observer landmarks, thereby presenting potential to save time in data collection, increase completeness of morphological quantification, eliminate observer error, and allow comparisons of shape diversity between different types of bones. We provide an R package for implementing this analysis.Item Open Access A NEW ZEROTH-ORDER ORACLE FOR DISTRIBUTED AND NON-STATIONARY LEARNING(2021) Zhang, YanZeroth-Order (ZO) methods have been applied to solve black-box or simulation-based optimization prroblems. These problems arise in many important applications nowa- days, e.g., generating adversarial attacks on machine learning systems, learning to control the system with complicated physics structure or human in the loop. In these problem settings, the objective function to optimize does not have an explicit math- ematical form and therefore its gradient cannot be obtained. This invalidates all gradient-based optimization approaches. On the other hand, ZO methods approxi- mate the gradient by using the objective function values. Many existing ZO methods adopt the two-point feedback scheme to approximate the unknown gradient due to its low estimation variance and fast convergence speed. Specifically, two-point ZO method estimates the gradient at the current iterate of the algorithm by querying the objective function value twice at two distinct neighbor points around the current iterate. Such scheme becomes infeasible or difficult to implement when the objective function is time-varying, or when multiple agents collaboratively optimize a global objective function that depends on all agents’ decisions, because the value of the objective function can be queried only once at a single decision point. However, con- ventional ZO method based on one-point feedback is subject to large variance of the gradient estimation and therefore slows down the convergence.

In this dissertation, we propose a novel one-point ZO method based on the residual feedback. Specifically, the residual feedback scheme estimates the gradient using the residual between the values of the objective function at two consecutive iterates of the algorithm. When optimizing a deterministic Lipschitz function, we show that the query complexity of ZO with the proposed one-point residual feedback matches that of ZO with the existing two-point schemes. Moreover, the query complexity of the proposed algorithm can be improved when the objective function has Lipschitz gradient. Then, for stochastic bandit optimization problems, we show that ZO with one-point residual feedback achieves the same convergence rate as that of two-point scheme with uncontrollable data samples.

Next, we apply the proposed one-point residual-feedback gradient estimator to solve online optimizaiton problems, where the objective function varies over time. In the online setting, since each objective function can only be evaluated once at a single decision point, existing two-point ZO methods are not feasible and only one-point ZO methods can be used. We develop regret bounds for ZO with the proposed one- point residual feedback scheme for both convex and nonconvex online optimization problems. Specifically, for both deterministic and stochastic problems and for both Lipschitz and smooth objective functions, we show that using residual feedback can produce gradient estimates with much smaller variance compared to conventional one- point feedback methods. As a result, our regret bounds are much tighter compared to existing regret bounds for ZO with conventional one-point feedback, which suggests that ZO with residual feedback can better track the optimizer of online optimization problems. Additionally, our regret bounds rely on weaker assumptions than those used in conventional one-point feedback methods.

The proposed residual-feedback scheme is next decentralized to conduct dis- tributed policy optimization in the multi-agent reinforcement learning (MARL) prob- lems. Existing MARL algorithms often assume that every agent can observe the states and actions of all the other agents in the network. This can be impractical in large-scale problems, where sharing the state and action information with multi-hop neighbors may incur significant communication overhead. The advantage of the pro- posed zeroth-order policy optimization method is that it allows the agents to compute the local policy gradients needed to update their local policy functions using local estimates of the global accumulated rewards that depend on partial state and action information only and can be obtained using consensus. Specifically, the local ZO policy gradients relying on one-point residual-feedback significantly reduces the vari- ance of the local policy gradient estimates compared to the conventional one-point policy gradient estimates, improving, in this way, the learning performance. We show that the proposed distributed zeroth-order policy optimization method with constant stepsize converges to a neighborhood of the global optimal policy that depends on the number of consensus steps used to calculate the local estimates of the global accumulated rewards.Another challenge in the distributed ZO optimization problems is that the agents may conduct local updates in an asynchronous fashion when they do not have ac- cess to a global clock. To deal with this challenge, we propose an asynchronous zeroth-order distributed optimization method that relies on the proposed one-point residual feedback gradient estimator. We show that this estimator is unbiased under asynchronous updating, and theoretically analyze its convergence. We demonstrate the effectiveness of all proposed algorithms via extensive numerical experiments.

Item Open Access A slicing obstruction from the $\frac {10}{8}$ theorem(Proceedings of the American Mathematical Society, 2016-08-29) Donald, A; Vafaee, F© 2016 American Mathematical Society. From Furuta’s 10/8 theorem, we derive a smooth slicing obstruction for knots in S3 using a spin 4-manifold whose boundary is 0-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly slice.Item Open Access A Spectral Deferred Correction Method for Solving Cardiac Models(2011) Bowen, Matthew M.Many numerical approaches exist to solving models of electrical activity in the heart. These models consist of a system of stiff nonlinear ordinary differential equations for the voltage and other variables governing channels, with the voltage coupled to a diffusion term. In this work, we propose a new algorithm that uses two common discretization methods, operator splitting and finite elements. Additionally, we incorporate a temporal integration process known as spectral deferred correction. Using these approaches,

we construct a numerical method that can achieve arbitrarily high order in both space and time in order to resolve important features of the models, while gaining accuracy and efficiency over lower order schemes.

Our algorithm employs an operator splitting technique, dividing the reaction-diffusion systems from the models into their constituent parts.

We integrate both the reaction and diffusion pieces via an implicit Euler method. We reduce the temporal and splitting errors by using a spectral deferred correction method, raising the temporal order and accuracy of the scheme with each correction iteration.

Our algorithm also uses continuous piecewise polynomials of high order on rectangular elements as our finite element approximation. This approximation improves the spatial discretization error over the piecewise linear polynomials typically used, especially when the spatial mesh is refined.

As part of these thesis work, we also present numerical simulations using our algorithm of one of the cardiac models mentioned, the Two-Current Model. We demonstrate the efficiency, accuracy and convergence rates of our numerical scheme by using mesh refinement studies and comparison of accuracy versus computational time. We conclude with a discussion of how our algorithm can be applied to more realistic models of cardiac electrical activity.

Item Open Access A stochastic-Lagrangian particle system for the Navier-Stokes equations(Nonlinearity, 2008-11-01) Iyer, Gautam; Mattingly, JonathanThis paper is based on a formulation of the Navier-Stokes equations developed by Constantin and the first author (Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with 1/N times the sum over these N copies. (We note that our formulation requires one to keep track of N stochastic flows of diffeomorphisms, and not just the motion of N particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space C1,α which consists of differentiable functions whose first derivative is α Hölder continuous (see section 3 for the precise definition). Further, we show that as N → ∞ the system converges to the solution of Navier-Stokes equations on any finite interval [0, T]. However for fixed N, we prove that this system retains roughly O(1/N) times its original energy as t → ∞. Hence the limit N → ∞ and T → ∞ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly. © 2008 IOP Publishing Ltd and London Mathematical Society.Item Open Access A Study of Edge Toric Ideals using Associated Graphs(2012-04-26) Shen, YingyiThis thesis studies properties of edge toric ideals and resolutions by analyzing the associated graphs of algebraic structures. It mainly focused on proving that the repeated edges in a graph wouldn't change some properties of its underlying algebraic structure. An application of this result is that when we study multi-edge graphs, we can simplify in nite numbers of graphs to a simple one by deleting all the repeated edges.Item Open Access A Third Order Numerical Method for Doubly Periodic Electromegnetic Scattering(2007-07-31) Nicholas, Michael JWe here developed a third-order accurate numerical method for scattering of 3D electromagnetic waves by doubly periodic structures. The method is an intuitively simple numerical scheme based on a boundary integral formulation. It involves smoothing the singular Green's functions in the integrands and finding correction terms to the resulting smooth integrals. The analytical method is based on the singular integral methods of J. Thomas Beale, while the scattering problem is motivated by the 2D work of Stephanos Venakides, Mansoor Haider, and Stephen Shipman. The 3D problem was done with boundary element methods by Andrew Barnes. We present a method that is both more straightforward and more accurate. In solving these problems, we have used the M\"{u}ller integral equation formulation of Maxwell's equations, since it is a Fredholm integral equation of the second kind and is well-posed. M\"{u}ller derived his equations for the case of a compact scatterer. We outline the derivation and adapt it to a periodic scatterer. The periodic Green's functions found in the integral equation contain singularities which make it difficult to evaluate them numerically with accuracy. These functions are also very time consuming to evaluate numerically. We use Ewald splitting to represent these functions in a way that can be computed rapidly.We present a method of smoothing the singularity of the Green's function while maintaining its periodicity. We do local analysis of the singularity in order to identify and eliminate the largest sources of error introduced by this smoothing. We prove that with our derived correction terms, we can replace the singular integrals with smooth integrals and only introduce a error that is third order in the grid spacing size. The derivation of the correction terms involves transforming to principal directions using concepts from differential geometry. The correction terms are necessarily invariant under this transformation and depend on geometric properties of the scatterer such as the mean curvature and the differential of the Gauss map. Able to evaluate the integrals to a higher order, we implement a \mbox{GMRES} algorithm to approximate solutions of the integral equation. From these solutions, M\"{u}ller's equations allow us to compute the scattered fields and transmission coefficients. We have also developed acceleration techniques that allow for more efficient computation.We provide results for various scatterers, including a test case for which exact solutions are known. The implemented method does indeed converge with third order accuracy. We present results for which the method successfully resolves Wood's anomaly resonances in transmission.Item Open Access A Variational Framework for Phase-Field Fracture Modeling with Applications to Fragmentation, Desiccation, Ductile Failure, and Spallation(2021) Hu, TianchenFracture is a common phenomenon in engineering applications. Many types of fracture exist, including, but not limited to, brittle fracture, quasi-brittle fracture, cohesive fracture, and ductile fracture. Predicting fracture has been one of the most challenging research topics in computational mechanics. The variational treatment of fracture and its associated phase-field regularization have been employed with great success for modeling fracture in brittle materials. Extending the variational statement to describe other types of fracture and coupled field phenomena has proven less straightforward. Main challenges that remain include how to best construct a total potential that is both mathematically sound and physically admissible, and how to properly describe the coupling between fracture and other phenomena.

The research presented in this dissertation aims at addressing the aforementioned challenges. A variational framework is proposed to describe fracture in general dissipative solids. In essence, the variational statement is extended to account for large deformation kinematics, inelastic deformation, dissipation mechanisms, dynamic effects, and thermal effects. The proposed variational framework is shown to be consistent with conservations and laws of thermodynamics, and it provides guidance and imposes restrictions on the construction of models for coupled field problems. Within the proposed variational framework, several models are instantiated to address practical engineering problems. A brittle and quasi-brittle fracture model is used to investigate fracture evolution in polycrystalline materials; a cohesive fracture model is applied to revisit soil desiccation; a novel ductile fracture model is proposed and successfully applied to simulate some challenging benchmark problems; and a creep fracture model is developed to simulate the spallation of oxide scale on high temperature heat exchangers.

Item Open Access Absolute Continuity of Singular SPDEs and Bayesian Inference on Dynamical Systems(2023) Su, LangxuanWe explore the interplay among probability, stochastic analysis, and dynamical systems through two lenses: (1) absolute continuity of singular stochastic partial differential equations (SPDEs); (2) Bayesian inference on dynamical systems.

In the first part, we prove that up to a certain singular regime, the law of the stochastic Burgers equation at a fixed time is absolutely continuous with respect to the corresponding stochastic heat equation with the nonlinearity removed. The results follow from an extension of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time. To deal with the singularity, we introduce a novel decomposition in the spirit of Da Prato-Debussche and Gaussian chaos decomposition in singular SPDEs, by separating out the noise into different levels of regularity, along with a number of renormalization techniques. The number of levels in this decomposition diverges to infinite as we move to the stochastic Burgers equation associated with the KPZ equation. This result illustrates the fundamental probabilistic structure of a class of singular SPDEs and a notion of ``ellipticity'' in the infinite-dimensional setting.

In the second part, we establish connections between large deviations and a class of generalized Bayesian inference procedures on dynamical systems. We show that posterior consistency can be derived using a combination of classical large deviation techniques, such as Varadhan's lemma, conditional/quenched large deviations, annealed large deviations, and exponential approximations. We identified the divergence term as the Donsker-Varadhan relative entropy rate, also related to the Kolmogorov-Sinai entropy in ergodic theory. As an application, we prove new quenched/annealed large deviation asymptotics and a new Bayesian posterior consistency result for a class of mixing stochastic processes. In the case of Markov processes, one can obtain explicit conditions for posterior consistency, when estimates for log-Sobolev constants are available, which makes our framework essentially a black box. We also recover state-of-the-art posterior consistency on classical dynamical systems with a simple proof. Our approach has the potential of proving posterior consistency for a wide range of Bayesian procedures in a unified way.

Item Open Access ADAPTIVE LOCAL REDUCED BASIS METHOD FOR RISK-AVERSE PDE CONSTRAINED OPTIMIZATION AND INVERSE PROBLEMS(2018) Zou, ZilongMany physical systems are modeled using partial dierential equations (PDEs) with uncertain or random inputs. For such systems, naively propagating a xed number of samples of the input probability law (or an approximation thereof) through the PDE is often inadequate to accurately quantify the risk associated with critical system responses. In addition, to manage the risk associated with system response and devise risk-averse controls for such PDEs, one must obtain the numerical solution of a risk-averse PDE-constrained optimization problem, which requires substantial computational eorts resulting from the discretization of the underlying PDE in both the physical and stochastic dimensions.

Bayesian Inverse problem, where unknown system parameters need to be inferred from some noisy data of the system response, is another important class of problems that suffer from excessive computational cost due to the discretization of the underlying PDE. To accurately characterize the inverse solution and quantify its uncertainty, tremendous computational eorts are typically required to sample from the posterior distribution of the system parameters given the data. Surrogate approximation of the PDE model is an important technique to expedite the inference process and tractably solve such problems.

In this thesis, we develop a goal-oriented, adaptive sampling and local reduced basis approximation for PDEs with random inputs. The method, which we denote by local RB, determines a set of samples and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. The local basis in a Voronoi cell is composed of the solutions at a xed number of closest samples as well as the gradient information in that cell. Thanks to the local nature of the method, computational cost of the approximation does not increase as more samples are included in the local RB model. We select the local RB samples in an adaptive and greedy manner using an a posteriori error indicator based on the residual of the approximation.

Additionally, we modify our adaptive sampling process using an error indicator that is specifically targeted for the approximation of coherent risk measures evaluated at quantities of interest depending on PDE solutions. This allow us to tailor our method to efficiently quantify the risk associated with the system responses. We then combine our local RB method with an inexact trust region method to eciently solve risk-averse optimization problems with PDE constraints. We propose a numerical framework for systematically constructing surrogate models for the trust-region subproblem and the objective function using local RB approximations.

Finally, we extend our local RB method to eciently approximate the Gibbs posterior distribution for inverse problems under uncertainty. The local RB method is employed to construct a cheap surrogate model for the loss function in the Gibbs posterior formula. To improve the accuracy of the surrogate approximation, we adopt a Sequential Monte Carlo framework to guide the progressive and adaptive construction of the local RB surrogate. The resulted method provides subjective and ecient inference of unknown system parameters under general distribution and noise assumptions.

We provide theoretical error bounds for our proposed local RB method and its extensions, and numerically demonstrate the performance of our methods through various examples.

Item Open Access Advances in Choquet theories(2022) Caprio, MicheleChoquet theory and the theory of capacities, both initiated by French mathematician Gustave Choquet, share the heuristic notion of studying the extrema of a convex set in order to give interesting results regarding its elements. In this work, we put to use Choquet theory in the study of finite mixture models and the theory of capacities in studying severe uncertainty.

In chapter 2, we show how by combining a classical non-parametric density estimator based on a Pólya tree with techniques from Choquet theory, it is possible to retrieve the weights of a finite mixture model. We also give the rate of convergence of the Pólya tree posterior to the Dirac measure on the weights.

In chapter 3, we introduce dynamic probability kinematics (DPK), a method for an agent to mechanically update subjective beliefs in the presence of partial information. We then generalize it to dynamic imprecise probability kinematics (DIPK), which allows the agent to express their initial beliefs via a set of probabilities. We provide bounds for the lower probability associated with the updated probability sets, and we study the behavior of the latter, in particular contraction, dilation, and sure loss. Examples are provided to illustrate how the methods work. We also formulate in chapter 4 an ergodic theory for the limit of the sequence of successive DIPK updates. As a consequence, we formulate a strong law of large numbers.

Finally, in chapter 5 we propose a new, more general definition of extended probability measures ("probabilities" whose codomain is the interval [-1,1]). We study their properties and provide a behavioral interpretation. We use them in an inference procedure, whose environment is canonically represented by a probability space, when both the probability measure and the composition of the state space are unknown. We develop an ex ante analysis - taking place before the statistical analysis requiring knowledge of the state space - in which we progressively learn its true composition. We describe how to update extended probabilities in this setting, and introduce the concept of lower extended probabilities. We provide two examples in the fields of ecology and opinion dynamics.

Item Open Access Advances in Log-concave Sampling and Domain Adaptation(2023) Wu, KeruIn the vast field of machine learning, the study of distributions and the innovation of algorithms driving from them serve as foundation pursuits. This dissertation delves into two critical topics tied to distributions: log-concave sampling and domain adaptation. Our research encompasses both theoretical and methodological developments in these fields, supported by thorough numerical experiments.

Beginning with log-concave sampling, we establish the minimax mixing time of the Metropolis-adjusted Langevin algorithm (MALA) for sampling from a log-smooth and strongly log-concave distribution. Our proof is divided into two main components: an upper bound and a lower bound. First, for a d-dimensional log-concave density with condition number $\kappa$, we show that MALA with a warm start mixes in $\tilde O(\kappa\sqrt(d))$ iterations up to logarithmic factors. Second, we prove a spectral gap based mixing time lower bound for reversible MCMC algorithms on general state spaces. We apply this lower bound result to construct a hard distribution for which MALA requires at least $\tilde\Omega(\kappa\sqrt{d})$ steps to mix. The lower bound aligns with our upper bound in terms of condition number and dimension, thereby demonstrating the minimax mixing time of MALA for log-concave sampling.

Shifting focus to Domain Adaptation (DA), we tackle the statistical learning problem where the distribution of the source data used to train a model differs from that of the target data used to evaluate the model. Our work is grounded in assumption of the presence of conditionally invariant components (CICs) --- feature representations that are relevant for prediction and remain conditionally invariant across the source and target distributions. We demonstrate three prominent roles of CICs in providing target risk guarantees in DA. First, we propose a new algorithm based on CICs, importance-weighted conditional invariant penalty (IW-CIP), which has target risk guarantees beyond simple settings such as covariate shift and label shift. Second, we show that CICs help identify large discrepancies between source and target risks of other DA algorithms. Finally, we demonstrate that incorporating CICs into the domain invariant projection (DIP) algorithm can address its failure scenario caused by label-flipping features.

Through rigorous analysis, we advance insights into log-concave sampling and domain adaptation. Our exploration underscores the importance of probabilistic understanding in designing algorithms tailored to intricate data distributions encountered in machine learning.

Item Open Access Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves(2018) Luo, MaTo study periods of fundamental groups of algebraic varieties, one requires an explicit algebraic de Rham theory for completions of fundamental groups. This thesis develops such a theory in two cases. In the first case, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity. In the second case, we develop an algebriac de Rham theory for relative completion of the fundamental group of the moduli space of elliptic curves with one marked point. This allows the construction of iterated integrals involving modular forms of the second kind, whereas previously Brown and Manin only studied iterated integrals of holomorphic modular forms.

Item Open Access Algorithms for the Reeb Graph and Related Concepts(2014) Parsa, SalmanThis thesis is concerned with a structure called the Reeb graph. There are three main problems considered. The first is devising an efficient algorithm for comnstructing the Reeb graph of a simplicial complex with respect to a generic simplex-wise linear real-valued function. We present an algorithm that builds the Reeb graph in almost optimal worst-case deterministic time. This was the first deterministic algorithm with the time bound which is linear up to a logarithmic factor. Without loss of generality, the complex is assumed to be 2-dimensional. The algorithm works by sweeping the function values and maintaining a spanning forest for the preimage, or the level-set, of the value. Using the observation that the operations that change the level-sets are off-line, we was able to achieve the above bound.

The second topic is the dynamic Reeb graphs. As the function changes its values, the Reeb graph also changes. We are interested in updating the Reeb graph. We reduce this problem to a graph problem that we call retroactive graph connectivity. We obtain algorithms for dynamic Reeb graphs, by using data structures that solve the graph problem.

The third topic is an argument regarding the complexity of computing Betti numbers. This problem is also related to the Reeb graph by means of the vertical Homology classes. The problem considered here is whether the realization of a simplicial complex in the Euclidean 4-space can result in an algorithm for computing its Homology groups faster than the usual matrix reduction methods. Using the observation that the vertical Betti numbers can always be computed more efficiently using the Reeb graph, we show that the answer to this question is in general negative. For instance, given a square matrix over the field with two elements, we construct a simplicial complex in linear time, realized in euclidean 4-space and a function on it, such that its Horizontal homology group, over the same field, is isomorphic with the null-space of the matrix. It follows that the Betti number computation for complexes realized in the 4-space is as hard as computing the rank for a sparse matrix.

Item Open Access An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability(IMA Journal of Numerical Analysis, 2007-01-01) Lamba, H; Mattingly, JC; Stuart, AMThe understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded, then explicit fixed time step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself. © The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.