1,039 research outputs found

    Semiparametric posterior corrections

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    We present a new approach to semiparametric inference using corrected posterior distributions. The method allows us to leverage the adaptivity, regularization and predictive power of nonparametric Bayesian procedures to estimate low-dimensional functionals of interest without being restricted by the holistic Bayesian formalism. Starting from a conventional nonparametric posterior, we target the functional of interest by transforming the entire distribution with a Bayesian bootstrap correction. We provide conditions for the resulting one-step posterior\textit{one-step posterior} to possess calibrated frequentist properties and specialize the results for several canonical examples: the integrated squared density, the mean of a missing-at-random outcome, and the average causal treatment effect on the treated. The procedure is computationally attractive, requiring only a simple, efficient post-processing step that can be attached onto any arbitrary posterior sampling algorithm. Using the ACIC 2016 causal data analysis competition, we illustrate that our approach can outperform the existing state-of-the-art through the propagation of Bayesian uncertainty.Comment: 53 page

    Asymptotics of stochastic learning in structured networks

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    Beam scanning by liquid-crystal biasing in a modified SIW structure

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    A fixed-frequency beam-scanning 1D antenna based on Liquid Crystals (LCs) is designed for application in 2D scanning with lateral alignment. The 2D array environment imposes full decoupling of adjacent 1D antennas, which often conflicts with the LC requirement of DC biasing: the proposed design accommodates both. The LC medium is placed inside a Substrate Integrated Waveguide (SIW) modified to work as a Groove Gap Waveguide, with radiating slots etched on the upper broad wall, that radiates as a Leaky-Wave Antenna (LWA). This allows effective application of the DC bias voltage needed for tuning the LCs. At the same time, the RF field remains laterally confined, enabling the possibility to lay several antennas in parallel and achieve 2D beam scanning. The design is validated by simulation employing the actual properties of a commercial LC medium

    Singularity Formation in the High-Dimensional Euler Equations and Sampling of High-Dimensional Distributions by Deep Generative Networks

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    High dimensionality brings both opportunities and challenges to the study of applied mathematics. This thesis consists of two parts. The first part explores the singularity formation of the axisymmetric incompressible Euler equations with no swirl in ℝⁿ, which is closely related to the Millennium Prize Problem on the global singularity of the Navier-Stokes equations. In this part, the high dimensionality contributes to the singularity formation in finite time by enhancing the strength of the vortex stretching term. The second part focuses on sampling from a high-dimensional distribution using deep generative networks, which has wide applications in the Bayesian inverse problem and the image synthesis task. The high dimensionality in this part becomes a significant challenge to the numerical algorithms, known as the curse of dimensionality. In the first part of this thesis, we consider the singularity formation in two scenarios. In the first scenario, for the axisymmetric Euler equations with no swirl, we consider the case when the initial condition for the angular vorticity is Cα Hölder continuous. We provide convincing numerical examples where the solutions develop potential self-similar blow-up in finite time when the Hölder exponent α &lt; α*, and this upper bound α* can asymptotically approach 1 - 2/n. This result supports a conjecture from Drivas and Elgindi [37], and generalizes it to the high-dimensional case. This potential blow-up is insensitive to the perturbation of initial data. Based on assumptions summarized from numerical experiments, we study a limiting case of the Euler equations, and obtain α* = 1 - 2/n which agrees with the numerical result. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the Euler equations. This one-dimensional model might suggest a possible way to show this finite-time blow-up scenario analytically. Compared to the first proved blow-up result of the 3D axisymmetric Euler equations with no swirl and Hölder continuous initial data by Elgindi in [40], our potential blow-up scenario has completely different scaling behavior and regularity of the initial condition. In the second scenario, we consider using smooth initial data, but modify the Euler equations by adding a factor Δ as the coefficient of the convection terms to weaken the convection effect. The new model is called the weak convection model. We provide convincing numerical examples of the weak convection model where the solutions develop potential self-similar blow-up in finite time when the convection strength Δ &lt; Δ*, and this upper bound Δ* should be close to 1 - 2/n. This result is closely related to the infinite-dimensional case of an open question [37] stated by Drivas and Elgindi. Our numerical observations also inspire us to approximate the weak convection model with a one-dimensional model. We give a rigorous proof that the one-dimensional model will develop finite-time blow-up if Δ &lt; 1 - 2/n, and study the approximation quality of the one-dimensional model to the weak convection model numerically, which could be beneficial to a rigorous proof of the potential finite-time blow-up. In the second part of the thesis, we propose the Multiscale Invertible Generative Network (MsIGN) to sample from high-dimensional distributions by exploring the low-dimensional structure in the target distribution. The MsIGN models a transport map from a known reference distribution to the target distribution, and thus is very efficient in generating uncorrelated samples compared to MCMC-type methods. The MsIGN captures multiple modes in the target distribution by generating new samples hierarchically from a coarse scale to a fine scale with the help of a novel prior conditioning layer. The hierarchical structure of the MsIGN also allows training in a coarse-to-fine scale manner. The Jeffreys divergence is used as the objective function in training to avoid mode collapse. Importance sampling based on the prior conditioning layer is leveraged to estimate the Jeffreys divergence, which is intractable in previous deep generative networks. Numerically, when applied to two Bayesian inverse problems, the MsIGN clearly captures multiple modes in the high-dimensional posterior and approximates the posterior accurately, demonstrating its superior performance compared with previous methods. We also provide an ablation study to show the necessity of our proposed network architecture and training algorithm for the good numerical performance. Moreover, we also apply the MsIGN to the image synthesis task, where it achieves superior performance in terms of bits-per-dimension value over other flow-based generative models and yields very good interpretability of its neurons in intermediate layers.</p

    Asymptotics of stochastic learning in structured networks

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    Tree-Based Diffusion Schr\"odinger Bridge with Applications to Wasserstein Barycenters

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    Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schr\"odinger Bridge (TreeDSB), an extension of the Diffusion Schr\"odinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multimarginal Sinkhorn algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion

    Asymptotic Analysis and Numerical Approximation of some Partial Differential Equations on Networks

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    In this thesis, we consider three different model problems on one-dimensional networks with applications in gas, water supply, and district heating networks, as well as bacterial chemotaxis. On each edge of the graph representing the network, the dynamics are described by partial differential equations. Additional coupling conditions at network junctions are needed to ensure basic physical principles and to obtain well-posed systems. Each of the model problems under consideration contains an asymptotic parameter epsilon>0, which is assumed to be small, describing either a singular perturbation, different modeling scales, or different physical regimes. A central objective of this work is the investigation of the asymptotic behavior of solutions for epsilon going to zero. Moreover, we focus on suitable numerical approximations based on Galerkin methods that are still viable in the asymptotic limit epsilon=0 and preserve the structure and basic properties of the underlying problems. In the first part, we consider singularly perturbed convection-diffusion equations on networks as well as the corresponding pure transport equations arising in the vanishing diffusion limit for epsilon going to zero, in which the coupling conditions change in number and type. This gives rise to interior boundary layers at network junctions. On a single interval, corresponding asymptotic estimates are well-established. A main contribution is the transfer of these results to networks. For an appropriate numerical approximation, we propose a hybrid discontinuous Galerkin method which is particularly suitable for dominating convection and coupling at network junctions. An approximation strategy is developed based on layer-adapted meshes, leading to epsilon-uniform error estimates. The second part is dedicated to a kinetic model of chemotaxis on networks describing the movement of bacteria being influenced by the presence of a chemical substance. Via a suitable scaling the classical Keller-Segel equations can be derived in the diffusion limit. We propose a proper set of coupling conditions that ensure the conservation of mass and lead to a well-posed problem. The local existence of solutions uniformly in the scaling can be established via fixed point arguments. Appropriate a-priori estimates then enable us to rigorously show the convergence of solutions to the diffusion limit. Via asymptotic expansions, we also establish a quantitative asymptotic estimate. In the last part, we focus on models for gas transport in pipe networks starting from the non-isothermal Euler equations with friction and heat exchange with the surroundings. An appropriate rescaling of the equations accounting for the large friction, large heat transfer, and low Mach regime leads to simplified isothermal models in the limit epsilon=0. We propose a fully discrete approximation of the isothermal Euler equations using a mixed finite element approach. Based on a reformulation of the equations and relative energy estimates, we derive convergence estimates that hold uniformly in the scaling to a parabolic gas model. We finally extend some ideas and results also to the non-isothermal regime

    Bayesian computation in astronomy: novel methods for parallel and gradient-free inference

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    The goal of this thesis is twofold; introduce the fundamentals of Bayesian inference and computation focusing on astronomical and cosmological applications, and present recent advances in probabilistic computational methods developed by the author that aim to facilitate Bayesian data analysis for the next generation of astronomical observations and theoretical models. The first part of this thesis familiarises the reader with the notion of probability and its relevance for science through the prism of Bayesian reasoning, by introducing the key constituents of the theory and discussing its best practices. The second part includes a pedagogical introduction to the principles of Bayesian computation motivated by the geometric characteristics of probability distributions and followed by a detailed exposition of various methods including Markov chain Monte Carlo (MCMC), Sequential Monte Carlo (SMC) and Nested Sampling (NS). Finally, the third part presents two novel computational methods and their respective software implementations. The first such development is Ensemble Slice Sampling (ESS), a new class of MCMC algorithms that extend the applicability of the standard Slice Sampler by adaptively tuning its only hyperparameter and utilising an ensemble of parallel walkers in order to efficiently handle strong correlations between parameters. The parallel, black–box and gradient-free nature of the method renders it ideal for use in combination with computationally expensive and non–differentiable models often met in astronomy. ESS is implemented in Python in the well–tested and open-source software package called zeus that is specifically designed to tackle the computational challenges posed by modern astronomical and cosmological analyses. In particular, use of the code requires minimal, if any, hand–tuning of hyperparameters while its performance is insensitive to linear correlations and it can scale up to thousands of CPUs without any extra effort. The next contribution includes the introduction of Preconditioned Monte Carlo (PMC), a novel Monte Carlo method for Bayesian inference that facilitates effective sampling of probability distributions with non–trivial geometry. PMC utilises a Normalising Flow (NF) in order to decorrelate the parameters of the distribution and then proceeds by sampling from the preconditioned target distribution using an adaptive SMC scheme. PMC, through its Python implementation pocoMC, achieves excellent sampling performance, including accurate estimation of the model evidence, for highly correlated, non–Gaussian, and multimodal target distributions. Finally, the code is directly parallelisable, manifesting linear scaling up to thousands of CPUs
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