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Random Measures, ANOVA Models and Quantifying Uncertainty in Randomized Controlled Trials

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arXiv 2023年

作者： Bastian, Caleb Deen Rabitz, Herschel Rempala, Grzegorz A. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Mathematics The Ohio State University ColumbusOH43210 United States Division of Biostatistics The Ohio State University ColumbusOH43210 United States

This short paper introduces a novel approach to global sensitivity analysis, grounded in the variance-covariance structure of random variables derived from random measures. The proposed methodology facilitates the application of information-theoretic rules for uncertainty quantification, offering several advantages. Specifically, the approach provides valuable insights into the decomposition of variance within discrete subspaces, similar to the standard ANOVA analysis. To illustrate this point, the method is applied to datasets obtained from the analysis of randomized controlled trials on evaluating the efficacy of the COVID-19 vaccine and assessing clinical endpoints in a lung cancer *** Codes 92-05, 92-11 Copyright © 2023, The Authors. All rights reserved.

关键词： Sensitivity analysis

A novel strong duality-based reformulation for trilevel infrastructure models in energy systems development

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arXiv 2023年

作者： Herrala, Olli Gabriel, Steven A. Oliveira, Fabricio Ekholm, Tommi Aalto University Department of Mathematics and Systems Analysis Espoo Finland University of Maryland Department of Mechanical Engineering/Applied Mathematics Statistics & Scientific Computing Program College ParkMD United States Norwegian University of Science and Technology Department of Industrial Economics and Technology Management Trondheim Norway Finnish Meteorological Institute Helsinki Finland

We explore the class of trilevel equilibrium problems with a focus on energy-environmental applications and present a novel single-level reformulation for such problems, based on strong duality. To the best of our knowledge, only one alternative single-level reformulation for trilevel problems exists. This reformulation uses a representation of the bottom-level solution set, whereas we propose a reformulation based on strong duality. Our novel reformulation is compared to this existing formulation, discussing both model sizes and computational performance. In particular, we apply this trilevel framework to a power market model, exploring the possibilities of an international policymaker in reducing emissions of the system. Using the proposed methods, we are able to obtain globally optimal solutions for a five-node case study representing the Nordic countries and assess the impact of a carbon tax on the electricity production portfolio. Copyright © 2023, The Authors. All rights reserved.

关键词： Energy resources

Data-Driven Mori-Zwanzig: Approaching a Reduced Order Model for Hypersonic Boundary Layer Transition

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arXiv 2023年

作者： Woodward, Michael Tian, Yifeng Mohan, Arvind Lin, Yen Ting Hader, Christoph Fasel, Hermann Chertkov, Misha Livescu, Daniel Graduate Interdisciplinary Program in Applied Mathematics Department of Mathematics University of Arizona TucsonAZ85721 United States Computer Computational and Statistical Sciences Division Los Alamos National Laboratory Los AlamosNM87544 United States Department of Aerospace and Mechanical Engineering University of Arizona TucsonAZ85721 United States

In this work, we apply, for the first time to spatially inhomogeneous flows, a recently developed data-driven learning algorithm of Mori-Zwanzig (MZ) operators, which is based on a generalized Koopman’s description of dynamical systems. The MZ formalism provides a mathematically exact procedure for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. The algorithm developed in this work applies Mori’s linear projection operator and an SVD based compression to the selection of the resolved variables (equivalently, a low rank approximation of the two time covariance matrices). We show that this MZ decomposition not only identifies the same spatio-temporal structures found by DMD, but it can also be used to extract spatio-temporal structures of the hysteresis effects present in the memory kernels. We perform an analysis of these structures in the context of a laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6, and show the dynamical relevance of the memory kernels. Additionally, by including these memory terms learned in our data-driven MZ approach, we show improvement in prediction accuracy over DMD at the same level of truncation and at a similar computational cost. Furthermore, an analysis of the spatio-temporal structures of the MZ operators shows identifiable structures associated with the nonlinear generation of the so-called "hot" streaks on the surface of the flared cone, which have previously been observed in experiments and direct numerical simulations. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamic mode decomposition

Bayesian Formulations for Graph Spectral Denoising

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arXiv 2023年

作者： Leone, Sam Sun, Xingzhi Perlmutter, Michael Krishnaswamy, Smita Program for Applied Mathematics Yale University United States Department of Computer Science Yale University United States Department of Mathematics Boise State University United States Department of Genetics Yale School of Medicine United States Wu Tsai Institute Yale University United States FAIR Meta AI United States Computational Biology and Bioinformatics Program Yale University United States

Here we consider the problem of denoising features associated to complex data, modeled as signals on a graph, via a smoothness prior. This is motivated in part by settings such as single-cell RNA where the data is very high-dimensional, but its structure can be captured via an affinity graph. This allows us to utilize ideas from graph signal processing. In particular, we present algorithms for the cases where the signal is perturbed by Gaussian noise, dropout, and uniformly distributed noise. The signals are assumed to follow a prior distribution defined in the frequency domain which favors signals which are smooth across the edges of the graph. By pairing this prior distribution with our three models of noise generation, we propose Maximum A Posteriori (M.A.P.) estimates of the true signal in the presence of noisy data and provide algorithms for computing the M.A.P. Finally, we demonstrate the algorithms' ability to effectively restore signals from white noise on image data and from severe dropout in single-cell RNA sequence data. © 2023, CC BY.

关键词： Frequency domain analysis

A physics informed neural network approach to simulating ice dynamics governed by the shallow ice approximation

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arXiv 2025年

作者： Chawla, Kapil Holmes, William Department of Mathematics Institute for Scientific Computing and Applied Mathematics Indiana University IN47405 United States Department of Mathematics Cognitive Science Program Indiana University IN47405 United States

In this article we develop a Physics Informed Neural Network (PINN) approach to simulate ice sheet dynamics governed by the Shallow Ice Approximation. This problem takes the form of a time-dependent parabolic obstacle problem [17]. Prior work has used this approach to address the stationary obstacle problem and here we extend it to the time dependent problem [4]. Through comprehensive 1D and 2D simulations, we validate the model’s effectiveness in capturing complex free-boundary conditions. By merging traditional mathematical modeling with cutting-edge deep learning methods, this approach provides a scalable and robust solution for predicting temporal variations in ice thickness. To illustrate this approach in a real world setting, we simulate the dynamics of the Devon Ice Cap, incorporating aerogeophysical data from 2000 [6] and 2018 [22]. © 2025, CC BY.

关键词： Approximation theory

Wigner non-negative states that verify the Wigner entropy conjecture

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arXiv 2024年

作者： Qian, Qipeng Gagatsos, Christos N. Program in Applied Mathematics The University of Arizona TucsonAZ85721 United States Department of Electrical and Computer Engineering The University of Arizona TucsonAZ85721 United States Wyant College of Optical Sciences The University of Arizona TucsonAZ85721 United States

We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth in [Phys. Rev. A 104, 042211 (2021)] and [J. Phys. A: Math. Theor. 50 385301]. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to 1 + ln π. We prove this conjecture for the qubits formed by Fock states |0) and |1) that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for the conjecture’s validity. Lastly, we elaborate on the states which are in accordance with our condition. Copyright © 2024, The Authors. All rights reserved.

关键词： Entropy

Viscoelastic fluid flow in a slowly varying planar contraction: the role of finite extensibility on the pressure drop

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arXiv 2024年

作者： Mahapatra, Bimalendu Ruangkriengsin, Tachin Stone, Howard A. Boyko, Evgeniy Faculty of Mechanical Engineering Technion – Israel Institute of Technology Haifa3200003 Israel Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mechanical and Aerospace Engineering Princeton University PrincetonNJ08544 United States

We analyze the steady viscoelastic fluid flow in slowly varying contracting channels of arbitrary shape and present a theory based on the lubrication approximation for calculating the flow rate–pressure drop relation at low and high Deborah (De) numbers. Unlike most prior theoretical studies leveraging the Oldroyd-B model, we describe the fluid viscoelasticity using a FENE-CR model and examine how the polymer chains’ finite extensibility impacts the pressure drop. We employ the low-Deborah-number lubrication analysis to provide analytical expressions for the pressure drop up to O(De4). We further consider the ultra-dilute limit and exploit a one-way coupling between the parabolic velocity and elastic stresses to calculate the pressure drop of the FENE-CR fluid for arbitrary values of the Deborah number. Such an approach allows us to elucidate elastic stress contributions governing the pressure drop variations and the effect of finite extensibility for all De. We validate our theoretical predictions with two-dimensional numerical simulations and find excellent agreement. We show that, at low Deborah numbers, the pressure drop of the FENE-CR fluid monotonically decreases with De, similar to the previous results for the Oldroyd-B and FENE-P fluids. However, at high Deborah numbers, in contrast to a linear decrease for the Oldroyd-B fluid, the pressure drop of the FENE-CR fluid exhibits a non-monotonic variation due to finite extensibility, first decreasing and then increasing with de. Nevertheless, even at sufficiently high Deborah numbers, the pressure drop of the FENE-CR fluid in the ultra-dilute and lubrication limits is lower than the corresponding Newtonian pressure drop. © 2024, CC BY.

关键词： Non Newtonian flow

Effective Dimension Aware Fractional-Order Stochastic Gradient Descent for Convex Optimization Problems

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arXiv 2025年

作者： Partohaghighi, Mohammad Marcia, Roummel Chen, YangQuan Electrical Engineering and Computer Science graduate program University of California Merced United States Department of Applied Mathematics University of California Merced MercedCA United States Lab Department of Mechanical Engineering School of Engineering University of California MercedCA95343 United States

Fractional-order stochastic gradient descent (FOSGD) leverages a fractional exponent to capture long-memory effects in optimization, yet its practical impact is often constrained by the difficulty of tuning and stabilizing this exponent. In this work, we introduce 2SED Fractional-Order Stochastic Gradient Descent (2SEDFOSGD), a novel method that synergistically combines the Two-Scale Effective Dimension (2SED) algorithm with FOSGD to automatically calibrate the fractional exponent in a data-driven manner. By continuously gauging model sensitivity and effective dimensionality, 2SED dynamically adjusts the exponent to curb erratic oscillations and enhance convergence rates. Theoretically, we demonstrate how this dimension-aware adaptation retains the benefits of fractional memory while averting the sluggish or unstable behaviors frequently observed in naive fractional SGD. Empirical evaluations across multiple benchmarks confirm that our 2SED-driven fractional exponent approach not only converges faster but also achieves more robust final performance, suggesting broad applicability for fractional-order methodologies in large-scale machine learning and related domains. © 2025, CC BY.

关键词： Convex optimization

English-Chinese Translation using Neural Machine Models

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English-Chinese Translation using Neural Machine Models

Telecommunications, Optics and Computer Science (TOCS), IEEE Conference on

作者： Qi Liu Pengfei Ou Applied and Computational Mathematical Sciences Program University of Washington Seattle the United States Department of Mathematics University of Illinois at Urbana Champaign Urbana the United States

ISBN: (纸本)9781665470544

During the last few years, neural machine translation (NMT) as a notable branch of machine translation has been increasing its popularity both in research and in practice. In particular, neural machine translation between English and Chinese, which are two of the most widely used languages all over the world, is receiving more and more attention. In this review, we discuss current mainstream models for neural machine translation, including recurrent neural network (RNN), convolution neural network (CNN), and self-attention network (SAN) or transformer. The mechanisms of these models are illustrated. Moreover, examples of studies and applications of them are analyzed. In addition, comparisons on the performance of different models are implemented.

关键词： Recurrent neural networks Training data Transformers Optics Tokenization Data models Telecommunications