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arXiv 2022年

作者： Cheng, Nuojin Malik, Osman Asif Xu, Yiming Becker, Stephen Doostan, Alireza Narayan, Akil Department of Applied Mathematics University of Colorado Boulder United States Applied Mathematics & Computational Research Division Lawrence Berkeley National Laboratory United States Corporate Model Risk Wells Fargo United States Smead Aerospace Engineering Sciences Department University of Colorado Boulder United States Scientific Computing and Imaging Institute Department of Mathematics University of Utah United States

Least squares regression is a ubiquitous tool for building emulators (a.k.a. surrogate models) of problems across science and engineering for purposes such as design space exploration and uncertainty quantification. When the regression data are generated using an experimental design process (e.g., a quadrature grid) involving computationally expensive models, or when the data size is large, sketching techniques have shown promise to reduce the cost of the construction of the regression model while ensuring accuracy comparable to that of the full data. However, random sketching strategies, such as those based on leverage scores, lead to regression errors that are random and may exhibit large variability. To mitigate this issue, we present a novel boosting approach that leverages cheaper, lower-fidelity data of the problem at hand to identify the best sketch among a set of candidate sketches. This in turn specifies the sketch of the intended high-fidelity model and the associated data. We provide theoretical analyses of this bi-fidelity boosting (BFB) approach and discuss the conditions the low- and high-fidelity data must satisfy for a successful boosting. In doing so, we derive a bound on the residual norm of the BFB sketched solution relating it to its ideal, but computationally expensive, high-fidelity boosted counterpart. Empirical results on both manufactured and PDE data corroborate the theoretical analyses and illustrate the efficacy of the BFB solution in reducing the regression error, as compared to the non-boosted solution. Copyright © 2022, The Authors. All rights reserved.

关键词： Random errors

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Mutual Information Maximizing Quantum Generative Adversarial Network and Its Applications in Finance

arXiv

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arXiv 2023年

作者： Lee, Mingyu Shin, Myeongjin Lee, Junseo Jeong, Kabgyun Department of Computer Science and Engineering Seoul National University Seoul08826 Korea Republic of QDQ Corp. Seoul06999 Korea Republic of School of Computing KAIST Daejeon34141 Korea Republic of School of Electrical and Electronic Engineering Yonsei University Seoul03722 Korea Republic of Quantum Computing R&D Norma Inc. Seoul04799 Korea Republic of Research Institute of Mathematics Seoul National University Seoul08826 Korea Republic of School of Computational Sciences Korea Institute for Advanced Study Seoul02455 Korea Republic of

One of the most promising applications in the era of NISQ (Noisy Intermediate-Scale Quantum) computing is quantum machine learning. Quantum machine learning offers significant quantum advantages over classical machine learning across various domains. Specifically, generative adversarial networks have been recognized for their potential utility in diverse fields such as image generation, finance, and probability distribution modeling. However, these networks necessitate solutions for inherent challenges like mode collapse. In this study, we capitalize on the concept that the estimation of mutual information between high-dimensional continuous random variables can be achieved through gradient descent using neural networks. We introduce a novel approach named InfoQGAN, which employs the Mutual Information Neural Estimator (MINE) within the framework of quantum generative adversarial networks to tackle the mode collapse issue. Furthermore, we elaborate on how this approach can be applied to a financial scenario, specifically addressing the problem of generating portfolio return distributions through dynamic asset allocation. This illustrates the potential practical applicability of InfoQGAN in real-world contexts. © 2023, CC BY.

关键词： Generative adversarial networks

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Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models

arXiv

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arXiv 2021年

作者： Guo, Ling Wu, Hao Zhou, Tao Department of Mathematics Shanghai Normal University Shanghai China School Odmathematical Sciences Tongji University Shanghai China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

We introduce in this work the normalizing field flows (NFF) for learning random fields from scattered measurements. More precisely, we construct a bijective transformation (a normalizing flow characterizing by neural networks) between a Gaussian random field with the Karhunen-Loève (KL) expansion structure and the target stochastic field, where the KL expansion coefficients and the invertible networks are trained by maximizing the sum of the log-likelihood on scattered measurements. This NFF model can be used to solve data-driven forward, inverse, and mixed forward/inverse stochastic partial differential equations in a unified framework. We demonstrate the capability of the proposed NFF model for learning Non Gaussian processes and different types of stochastic partial differential equations. Copyright © 2021, The Authors. All rights reserved.

关键词： Uncertainty analysis

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An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

arXiv

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arXiv 2021年

作者： Huang, Jianguo Wang, Haoqin Zhou, Tao School of Mathematical Sciences MOE-LSC Shanghai Jiao Tong University Shanghai China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method. © 2021, CC BY.

关键词： Deep neural networks

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Convergent and orthogonality preserving schemes for approximating the Kohn-Sham orbitals

arXiv

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arXiv 2021年

作者： Dai, Xiaoying Zhang, Liwei Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 Chinaand School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations. In this paper, we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model, with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals exponentially without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly. In addition, we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our *** Codes 37M05, 81Q05, 81Q35 © 2021, CC BY-NC-SA.

关键词： Density functional theory

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An inverse averaging finite element method for solving the size-modified Poisson-Nernst-Planck equations in ion channel simulations

arXiv

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arXiv 2021年

作者： Shen, Ruigang Zhang, Qianru Lu, Benzhuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this work, we introduce an inverse averaging finite element method (IAFEM) for solving the size-modified Poisson-Nernst-Planck (SMPNP) equations. Comparing with the classical Poisson-Nernst-Planck (PNP) equations, the SMPNP equations add a nonlinear term to each of the Nernst-Planck (NP) fluxes to describe the steric repulsion which can treat multiple nonuniform particle sizes in simulations. Since the new terms include sums and gradients of ion concentrations, the nonlinear coupling of SMPNP equations is much stronger than that of PNP equations. By introducing a generalized Slotboom transform, each of the size-modified NP equation is transformed into a self-adjoint equation with exponentially behaved coefficient, which has similar simple form to the standard NP equation with the Slotboom transformation. This treatment enables employing our recently developed inverse averaging technique to deal with the exponential coefficients of the reformulated formulations, featured with advantages of numerical stability and flux conservation especially in strong nonlinear and convection-dominated cases. Comparing with previous stabilization methods, the IAFEM proposed in this paper can still possess the numerical stability when dealing with convection-dominated problems. And it is more concise and easier to be numerically implemented. Numerical experiments about a model problem with analytic solutions are presented to verify the accuracy and order of IAFEM for SMPNP equations. Studies about the size-effects of a sphere model and an ion channel system are presented to show that our IAFEM is more effective and robust than the traditional finite element method (FEM) when solving SMPNP equations in simulations of biological *** Codes 35J61, 65N30, 92C40 Copyright © 2021, The Authors. All rights reserved.

关键词： Finite element method

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What we should learn from pandemic publishing

arXiv

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arXiv 2024年

作者： Sikdar, Satyaki Venturini, Sara Charpignon, Marie-Laure Kumar, Sagar Rinaldi, Francesco Tudisco, Francesco Fortunato, Santo Majumder, Maimuna S. Luddy School of Informatics Computing and Engineering Indiana University BloomingtonIN United States Department of Computer Science Loyola University Chicago ChicagoIL United States Senseable City Laboratory Massachusetts Institute of Technology CambridgeMA United States Department of Mathematics "Tullio Levi-Civita" University of Padova Padova Italy Institute for Data Systems and Society Massachusetts Institute of Technology CambridgeMA United States Network Science Institute Northeastern University BostonMA United States School of Mathematics The University of Edinburgh Edinburgh United Kingdom School of Mathematics Gran Sasso Science Institute L’Aquila Italy Department of Pediatrics Harvard Medical School BostonMA United States Computational Health Informatics Program Boston Children’s Hospital BostonMA United States

Since the emergence of COVID-19, discussions of ongoing pandemic-related research have accounted for an unprecedented share of media coverage and debate in the public sphere1. The urgency of the pandemic forced researchers to operate on an accelerated timeline, as both policymakers and the public relied on the most current evidence to guide their decisions and behaviors. With high demand for rapid pandemic-related insights and lower barriers to entry via preprint servers, the volume of COVID-19 articles skyrocketed2. The pressing need for research triggered the participation of many researchers with expertise in the science of infectious disease outbreaks (‘outbreak scientists’), who were joined by researchers from other disciplines (‘bellwethers’) and more junior researchers still in training (‘newcomers’) with the common goal of advancing the frontiers of pandemic science and informing policy decisions3. Please see Box 1 for details of this taxonomy. © 2024, CC BY-NC-ND.

关键词： COVID-19

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Personalized and uncertainty-aware coronary hemodynamics simulations: From Bayesian estimation to improved multi-fidelity uncertainty quantification

arXiv

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arXiv 2024年

作者： Menon, Karthik Zanoni, Andrea Khan, Owais Geraci, Gianluca Nieman, Koen Schiavazzi, Daniele E. Marsden, Alison L. Stanford School of Medicine StanfordCA United States Institute for Computational and Mathematical Engineering Stanford University StanfordCA United States Department of Electrical Computer and Biomedical Engineering Toronto Metropolitan University TorontoON Canada Center for Computing Research Sandia National Laboratories AlbuquerqueNM United States Division of Cardiovascular Medicine Stanford School of Medicine StanfordCA United States Department of Radiology Stanford School of Medicine StanfordCA United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN United States Department of Bioengineering Stanford University StanfordCA United States

Background: Non-invasive simulations of coronary hemodynamics have improved clinical risk stratification and treatment outcomes for coronary artery disease, compared to relying on anatomical imaging alone. However, simulations typically use empirical approaches to distribute total coronary flow amongst the arteries in the coronary tree, which ignores patient variability, the presence of disease, and other clinical factors. Further, uncertainty in the clinical data often remains unaccounted for in the modeling pipeline. Objective: We present an end-to-end uncertainty-aware pipeline to (1) personalize coronary flow simulations by incorporating vessel-specific coronary flows as well as cardiac function;and (2) predict clinical and biomechanical quantities of interest with improved precision, while accounting for uncertainty in the clinical data. Methods: We assimilate patient-specific measurements of myocardial blood flow from clinical CT myocardial perfusion imaging to estimate branch-specific coronary artery flows. Simulated noise in the clinical data is used to estimate the joint posterior distributions of the model parameters using adaptive Markov Chain Monte Carlo sampling. Additionally, the posterior predictive distribution for the relevant quantities of interest is determined using a new approach combining multi-fidelity Monte Carlo estimation with non-linear, data-driven dimensionality reduction. This leads to improved correlations between high- and low-fidelity model outputs. Results: Our framework accurately recapitulates clinically measured cardiac function as well as branch-specific coronary flows under measurement noise uncertainty. We observe substantial reductions in confidence intervals for estimated quantities of interest compared to single-fidelity Monte Carlo estimation and state-of-the-art multi-fidelity Monte Carlo methods. This holds especially true for quantities of interest that showed limited correlation between the low- and high-fidelity model

关键词： Monte Carlo methods

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On the convergence of stochastic gradient descent with bandwidth-based step size

arXiv

引用

arXiv 2021年

作者： Wang, Xiaoyu Yuan, Ya-Xiang Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences No.19A Yuquan Road Beijing100049 China State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

We investigate the stochastic gradient descent (SGD) method where the step size lies within a banded region instead of being given by a fixed formula. The optimal convergence rate under mild conditions and large initial step size is proved. Our analysis provides comparable theoretical error bounds for SGD associated with a variety of step sizes. In addition, the convergence rates for some existing step size strategies, e.g., triangular policy and cosine-wave, can be revealed by our analytical framework under the boundary constraints. The bandwidth-based step size provides efficient and flexible step size selection in optimization. We also propose a 1/t up-down policy and give several non-monotonic step sizes. Numerical experiments demonstrate the efficiency and significant potential of the bandwidth-based step-size in many *** Codes 65K05, 65K10, 90C15 Copyright © 2021, The Authors. All rights reserved.

关键词： Bandwidth

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Mathematical analysis and numerical approximations of density functional theory models for metallic systems

arXiv

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arXiv 2022年

作者： Dai, Xiaoying De Gironcoli, Stefano Yang, Bin Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China CNR-IOM DEMOCRITOS Simulation Centre Via Bononea 265 Trieste34146 Italy NCMIS Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this paper, we investigate the energy minimization model of the ensemble Kohn-Sham density functional theory for metallic systems, in which a pseudo-eigenvalue matrix and a general smearing approach are involved. We study the invariance and the existence of the minimizer of the energy functional. We propose an adaptive double step size strategy and the corresponding preconditioned conjugate gradient methods for solving the energy minimization model. Under some mild but reasonable assumptions, we prove the global convergence of our algorithms. Numerical experiments show that our algorithms are efficient, especially for large scale metallic systems. In particular, our algorithms produce convergent numerical approximations for some metallic systems, for which the traditional self-consistent field iterations fail to converge. Copyright © 2022, The Authors. All rights reserved.

关键词： Conjugate gradient method

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