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Two-Stage Maximum Score Estimator

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arXiv 2020年

作者： Gao, Wayne Yuan Xu, Sheng Xu, Kan Department of Economics University of Pennsylvania 133 S 36th St. PhiladelphiaPA19104 United States The Program in Applied and Computational Mathematics Princeton University Fine Hall Washington Road PrincetonNJ08544 United States

This paper considers the asymptotic theory of a semiparametric M-estimator that is generally applicable to models that satisfy a monotonicity condition in one or several parametric indexes. We call this estimator the two-stage maximum score (TSMS) estimator, since our estimator involves a first-stage nonparametric regression when applied to the binary choice model of Manski (1975, 1985). We characterize the asymptotic distribution of the TSMS estimator, which features phase transitions depending on the dimension of the first-stage estimation. Effectively, the first-stage nonparametric estimator serves as an imperfect smoothing function on a non-smooth criterion function, leading to the pivotality of the first-stage estimation error with respect to the second-stage convergence rate and asymptotic distribution. Copyright © 2020, The Authors. All rights reserved.

关键词： Functions

Phase equilibrium of water with hexagonal and cubic ice using the SCAN functional

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arXiv 2021年

作者： Piaggi, Pablo Miguel Panagiotopoulos, Athanassios Z. Debenedetti, Pablo G. Car, Roberto Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemical and Biological Engineering Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Machine learning models are rapidly becoming widely used to simulate complex physicochemical phenomena with ab initio accuracy. Here, we use one such model as well as direct density functional theory (DFT) calculations to investigate the phase equilibrium of water, hexagonal ice (Ih), and cubic ice (Ic), with an eye towards studying ice nucleation. The machine learning model is based on deep neural networks and has been trained on DFT data obtained using the SCAN exchange and correlation functional. We use this model to drive enhanced sampling simulations aimed at calculating a number of complex properties that are out of reach of DFT-driven simulations and then employ an appropriate reweighting procedure to compute the corresponding properties for the SCAN functional. This approach allows us to calculate the melting temperature of both ice polymorphs, the driving force for nucleation, the heat of fusion, the densities at the melting temperature, the relative stability of ice Ih and Ic, and other properties. We find a correct qualitative prediction of all properties of interest. In some cases, quantitative agreement with experiment is better than for state-of-the-art semiempirical potentials for water. Our results also show that SCAN correctly predicts that ice Ih is more stable than ice Ic. Copyright © 2021, The Authors. All rights reserved.

关键词： Ice

Reliability and efficiency of DWR-type a posteriori error estimates with smart sensitivity weight recovering

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arXiv 2020年

作者： Endtmayer, B. Langer, U. Wick, T. Doctoral Program on Computational Mathematics Johannes Kepler University Altenbergerstr. 69 LinzA-4040 Austria Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstr. 69 LinzA-4040 Austria Leibniz Universität Hannover Institut für Angewandte Mathematik AG Wissenschaftliches Rechnen Welfengarten 1 Hannover30167 Germany Leibniz Universität Hannover Germany

We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work [SIAM J. Sci. Comput., 42(1), A371–A394, 2020], we showed efficiency and reliability for error estimators based on enriched finite element spaces. However, the solution of problems on a enriched finite element space is expensive. In the literature, it is well known that one can use some higher-order interpolation to overcome this bottleneck. Using a saturation assumption, we extend the proofs of efficiency and reliability to such higher-order interpolations. The results can be used to create a new family of algorithms, where one of them is tested on three numerical examples (Poisson problem, p-Laplace equation, Navier-Stokes benschmark), and is compared to our previous algorithm. Copyright © 2020, The Authors. All rights reserved.

关键词： Interpolation

Nonuniform 3D finite difference elastic wave simulation on staggered grids

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arXiv 2020年

作者： Gao, Longfei Ghattas, Omar Keyes, David Oden Institute for Computational Engineering and Sciences The University of Texas at Austin AustinTX78712 United States Oden Institute for Computational Engineering and Sciences Jackson School of Geosciences Department of Mechanical Engineering The University of Texas at Austin AustinTX78712 United States Applied Mathematics and Computational Science Program Extreme Computing Research Center King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

We present an approach to simulate the 3D isotropic elastic wave propagation using nonuniform finite difference discretization on staggered grids. Specifically, we consider simulation domains composed of layers of uniform grids with different grid spacings, separated by nonconforming interfaces. We demonstrate that this layer-wise finite difference discretization has the potential to significantly reduce the simulation cost, compared to its fully uniform counterpart. Stability of such a discretization is achieved by using specially designed difference operators, which are variants of the standard difference operators with adaptations near boundaries or interfaces, and penalty terms, which are appended to the discretized wave system to weakly impose boundary or interface conditions. Combined with specially designed interpolation operators, the discretized wave system is shown to preserve the energy conserving property of the continuous elastic wave equation, and a fortiori ensure the stability of the simulation. Numerical examples are presented to demonstrate the efficacy of the proposed simulation approach. © 2020, CC BY.

关键词： Elastic waves

Models for plasma kinetics during simultaneous therapeutic plasma exchange and extracorporeal membrane oxygenation

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arXiv 2020年

作者： Puelz, Charles Danial, Zach Marinaro, Jon Raval, Jay Griffith, Boyce E. Peskin, Charles S. Courant Institute of Mathematical Sciences New York University Department of Emergency Medicine University of New Mexico Department of Pathology University of New Mexico Departments of Mathematics Applied Physical Sciences and Biomedical Engineering University of North Carolina Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Computational Medicine Program University of North Carolina McAllister Heart Institute University of North Carolina

This paper focuses on the derivation and simulation of mathematical models describing new plasma fraction in blood for patients undergoing simultaneous extracorporeal membrane oxygenation and therapeutic plasma exchange. Models for plasma exchange with either veno–arterial or veno–venous extracorporeal membrane oxygenation are considered. Two classes of models are derived for each case, one in the form of an algebraic delay equation and another in the form of a system of delay differential equations. In special cases, our models reduce to single compartment ones for plasma exchange that have been validated with experimental data [15]. We also show that the algebraic delay equations are forward Euler discretizations of the delay differential equations, with timesteps equal to transit times through model compartments. Numerical simulations are performed to compare different model types, to investigate the impact of plasma device port switching on the efficiency of the exchange process, and to study the sensitivity of the models to their parameters. Copyright © 2020, The Authors. All rights reserved.

关键词： Respiratory therapy

A trace theorem for sobolev spaces on the sierpinski gasket

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arXiv 2019年

作者： CAO, SHIPING LI, SHUANGPING STRICHARTZ, ROBERT S. TALWAI, PREM Department of Mathematics Cornell University Ithaca14853 United States Program in Applied and Computational Mathematics Princeton University NJ08544-1000 United States

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the L2 domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the *** Codes 28A80 Copyright © 2019, The Authors. All rights reserved.

关键词： Sobolev spaces

Cryo-EM reconstruction of continuous heterogeneity by Laplacian spectral volumes

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arXiv 2019年

作者： Moscovich, Amit Halevi, Amit Andén, Joakim Singer, Amit Program in Applied & Computational Mathematics Princeton University Princeton NJ United States Center for Computational Mathematics Flatiron Institute New YorkNY United States Department of Mathematics Princeton University Princeton NJ United States

Single-particle electron cryomicroscopy is an essential tool for high-resolution 3D reconstruction of proteins and other biological macromolecules. An important challenge in cryo-EM is the reconstruction of non-rigid molecules with parts that move and deform. Traditional reconstruction methods fail in these cases, resulting in smeared reconstructions of the moving parts. This poses a major obstacle for structural biologists, who need high-resolution reconstructions of entire macromolecules, moving parts included. To address this challenge, we present a new method for the reconstruction of macromolecules exhibiting continuous heterogeneity. The proposed method uses projection images from multiple viewing directions to construct a graph Laplacian through which the manifold of three-dimensional conformations is analyzed. The 3D molecular structures are then expanded in a basis of Laplacian eigenvectors, using a novel generalized tomographic reconstruction algorithm to compute the expansion coefficients. These coefficients, which we name spectral volumes, provide a high-resolution visualization of the molecular dynamics. We provide a theoretical analysis and evaluate the method empirically on several simulated data *** Codes 62P10, 65R32, 94A08 Copyright © 2019, The Authors. All rights reserved.

关键词： Macromolecules

An immersed interface method for incompressible flows and geometries with sharp features

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Journal of computational Physics 2025年 537卷

作者： Facci, Michael J. Kolahdouz, Ebrahim M. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Flatiron Institute Simons Foundation New YorkNY United States Department of Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

The immersed interface method (IIM) for models of fluid flow and fluid-structure interaction imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Most prior work using the IIM for fluid dynamics applications has focused on smooth interfaces, but boundaries with sharp features such as corners and edges can appear in practical analyses, particularly on engineered structures. The present study builds on our work to integrate finite element-type representations of interface geometries with the IIM. Initial realizations of this approach used a continuous Galerkin (CG) finite element discretization for the boundary, but as we show herein, these approaches generate large errors near sharp geometrical features. To overcome this difficulty, this study introduces an IIM approach using a discontinuous Galerkin (DG) representation of the jump conditions. Numerical examples explore the impacts of different interface representations on accuracy for both smooth and sharp boundaries, particularly flows interacting with fixed interface configurations. We demonstrate that using a DG approach provides accuracy that is comparable to the CG method for smooth cases. Further, we identify a time step size restriction for the CG representation that is directly related to the sharpness of the geometry. In contrast, time step size restrictions imposed by DG representations are demonstrated to be nearly insensitive to the presence of sharp features. © 2025

关键词： Incompressible flow

NUMERICAL ANALYSIS OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES FOR SEMIPERMEABLE MEDIA

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Journal of computational mathematics 2019年第4期37卷 506-523页

作者： Weimin Han Ziping Huang Cheng Wang Wei Xu Program in Applied Mathematical and Computational Sciences (A MCS)& Department of Mathematics University of Iowa Iowa City IA 522/2 USA School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College J taxing 314051 China School of Mathematical Sciences Tojigji University Shanghai 200092 China School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College Jiaxing 314051 China

In this paper, we consider elliptic hemivariational inequalities arising in applications in semipermeable media. In its general form, the model includes both interior and boundary semipermeability terms. Detailed study is given on the hemivariational inequality in the case of isotropic and homogeneous semipermeable media. Solution existence and uniqueness of the problem are explored. Convergence of the Galerkin method is shown under the basic solution regularity available from the existence result. An optimal order error estimate is derived for the linear finite element solution under suitable solution regularity assumptions. The results can be readily extended to the study of more general hemivariational inequalities for non-isotropic and heterogeneous semipermeable media with interior semipermeability and/or boundary semiperrneability. Numerical examples are presented to show the performance of the finite element approximations;in particular, the theoretically predicted optimal first order convergence in H' norm of the linear element solutions is clearly observed.

关键词： Hemivariational inequality Interior semipermeability Boundary semipermeability Finite element method Error estimate