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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

Publisher's Note: “Reacting and non-reacting, three-dimensional shear layers with spanwise stretching” [Phys. Fluids 34, 123602 (2022)]

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Physics of Fluids 2023年第1期35卷

作者： Jonathan L. Palafoutas William A. Sirignano 1Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08540 USA 2Department of Mechanical and Aerospace Engineering University of California Irvine Irvine California 92697 USA

This article was published online on 2 December 2022 with errors throughout the paper. All the derivative terms had the wrong denominator; the denominators were

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Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs

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arXiv 2023年

作者： Leung, Tim Park, Hyungbin Yeo, Heejun Department of Applied Mathematics Computational Finance & Risk Management Program University of Washington SeattleWA98195 United States Department of Mathematical Sciences Research Institute of Mathematics Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of Department of Mathematical Sciences Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund (LETF). When the Markovian model parameters in the reference asset are uncertain, the robust long-term growth rate is derived by analyzing the worst-case parameters among an uncertainty set. We compute the growth rate and describe the optimal leverage ratio maximizing the robust long-term growth rate. To achieve this, the worst-case parameters are analyzed by the comparison principle, and the growth rate of the worst-case is computed using the martingale extraction method. The robust long-term growth rates are obtained explicitly under a number of models for the reference asset, including the geometric Brownian motion (GBM), Cox–Ingersoll–Ross (CIR), 3/2, and Heston and 3/2 stochastic volatility models. Additionally, we demonstrate the impact of stochastic interest rates, such as the Vasicek and inverse GARCH short rate models. This paper is an extended work of Leung and Park [2017]. Copyright © 2023, The Authors. All rights reserved.

关键词： Growth rate

Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems

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arXiv 2017年

作者： Hubmer, Simon Sherina, Ekaterina Neubauer, Andreas Scherzer, Otmar Johannes Kepler University Linz Doctoral Program Computational Mathematics Altenbergerstraße 69 LinzA-4040 Austria Technical University of Denmark Department of Applied Mathematics and Computer Science Asmussens Allé Kongens Lyngby 2800 Denmark Johannes Kepler University Linz Industrial Mathematics Institute Altenbergerstraße 69 LinzA-4040 Austria University of Vienna Computational Science Center Oskar Morgenstern-Platz 1 Vienna1090 Austria Johann Radon Institute Linz Altenbergerstraße 69 LinzA-4040 Austria

We consider a problem of quantitative static elastography, the estimation of the Lamé parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lamé parameters from displacement data simulating a static elastography experiment are *** Codes 65J22, 65J15, 74G75 Copyright © 2017, The Authors. All rights reserved.

关键词： Inverse problems

Realizability of Iso-g2 Processes via Effective Pair Interactions

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arXiv 2022年

作者： Wang, Haina Stillinger, Frank H. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Institute of Materials and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Natural Sciences Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by d-dimensional many-body systems. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. In light of this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse methodology that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably numerically realizable up to the packing fraction = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal" packing fraction c = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through c. For near but below c, the large-r behaviors of the effective potentials are given exactly by the functional forms exp[−κ()r] for d = 1, r−1/2 exp[−κ()r] for d = 2, and r−1 exp[−κ()r] (Yukawa form) for d = 3, where κ−1() is a screening length, and for = c, the potentials at large r are given by the pure Coulomb forms in the respective dimensions, as predicted by Torquato and Stillinger [Phys. Rev. E, 68, 041113 1-25 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost" random sequential addition at the saturation density c = 1/8 is much shorter-ranged than that for the 3D unit-step-function g2 at c, and thus does not constrain the realizability

关键词： Inverse problems

Designer Pair Statistics of Disordered Many-Particle Systems with Novel Properties

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arXiv 2023年

作者： Wang, Haina Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Materials Institute Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Natural Sciences Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States

Knowledge of exact analytical functional forms for the pair correlation function g2(r) and its corresponding structure factor S(k) of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing data base of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperature that precisely match targeted forms for both g2(r) and S(k). Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve systems with a wide range of translational order and self-diffusion coefficients D, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics. © 2023, CC BY.

关键词： Statistics

Precise Determination of Pair Interactions from Pair Statistics of Many-Body Systems In and Out of Equilibrium

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arXiv 2022年

作者： Torquato, Salvatore Wang, Haina Department of Chemistry Department of Physics Princeton Institute of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Natural Sciences Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States Department of Chemistry Princeton University PrincetonNJ08544 United States

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g2(r) or corresponding structure factor S(k) in d-dimensional Euclidean space Rd is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato conjectured that any realizable g2(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised a novel optimization algorithm to find effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g2(r) and structure factor S(k) so that the small- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point;(2) liquid under the Dzugutov potential;(3) nonequilibrium random sequential addition packing;and (4) and a nonequilibrium hyperuniform "cloaked" uniformly randomized lattice (URL). We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total L2-norm errors

关键词： Statistics

Asymmetric profiles observed in the recombination of Bi79+: A benchmark for relativistic theories involving interference

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Physical Review A 2009年第1期80卷 014503-014503页

作者： Nobuyuki Nakamura Anthony P. Kavanagh Hirofumi Watanabe Hiroyuki A. Sakaue Yueming Li Daiji Kato Fred J. Currell Xiao-Min Tong Tsutomu Watanabe Shunsuke Ohtani Institute for Laser Science The University of Electro-Communications Chofu Tokyo 182-8585 Japan Queen’s University Belfast Belfast BT7 1NN United Kingdom Chubu University Kasugai-shi Aichi 487-8501 Japan National Institute for Fusion Science Toki Gifu 509-5292 Japan Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing 100088 China Doctoral Program in Materials Science Graduate School of Pure and Applied Sciences University of Tsukuba Tsukuba Ibaraki 305-8573 Japan Center for Computational Sciences University of Tsukuba Tsukuba Ibaraki 305-8577 Japan

Asymmetric profiles have been observed in the recombination cross section of Be-like Bi obtained by measuring the electron energy dependence of the ion abundance ratio in an electron-beam ion trap. In contrast to the previous x-ray measurements, the present measurement gives the integrated recombination cross section with higher statistical quality, which provides a benchmark to test the relativistic theory involving the interference between the resonant and continuum states. The comparison with our theoretical study shows that the Breit interaction plays an important role in this case.

关键词： Benchmarking

Sequential Markov Chain Monte Carlo for Lagrangian Data Assimilation with Applications to Unknown Data Locations

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arXiv 2023年

作者： Ruzayqat, Hamza Beskos, Alexandros Crisan, Dan Jasra, Ajay Kantas, Nikolas Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Statistical Science University College London LondonWC1E 6BT United Kingdom Department of Mathematics Imperial College London LondonSW7 2AZ United Kingdom

We consider a class of high-dimensional spatial filtering problems, where the spatial locations of observations are unknown and driven by the partially observed hidden signal. This problem is exceptionally challenging as not only is high-dimensional, but the model for the signal yields longer-range time dependencies through the observation locations. Motivated by this model we revisit a lesser-known and provably convergent computational methodology from [4, 11, 28] that uses sequential Markov Chain Monte Carlo (MCMC) chains. We extend this methodology for data filtering problems with unknown observation locations. We benchmark our algorithms on Linear Gaussian state space models against competing ensemble methods and demonstrate a significant improvement in both execution speed and accuracy. Finally, we implement a realistic case study on a high-dimensional rotating shallow water model (of about 104 - 105 dimensions) with real and synthetic data. The data is provided by the National Oceanic and Atmospheric Administration (NOAA) and contains observations from ocean drifters in a domain of the Atlantic Ocean restricted to the longitude and latitude intervals [-51◦,-41◦], [17◦,27◦] *** Codes 62M20, 60G35, 60J20, 94A12, 93E11, 65C40 © 2023, CC BY.

关键词： Location