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Optimal control for a discrete time influenza model

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2nd Colombian Congress on computational Biology and Bioinformatics, CCBCOL 2013

作者： Parra, Paula Andrea Gonzalez Ceberio, Martine Lee, Sunmi Castillo-Chavez, Carlos Universidad Autónoma de Occidente Departamento de Matematicas Cali Colombia The University of Texas at El Paso Computer Science Department El Paso TX 79968-0514 United States Kyung Hee University Department of Applied Mathematics Yongin 446-701 Korea Republic of Arizona State University Mathematical Computational and Modeling Sciences Center Tempe AZ 85287 United States

ISBN: (纸本)9783319015675

We formulated a discrete time model in order to study optimal control strategies for a single influenza outbreak. In our model, we divided the population into four classes: susceptible, infectious, treated, and recovered individuals. The total population was divided into subgroups according to activity or susceptibility levels. The goal was to determine how treatment doses should be distributed in each group in order to reduce the final epidemic size. The case of limited resources is considered by including an isoperimetric constraint. We found that the use of antiviral treatment resulted in reductions in the cumulative number of infected individuals. We proposed to solve the problem by using the primal-dual interior-point method that enforces epidemiological constraints explicitly. © Springer International Publishing Switzerland 2014.

关键词： Optimization

A fourth-order accurate finite volume method for ideal mhd via upwind constrained transport

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arXiv 2017年

作者： Felker, Kyle Gerard Stone, James Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Astrophysical Sciences Princeton University PrincetonNJ08544 United States

We present a fourth-order accurate finite volume method for the solution of ideal magnetohydrodynamics (MHD). The numerical method combines high-order quadrature rules in the solution of semi-discrete formulations of hyperbolic conservation laws with the upwind constrained transport (UCT) framework to ensure that the divergence-free constraint of the magnetic field is satisfied. A novel implementation of UCT that uses the piecewise parabolic method (PPM) for the reconstruction of magnetic fields at cell corners in 2D is introduced. The resulting scheme can be expressed as the extension of the second-order accurate constrained transport (CT) Godunov-type scheme that is currently used in the Athena astrophysics code. After validating the base algorithm on a series of hydrodynamics test problems, we present the results of multidimensional MHD test problems which demonstrate formal fourth-order convergence for smooth problems, robustness for discontinuous problems, and improved accuracy relative to the second-order scheme. Copyright © 2017, The Authors. All rights reserved.

关键词： Numerical methods

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

Energy stable arbitrary order ETD-MS method for gradient flows with Lipschitz nonlinearity

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arXiv 2021年

作者： Chen, Wenbin Wang, Shufen Wang, Xiaoming Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University Shanghai200433 China School of Mathematical Sciences Fudan University Shanghai200433 China SUSTech International Center for Mathematics Department of Mathematics Guangdong Provincial Key Laboratory of Computational Science and Material Design National Center for Applied Mathematics Shenzhen Southern University of Science and Technology Shenzhen518055 China

We present a methodology to construct efficient high-order in time accurate numerical schemes for a class of gradient flows with appropriate Lipschitz continuous nonlinearity. There are several ingredients to the strategy: the exponential time differencing (ETD), the multi-step (MS) methods, the idea of stabilization, and the technique of interpolation. They are synthesized to develop a generic kth order in time efficient linear numerical scheme with the help of an artificial regularization term of the form Aτk∂t∂ Lp(k)u where L is the positive definite linear part of the flow, τ is the uniform time step-size. The exponent p(k) is determined explicitly by the strength of the Lipschitz nonlinear term in relation to L together with the desired temporal order of accuracy k. To validate our theoretical analysis, the thin film epitaxial growth without slope selection model is examined with a fourth-order ETD-MS discretization in time and Fourier pseudo-spectral in space discretization. Our numerical results on convergence and energy stability are in accordance with our theoretical results. Copyright © 2021, The Authors. All rights reserved.

关键词： Film growth

A fitted scheme for a Caputo initial-boundary value problem

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arXiv 2024年

作者： Gracia, Jose Luis O’Riordan, E. Stynes, M. Department of Applied Mathematics University of Zaragoza Torres Quevedo Building Campus Rio Ebro Zaragoza50018 Spain School of Mathematical Sciences Dublin City University Glasnevin Dublin 9 Ireland Applied and Computational Mathematics Division Beijing Computational Science Research Center Haidian District Beijing100193 China

In this paper we consider an initial-boundary value problem with a Caputo time derivative of order α ∈ (0, 1). The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results. © 2024, CC BY-NC-ND.

关键词： Grading

SUPERCONVERGENT P1 HONEYCOMB VIRTUAL ELEMENTS AND LIFTED P3 SOLUTIONS

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arXiv 2023年

作者： Lin, Yanping Xu, Xuejun Zhang, Shangyou Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Hong Kong School of Mathematical Science Tongji University Shanghai200092 China Institute of Computational Mathematics AMSS Chinese Academy of Sciences Beijing100190 China Department of Mathematical Sciences University of Delaware NewarkDE19716 United States

When solving the Poisson equation on honeycomb hexagonal grids, we show that the P1 virtual element is three-order superconvergent in H1-norm, and two-order superconvergent in L2 and L∞ norms. We define a local post-process which lifts the superconvergent P1 solution to a P3 solution of the optimal-order approximation. The theory is confirmed by a numerical *** Codes 65N15, 65N30 © 2023, CC0.

关键词： Poisson equation

Composing Optimized Stepsize Schedules for Gradient Descent

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arXiv 2024年

作者： Grimmer, Benjamin Shu, Kevin Wang, Alex L. Johns Hopkins University Department of Applied Mathematics and Statistics United States California Institute of Technology Computational and Mathematical Sciences United States Purdue University Daniels School of Business United States

Recent works by Altschuler and Parrilo [2] and Grimmer et al. [5] have shown that it is possible to accelerate the convergence of gradient descent on smooth convex functions, even without momentum, just by picking special stepsizes. In this paper, we provide a general theory for composing stepsize schedules capturing all recent advances in this area and more. We propose three notions of "composable" stepsize schedules with elementary associated composition operations for combining them. From these operations, in addition to recovering recent works, we construct three highly optimized sequences of stepsize schedules. We first construct optimized stepsize schedules of every length generalizing the exponentially spaced silver stepsizes of [2]. We then construct highly optimized stepsizes schedules for minimizing final objective gap or gradient norm, improving on prior rates by constants and, more importantly, matching or beating the numerically computed minimax optimal schedules of [7]. We conjecture these schedules are in fact minimax (information theoretic) optimal. Several novel tertiary results follow from our theory including recovery of the recent dynamic gradient norm minimizing short stepsizes of [14] and extending them to objective gap minimization. Copyright © 2024, The Authors. All rights reserved.

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Emergent spaces for coupled oscillators

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arXiv 2020年

作者： Thiem, Thomas N. Kooshkbaghi, Mahdi Bertalan, Tom Laing, Carlo R. Kevrekidis, Ioannis G. Department of Chemical and Biological Engineering Princeton University United States Program in Applied and Computational Mathematics Princeton University United States Department of Mechanical Engineering Massachusetts Institute of Technology United States School of Natural and Computational Sciences Massey University New Zealand Department of Applied Mathematics and Statistics Johns Hopkins University United States

Systems of coupled dynamical units (e.g. oscillators or neurons) are known to exhibit complex, emergent behaviors that may be simplified through coarse-graining: a process in which one discovers coarse variables and derives equations for their evolution. Such coarse-graining procedures often require extensive experience and/or a deep understanding of the system dynamics. In this paper we present a systematic, data-driven approach to discovering "bespoke" coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical time integrators (initial value solvers). This approach allows us to learn accurate approximations of time derivatives of state variables from sparse flow data, and hence discover useful approximate differential equation descriptions of their dynamic behavior. We demonstrate this capability by learning ODEs that agree with the known analytical expression for the Kuramoto order parameter dynamics at the continuum limit. We then show how this approach can also be used to learn the dynamics of coarse variables discovered through our manifold learning methodology. In both of these examples, we compare the results of our neural network based method to typical finite differences complemented with geometric harmonics. Finally, we present a series of computational examples illustrating how a variation of our manifold learning methodology can be used to discover sets of "effective" parameters, reduced parameter combinations, for multi-parameter models with complex coupling. We conclude with a discussion of possible ext

关键词： Neural networks

A new note on spectra of optimal and superoptimal preconditioned matrices

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A new note on spectra of optimal and superoptimal preconditi...

The 2010 International Conference on Computer Application and System Modeling(2010计算机应用与系统建模国际会议 ICCASM 2010)

作者： Shao-Hua Cheng Li-Tao Zhang Ting-Zhu Huang Tong-Xiang Gu Department of Mathematics and Physics Zhengzhou Institute of Aeronautical Industry Management Henan School of Mathematical Sciences University of Electronic Science Technology of China Chengdu China Laboratory of Computationary Physics Institute of Applied Physics and Computational Mathematics Beij

Recently, Cheng et al. [Lin. Alg. Appl. 422 (2007): 482-485] proposed the spectral comparison of optimal preconditioner in Chan [SIAM J. Sci. Statist. Comput. 9 (1988): 766-771] and superoptimal preconditioner in Tyrtyshnikov [SIAM J. Matrix Anal. Appl. 13 (1992): 459-473). In this paper, based on the work of Cheng et al., we further compare the spectra of optimal and superoptimal preconditioned matrices.

关键词： Artificial neural networks