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On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations

arXiv

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arXiv 2017年

作者： Weinan, E. Hutzenthaler, Martin Jentzen, Arnulf Kruse, Thomas Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States BICMR and School of Mathematical Sciences Peking University Beijing100870 China Faculty of Mathematics University of Duisburg-Essen Essen45117 Germany Seminar für Angewandte Mathematik ETH Zurich Zürich8092 Switzerland

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in the state-of-the-art pricing and hedging of financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article [E, W., Hutzenthaler, M., Jentzen, A., & Kruse, T. Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)] we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution for semilinear heat equations that the computational complexity is bounded by O(d ϵ-(4+δ)) for any δ 2 (0,∞), where d is the dimensionality of the problem and ϵ ∈ (0,∞) is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of 100-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for these 100-dimensional example PDEs are very satisfactory in terms of accuracy and speed. In addition, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the literature. Copyright © 2017, The Authors. All rights reserved.

关键词： Monte Carlo methods

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DECOUPLING FOR CONVEX HYPERSURFACES OF FINITE TYPE IN HIGHER DIMENSIONS

arXiv

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arXiv 2022年

作者： Gao, Chuanwei Li, Zhuoran Zhao, Tengfei Zheng, Jiqiang School of Mathematical Sciences Capital Normal University Beijing100048 China Department of Mathematics Taizhou University Taizhou225300 China School of Mathematics and Physics University of Science and Technology Beijing Beijing100083 China Institute of Applied Physics and Computational Mathematics Beijing100088 China

In this paper, we establish an 2 decoupling inequality for the convex hypersurface{ (ξ1, ..., ξn−1, ξ1m + ... + ξmn−1): (ξ1, ..., ξn−1) ∈ [0, 1]n−1} associated with the decomposition adapted to hypersurfaces of finite type, where n ≥ 2 and m ≥ 4 is an even number. The key ingredients of the proof include 2 decoupling inequalities for convex hypersurfaces of the form { (ξ1, ..., ξn−1, φ1(ξ1)+...+φs(ξs)+ξms+1+...+ξmn−1): (ξ1, ..., ξn−1) ∈ [0, 1]n−1} , 0 ≤ s ≤ n − 1, with φ1, ..., φs being m-nondegenerate. © 2022, CC BY.

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Optimal planning of slow-ramping power production in energy systems with renewables forecasts and limited storage

Optimal planning of slow-ramping power production in energy ...

引用

International Conference on Probabilistic Methods applied to Power Systems (PMAPS)

作者： Nathaniel Richmond Peter Jacko Armand M. Makowski BCAM — Basque Center for Applied Mathematics Bilbao Basque Country Spain Applied Mathematical & Computational Sciences University of Iowa USA Lancaster University Management School Lancaster UK Department of Electrical & Computer Engineering University of Maryland College Park USA

We address the cost-efficient operation of an energy production system under renewables uncertainty. We develop an MDP model for an idealized system with the following features: (1) perfectly predictable power demand, (2) a renewable power source subject to uncertain forecast, (3) limited energy storage, (4) an unlimited fast-ramping power source, and (5) a slow-ramping power source which requires (optimal) planning. A finite-horizon stochastic optimization problem is introduced to minimize the overall cost of operating the system, and then solved numerically using standard approaches (based on backward induction) and available data. In contrast with the unit commitment problem which is traditionally optimized for a single planning frame, we show in simple scenarios that it may be beneficial to optimize over a few planning frames, and that there is no benefit to considering longer (e.g., infinite) horizons. We discretize the state space in an attempt to mitigate the curse of dimensionality usually associated with numerically solving MDPs. We note that few discretization states already yield a significant decrease in the total cost.

关键词： Production Planning Wind forecasting computational modeling Energy storage Wind power generation

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Hypocoercivity-compatible finite element methods for the long-time computation of Kolmogorov’s equation

arXiv

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arXiv 2018年

作者： Georgoulis, Emmanuil H. Department of Mathematics University of Leicester LeicesterLE1 7RH United Kingdom Department of Mathematics School of Applied Mathematical and Physical Sciences National Technical University of Athens Zografou15780 Greece Institute of Applied and Computational Mathematics Foundation of Research and Technology – Hellas Vassilika Vouton 700 13 Heraklion Crete Greece

This work is concerned with the development of a family of Galerkin finite element methods for the classical Kolmogorov’s equation. Kolmogorov’s equation serves as a sufficiently rich, for our purposes, model problem for kinetic-type equations and is characterised by diffusion in one of the two (or three) spatial directions only. Nonetheless, its solution admits typically decay properties to some long time equilibrium, depending on closure by suitable boundary/decay-at-infinity conditions. A key attribute of the proposed family of methods is that they also admit similar decay properties at the (semi)discrete level for very general families of triangulations. The method construction uses ideas by the general theory of hypocoercivity developed by Villani [23], along with judicious choice of numerical flux functions. These developments turn out to be sufficient to imply that the proposed finite element methods admit a priori error bounds with constants independent of the final time, despite Kolmogorov equation’s degenerate diffusion nature. Thus, the new methods provably allow for robust error analysis for final times tending to infinity. The extension to three spatial dimensions is also briefly discussed. Copyright © 2018, The Authors. All rights reserved.

关键词： computational complexity

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Structured time-delay models for dynamical systems with connections to Frenet-Serret frame

arXiv

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arXiv 2021年

作者： Hirsh, Seth M. Ichinaga, Sara M. Brunton, Steven L. Kutz, J. Nathan Brunton, Bingni W. Department of Physics University of Washington SeattleWA United States Applied and Computational Mathematical Sciences Program University of Washington SeattleWA United States Department of Mechanical Engineering University of Washington SeattleWA United States Department of Applied Mathematics University of Washington SeattleWA United States Department of Biology University of Washington SeattleWA United States

Time-delay embeddings and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition (DMD) on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal with a zero diagonal;for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in Frenet-Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modeling procedure on data from several nonlinear synthetic and real-world examples. © 2021, CC BY.

关键词： Time delay

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Gegenbauer expansions and addition theorems for a binomial and logarithmic fundamental solution of the even-dimensional Euclidean polyharmonic equation

arXiv

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arXiv 2022年

作者： Cohl, Howard S. Hirtenstein, Jessie E. Lawrence, Jim Ritter, Lisa Applied and Computational Mathematics Division National Institute of Standards and Technology Mission ViejoCA92694 United States Department of Physics University of California DavisCA95616 United States Department of Mathematical Sciences George Mason University FairfaxVA22030 United States Applied and Computational Mathematics Division National Institute of Standards and Technology GaithersburgMD20878 United States

On even-dimensional Euclidean space for integer powers of the Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on spheres in arbitrary even dimensions. © 2022, CC0.

关键词： Fourier analysis

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Unbiased Approximations for Stationary Distributions of McKean-Vlasov SDEs

arXiv

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arXiv 2024年

作者： Awadelkarim, Elsiddig Chada, Neil K. Jasra, Ajay 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 Mathematics City University of Hong Kong Hong Kong School of Data Science The Chinese University of Hong Kong Shenzhen Shenzhen China

We consider the development of unbiased estimators, to approximate the stationary distribution of Mckean-Vlasov stochastic differential equations (MVSDEs). These are an important class of processes, which frequently appear in applications such as mathematical finance, biology and opinion dynamics. Typically the stationary distribution is unknown and indeed one cannot simulate such processes exactly. As a result one commonly requires a time-discretization scheme which results in a discretization bias and a bias from not being able to simulate the associated stationary distribution. To overcome this bias, we present a new unbiased estimator taking motivation from the literature on unbiased Monte Carlo. We prove the unbiasedness of our estimator, under assumptions. In order to prove this we require developing ergodicity results of various discrete time processes, through an appropriate discretization scheme, towards the invariant measure. Numerous numerical experiments are provided, on a range of MVSDEs, to demonstrate the effectiveness of our unbiased estimator. Such examples include the Currie-Weiss model, a 3D neuroscience model and a parameter estimation problem. © 2024, CC BY-NC-SA.

关键词： Stochastic systems

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An Energy-Based Discontinuous Galerkin Method with Tame CFL Numbers for the Wave Equation

arXiv

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arXiv 2021年

作者： Appelö, Daniel Zhang, Lu Hagstrom, Thomas Li, Fengyan Department of Computational Mathematics Science & Engineering Department of Mathematics Michigan State University East LansingMI48824 United States Department of Applied Physics and Applied Mathematics Columbia University New YorkNY10027 United States Department of Mathematics Southern Methodist University DallasTX75275 United States Department of Mathematical Sciences Rensselaer Polytechnic Institute TroyNY12180 United States

We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the typical numerical stiffness associated with high order piecewise polynomial approximations. In one space dimension with periodic boundary conditions and suitably chosen numerical fluxes, we prove bounds on the spatial operators that establish stability for CFL numbers c∆ht MSC Codes 65M12, 65M60 Copyright © 2021, The Authors. All rights reserved.

关键词： Galerkin methods

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Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

arXiv

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arXiv 2018年

作者： Liu, Yuan Cheng, Yingda Chen, Shanqin Zhang, Yong-Tao Department of Mathematics and Statistics Mississippi State University Mississippi StateMS39762 United States Department of Mathematics Department of Computational Mathematics Science and Engineering Michigan State University East LansingMI48824 United States Department of Mathematical Sciences Indiana University South Bend South BendIN46615 United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained. Copyright © 2018, The Authors. All rights reserved.

关键词： Linear equations

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Multi-scale Simulation of Carbon Nanotubes Interactions with Cell Membrane: DFT Calculations and Molecular Dynamic Simulation

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Procedia Materials Science 2015年 11卷 423-427页

作者： S.H. Tabari Y. Jamali R. Poursalehi Nanomaterials group Department of Materials Engineering Tarbiat Modares University Tehran 14115-143 Iran Department of Applied Mathematics School of Mathematical Sciences Tarbiat Modares University 14115-134 Iran Computational Physics Research Laboratory School of Nano-Science Institute for research in Fundamental Sciences (IPM) Tehran Iran

Due to unique physiochemical properties of carbon nanotubes (CNTs), they are utilized in biomedicine applications including biomedical engineering, tissue engineering, drug delivery, gene therapy and biosensor engineering. Studies have been shown that carbon nanotube have a large propensity to interact with biomembranes; therefore, the toxicity of these nanomaterials and understanding the way CNTs interact with cell become more important. Recently, enormous efforts have been made to understand the molecular mechanism of CNTs and lipid bilayer interaction. Because of the relative large scale of lipid bilayer, quantum ab-initio methods are not common for such systems; consequently, both all atom and coarse grained molecular dynamic simulation (MD) are widely performed. For simulation of such system it is important to apply the correct physical and structural properties of carbon nanotube in the simulation, for instance with density functional theory (DFT) modeling it has been clear that, for open-ended CNTs not only is there unsaturated bonds at the end of tube, but also has a significant dipole moment which should not neglected and their quantity correspond to nanotube chirality and diameter. The aim of this work is to discuss the partial charges distribution along the carbon nanotubes and describe how they affect in CNTs -lipid bilayer interaction by free energy calculation. Our Simulation consist of two stage, first partial charges for pristine and functional carbon nanotube were determined by DFT calculation then results were applied to molecular dynamic force field. The DFT simulation computes partial charges quantity at the ends of nanotube and its results indicate that partial charges for functional nanotubes are higher than pristine nanotubes. The MD simulation determines that nanotubes interaction energy with biomembrane is highly strong, about 120 kcal/mol.

关键词： molecular dynamic partial charge free energy carbon nanotube cell membrane density functional theory

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