Search Results - numerical approximation of high-dimensional functions

A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations by Grohs, Philipp, Hornung, Fabian, Jentzen, Arnulf, von Wurstemberger, Philippe

“…). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power…”
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Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions by Hutzenthaler, Martin, Nguyen, Tuan Anh

“…The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs…”
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Quasi-interpolation for high-dimensional function approximation by Gao, Wenwu, Wang, Jiecheng, Sun, Zhengjie, Fasshauer, Gregory E.

“…The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation…”
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Numerical approximation based on deep convolutional neural network for high‐dimensional fully nonlinear merged PDEs and 2BSDEs by Xiao, Xu, Qiu, Wenlin, Nikan, Omid

“…This paper proposes two efficient approximation methods to solve high‐dimensional fully nonlinear partial differential equations (NPDEs) and second…”
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O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling by Khoromskij, Boris N.

“…In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high-dimensional applications…”
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Correcting for unknown errors in sparse high-dimensional function approximation by Adcock, Ben, Bao, Anyi, Brugiapaglia, Simone

“…We consider sparsity-based techniques for the approximation of high-dimensional functions from random pointwise evaluations…”
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The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions by Dehghan, Mehdi, Abbaszadeh, Mostafa, Mohebbi, Akbar

“…In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation…”
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A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge–Kutta method by Dehghan, Mehdi, Mohammadi, Vahid

“… The numerical meshless method which will be used here is RBF-FD technique. The main advantage of this method is the approximation of the required derivatives based on finite difference technique at each local-support domain as Ωi. At each Ω…”
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High-dimensional approximation with kernel-based multilevel methods on sparse grids by Kempf, Rüdiger, Wendland, Holger

“…Moderately high-dimensional approximation problems can successfully be solved by combining univariate approximation processes using an intelligent combination technique…”
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DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations by Tang, Kejun, Wan, Xiaoliang, Yang, Chao

“… In particular, we treat the residual as a probability density function and approximate it with a deep generative model, called KRnet…”
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Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation by Potts, Daniel, Weidensager, Laura

“…We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions…”
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Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions by Shan, Songqing, Wang, G. Gary

“… Such integration, however, faces multiple challenges. The most eminent challenges arise from high-dimensionality of problems, computationally-expensive analysis/simulation, and unknown function properties (i.e…”
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Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs by Dektor, Alec, Rodgers, Abram, Venturi, Daniele

“…We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs…”
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An adaptive PCE-HDMR metamodeling approach for high-dimensional problems by Yue, Xinxin, Zhang, Jian, Gong, Weijie, Luo, Min, Duan, Libin

“…Metamodel-based high-dimensional model representation (HDMR) has recently been developed as a promising tool for approximating high-dimensional and computationally expensive problems in engineering design and optimization…”
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DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option Pricing by Elbrächter, Dennis, Grohs, Philipp, Jentzen, Arnulf, Schwab, Christoph

“…) univariate functions, this provides insights into rates of deep ReLU approximation of multivariate functions with tensor structures…”
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Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance by Naito, Riu, Yamada, Toshihiro

“…The paper introduces a new deep learning-based high-order spatial approximation for a solution of a high-dimensional Kolmogorov equation where the initial condition is only assumed to be a…”
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Split S-ROCK Methods for High-Dimensional Stochastic Differential Equations by Komori, Yoshio, Burrage, Kevin

“…We propose explicit stochastic Runge–Kutta (RK) methods for high-dimensional Itô…”
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Visual Exploration of High Dimensional Scalar Functions by Gerber, S, Bremer, P, Pascucci, V, Whitaker, R

“… In many instances it is possible to observe both input parameters and system outputs, and characterize the system as a high-dimensional function…”
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Fast and flexible Bayesian species distribution modelling using Gaussian processes by Golding, Nick, Purse, Bethan V., Warton, David

“…‐dimensional interactions between predictors. We propose fitting GP SDMs using deterministic numerical approximations, rather than Markov chain Monte Carlo methods in order to make GPs more computationally efficient and easy to use…”
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Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems by Ehring, Tobias, Haasdonk, Bernard

“…Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality…”
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