本文考虑了给定的Quantity of Interest（QoI）的统计量近似问题，该QoI依赖于由N个随机变量参数化的随机域上线性椭圆型PDE的解。椭圆问题被映射到具有固定确定性域的相应PDE。我们证明了解可以解析延拓到随机变量的复平面上的一个明确区域。然后使用稀疏网格列点法计算QoI的均值和标准差。最后推导并比较了QoI的均值和方差的收敛率与数值实验结果。
英文摘要：
In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables. The elliptic problem is remapped on to a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in $\C^{N}$ with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and standard deviation of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.
Hermite spectral collocation methods for fractional PDEs in unbounded domains
Southern University of Sciences and Technology（南方科技大学）；Hong Kong Baptist University（香港浸会大学）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）
Based on the Fourier extension, we propose an oversampling collocation method for solving the elliptic partial differential equations with variable coefficients over arbitrary irregular domains. This method only uses the function values on the equispaced nodes, which has low computational cost and versatility. While a variety of numerical experiments are presented to demonstrate the effectiveness of this method, it shows that the approximation error fast reaches a plateau with increasing the degrees of freedom, due to the inherent ill-conditioned of frames.
We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of t…
We obtained convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ from the non-compact set $\mathbb{R}^\infty$. The approximation error is measured in the norm of the Bochner space $L_2(\mathbb{R}^\infty, V, γ)$, where $γ$ is the infinite tensor product standard Gaussian probability measure on $\mathbb{R}^\infty$ and $V$ is the energy space. We also obtained similar results for the case when the lognormal inputs are parametrized on $\mathbb{R}^M$ with very large dimension $M$, and th…
This work is a follow-up to our previous contribution ("Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)", SIAM J. Numer. Anal., 2018), and contains further insights on some aspects of the solution of elliptic PDEs with lognormal diffusion coefficients using sparse grids. Specifically, we first focus on the choice of univariate interpolation rules, advocating the use of Gaussian Leja points as introduced by Narayan and Jakeman ("Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional …
IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Consiglio Nazionale delle Ricerche - Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR-IMATI)（意大利国家研究委员会应用数学与信息技术研究所）；RWTH Aachen University（亚琛工业大学）
We introduce a new numerical method based on machine learning to approximate the solution of elliptic partial differential equations with collocation using a set of sigmoidal functions. We show that a feedforward neural network with a single hidden layer with sigmoidal functions and fixed, random, internal weights and biases can be used to compute accurately a collocation solution. The choice to fix internal weights and bias leads to the so-called Extreme Learning Machine network. We discuss how to determine the range for both internal weights and biases in order to obtain a good underlining a…
Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Università di Pavia（帕维亚大学）；CNR-IMATI（意大利国家研究委员会应用数学研究所）
We consider the isogeometric analysis for fractional PDEs involving the fractional Laplacian in two dimensions. An isogeometric collocation method is developed to discretize the fractional Laplacian and applied to the fractional Poisson problem and the time-dependent fractional porous media equation. Numerical studies exhibit monotonous convergence with a rate of $\mathcal{O}(N^{-1})$, where $N$ is the degrees of freedom. A comparison with finite element analysis shows that the method enjoys higher accuracy per degree of freedom and has a better convergence rate. We demonstrate that isogeometr…
Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity
Department of Computer Science and Technology（计算机科学与技术系）；Department of Applied Mathematics and Theoretical Physics（应用数学与理论物理系）；University of Cambridge（剑桥大学）
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers fo…
Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severely ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyse and characterise the convergen…
We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $θ$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties…
Predictive Moving Sample Method for Physics-Informed Neural Solvers of Time-Dependent PDEs
The School of Information Science and Technology and the Institute of Mathematical Sciences ShanghaiTech University（上海科技大学信息科学与技术学院与数学科学学院）；SKLMS & LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所，数学重点实验室）；School of Mathematical Sciences University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Research Center for Mathematics and Interdisciplinary Sciences Shandong University（山东大学数学交叉前沿科学中心）；Frontiers Science Center for Nonlinear Expectations, Minister of Education Shandong University（教育部非线性期望前沿科学中心）；The Institute of Mathematical Sciences ShanghaiTech University（上海科技大学数学科学学院）
Local Gevrey regularity and Quasi-Monte Carlo quadrature for PDEs parameterized on non-compact domains
Vietnam National Foundation for Science and Technology Development (NAFOSTED)（越南国家科学技术基金会）；Vietnam Institute for Advanced Study in Mathematics (VIASM)（越南高等数学研究所）；Hausdorff Research Institute for Mathematics（豪斯多夫数学研究所）；University of Oldenburg（奥尔登堡大学）
This paper considers the approximation of partial differential equations with a point collocation framework based on high-order local maximum-entropy schemes (HOLMES). In this approach, smooth basis functions are computed through an optimization procedure and the strong form of the problem is directly imposed at the collocation points, reducing significantly the computational times with respect to the Galerkin formulation. Furthermore, such a method is truly meshless, since no background integration grids are necessary. The validity of the proposed methodology is verified with supportive numer…
This paper is concerned with the regularity of solutions to parabolic evolution equations. Special attention is paid to the smoothness in the specific anisotropic scale $\ B^{r\mathbf{a}}_{τ,τ}, \ \frac{1}τ=\frac{r}{d}+\frac{1}{p}\ $ of Besov spaces where $\mathbf{a}$ measures the anisotropy. The regularity in these spaces determines the approximation order that can be achieved by fully space-time adaptive approximation schemes. In particular, we show that for the heat equation our results significantly improve previous results by Aimar and Gomez [3].
Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability an…
This paper focuses on the spectral properties of the mass and stiffness matrices for acoustic wave problems discretized with Isogeometric analysis (IGA) collocation methods in space and Newmark methods in time. Extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann and absorbing boundary conditions, varying the polynomial degree $p$, mesh size $h$, regularity $k$, of the IGA discretization and the time step $Δt$ and parameter $β$ of the Newmark method. Results on the s…
