237 research outputs found
A Spectral Approach for Learning Spatiotemporal Neural Differential Equations
Rapidly developing machine learning methods has stimulated research interest
in computationally reconstructing differential equations (DEs) from
observational data which may provide additional insight into underlying
causative mechanisms. In this paper, we propose a novel neural-ODE based method
that uses spectral expansions in space to learn spatiotemporal DEs. The major
advantage of our spectral neural DE learning approach is that it does not rely
on spatial discretization, thus allowing the target spatiotemporal equations to
contain long range, nonlocal spatial interactions that act on unbounded spatial
domains. Our spectral approach is shown to be as accurate as some of the latest
machine learning approaches for learning PDEs operating on bounded domains. By
developing a spectral framework for learning both PDEs and integro-differential
equations, we extend machine learning methods to apply to unbounded DEs and a
larger class of problems.Comment: 21 pages, 5 figure
Numerical Methods for PDE Constrained Optimization with Uncertain Data
Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization.
The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods
A symmetric integrated radial basis function method for solving differential equations
In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation-based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high-order accuracy can be achieved together. Cartesian-grid-based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary-value problems governed by the Poisson and convection-diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of āgeneralized polynomial chaos' (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficient
Adaptive sparse grid discontinuous Galerkin method: review and software implementation
This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG)
method for computing high dimensional partial differential equations (PDEs) and
its software implementation. The C\texttt{++} software package called AdaM-DG,
implementing the aSG-DG method, is available on Github at
\url{https://github.com/JuntaoHuang/adaptive-multiresolution-DG}. The package
is capable of treating a large class of high dimensional linear and nonlinear
PDEs. We review the essential components of the algorithm and the functionality
of the software, including the multiwavelets used, assembling of bilinear
operators, fast matrix-vector product for data with hierarchical structures. We
further demonstrate the performance of the package by reporting numerical error
and CPU cost for several benchmark test, including linear transport equations,
wave equations and Hamilton-Jacobi equations
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
Discrete Sparse Fourier Hermite Approximations in High Dimensions
In this dissertation, the discrete sparse Fourier Hermite approximation of a function in a specified Hilbert space of arbitrary dimension is defined, and theoretical error bounds of the numerically computed approximation are proven. Computing the Fourier Hermite approximation in high dimensions suffers from the well-known curse of dimensionality. In short, as the ambient dimension increases, the complexity of the problem grows until it is impossible to numerically compute a solution. To circumvent this difficulty, a sparse, hyperbolic cross shaped set, that takes advantage of the natural decaying nature of the Fourier Hermite coefficients, is used to serve as an index set for the approximation. The Fourier Hermite coefficients must be numerically estimated since they are nearly impossible to compute exactly, except in trivial cases. However, care must be taken to compute them numerically, since the integrals involve oscillatory terms. To closely approximate the integrals that appear in the approximated Fourier Hermite coefficients, a multiscale quadrature method is used. This quadrature method is implemented through an algorithm that takes advantage of the natural properties of the Hermite polynomials for fast results.
The definitions of the sparse index set and of the quadrature method used will each introduce many interdependent parameters. These parameters give a user many degrees of freedom to tailor the numerical procedure to meet his or her desired speed and accuracy goals. Default guidelines of how to choose these parameters for a general function f that will significantly reduce the computational cost over any naive computational methods without sacrificing accuracy are presented. Additionally, many numerical examples are included to support the complexity and accuracy claims of the proposed algorithm
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