International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 30th Nov 2022, 10:34:19pm CET
MS 19b: Efficient Spectral and High-Order Methods for High-Dimensional Applications
In recent years, there has been a significant growth of interests in numerical methods and sci-
entific computing of high-dimensional problems arising from different areas, including the
classical gas kinetic theory, quantum mechanics, multi-asset option pricing problem, uncer-
tainty quantification, etc. For those high-dimensional applications, spectral and high-order
methods are often preferred to low-order methods due to high-accuracy and lower memory
request. E.g. spectral sparse grid methods and polynomial chaos are two widely used spectral
methods for those high dimensional problems. Recently, deep neural network method is also
found to be efficient for solving high dimensional partial differential equations and the depth
of a neural network is related to the “spectral order” of the neural network approximation.
This minisymposium aims to bring together active researchers in related areas to present
and discuss their newest advances in both mathematical theory and numerical algorithm of
efficient numerical approximations for high-dimensional problems.
2:00pm - 2:30pm
A Domain Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients
Oak Ridge National Laboratory, United States of America
We developed a domain-decomposition model reduction method for linear steady- state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equation with random diffusivity, and the convection-dominated transport equation with random velocity. We investigated two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence in practice. The motivation is to exploit domain decomposition to reduce the parametric dimension of local problems in sub-domains, such that entire parametric map can be approximated with a small number of expen- sive PDE simulations. The new method combines domain decomposition with model reduction and sparse polynomial approximation, so as to simultaneously handle high-dimensionality and irregular behavior of the PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) sparse approximation of operators involving non-affine high-dimensional random fields; (iii) effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solutions. Two numerical examples are provided to demonstrate the advantageous performance of our method.
2:30pm - 3:00pm
VAE-KRnet and its applications to variational Bayes
Louisiana State University, United States of America
In this work, we have proposed a generative model for density estimation, called VAE-KRnet, which combines the canonical variational autoencoder (VAE) with our recently developed flow-based generative model, called KRnet. VAE is used as a dimension reduction technique to capture the latent space, and KRnet is used to model the distribution of the latent variables. Using a linear model between the data and the latent variables, we show that VAE-KRnet can be more effective and robust than the canonical VAE. We use the developed VAE-KRnet as a density model to approximate either data distribution or an arbitrary probability density function (PDF) known up to a constant. VAE-KRnet provides a flexible density model in terms of dimensionality. When the number of dimensions is relatively small, KRnet can be used to approximate the posterior effectively with respect to the original random variable. For high-dimensional cases, we consider VAE-KRnet to incorporate the dimension reduction. One important application is the variational Bayes for the approximation of the posterior distribution. The variational Bayes approaches are usually based on the minimization of the Kullback-Leibler (KL) divergence between the model and the posterior, which often underestimates the variance if the model capability is not sufficiently strong. However, for high-dimensional distributions, it is very challenging to construct an accurate model due to the curse of dimensionality, where extra assumptions are often introduced for efficiency, e.g., the classical mean-field approach assumes mutual independence between dimensions. To alleviate the underestimation of variance, we include into the loss the maximization of the mutual information between the latent random variable and the original one. Numerical experiments have been presented to demonstrate the effectiveness of our model.
3:00pm - 3:30pm
Spectral methods for nonlinear functionals and functional differential equations
University of California, Santa Cruz, United States of America
We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE.
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