ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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

Session Overview 
Session  
MS 19b: Efficient Spectral and HighOrder Methods for HighDimensional Applications
 
Session Abstract  
In recent years, there has been a significant growth of interests in numerical methods and sci entific computing of highdimensional problems arising from different areas, including the classical gas kinetic theory, quantum mechanics, multiasset option pricing problem, uncer tainty quantification, etc. For those highdimensional applications, spectral and highorder methods are often preferred to loworder methods due to highaccuracy 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 highdimensional problems.  
Presentations  
2:00pm  2:30pm
A Domain Decomposition Model Reduction Method for Linear ConvectionDiffusion Equations with Random Coefficients Oak Ridge National Laboratory, United States of America We developed a domaindecomposition model reduction method for linear steady state convectiondiffusion equations with random coefficients. Of particular interest to this effort are the diffusion equation with random diffusivity, and the convectiondominated 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 highdimensional parametric dependence in practice. The motivation is to exploit domain decomposition to reduce the parametric dimension of local problems in subdomains, 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 highdimensionality and irregular behavior of the PDEs under consideration. The advantages of our method lie in three aspects: (i) onlineoffline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) sparse approximation of operators involving nonaffine highdimensional 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
VAEKRnet 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 VAEKRnet, which combines the canonical variational autoencoder (VAE) with our recently developed flowbased 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 VAEKRnet can be more effective and robust than the canonical VAE. We use the developed VAEKRnet as a density model to approximate either data distribution or an arbitrary probability density function (PDF) known up to a constant. VAEKRnet 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 highdimensional cases, we consider VAEKRnet 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 KullbackLeibler (KL) divergence between the model and the posterior, which often underestimates the variance if the model capability is not sufficiently strong. However, for highdimensional 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 meanfield 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 highdimensional multivariate functions and highdimensional 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 highdimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of highdimensional functions, and compute approximate solutions to FDEs by solving highdimensional 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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