Related papers: Theory and numerics of subspace approximation of e…
The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm…
Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…
This book is about solving matrix nearness problems that are related to eigenvalues or singular values or pseudospectra. These problems arise in great diversity in various fields, be they related to dynamics, as in questions of robust…
We present a unified theoretical framework for parametric low-rank approximation, a research area devoted to the development of efficient algorithms that act as adaptive alternatives of traditional methods such as Singular Value…
We study the linear subspace fitting problem in the overparameterized setting, where the estimated subspace can perfectly interpolate the training examples. Our scope includes the least-squares solutions to subspace fitting tasks with…
In this paper, we introduce a new domain adaptation (DA) algorithm where the source and target domains are represented by subspaces spanned by eigenvectors. Our method seeks a domain invariant feature space by learning a mapping function…
We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=\lambda\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter.…
In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the…
Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of an operator where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several…
The numerical analysis for the small amplitude motion of an elastic beam with internal damping is investigated in domain with moving ends. An efficient numerical method is constructed to solve this moving boundary problem. The stability and…
How do the geometric properties of a domain impact the spectrum of an operator defined on it? How do we compute accurate and reliable approximations of these spectra? The former question is studied in spectral geometry, and the latter is a…
The parallel orbital-updating approach is an orbital/eigenfunction iteration based approach for solving eigenvalue problems when many eigenpairs are required. It has been proven to be efficient, for instance, in electronic structure…
We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that…
We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in…
In this paper, we give a numerical analysis for the transmission eigenvalue problem by the finite element method. A type of multilevel correction method is proposed to solve the transmission eigenvalue problem. The multilevel correction…
Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of…
In this paper, we present a novel parallel augmented subspace method and build a package Parallel Augmented Subspace Eigensolver (PASE) for solving large scale eigenvalue problems by the massively parallel finite element discretization.…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…
An analysis of high-dimensional data can offer a detailed description of a system but is often challenged by the curse of dimensionality. General dimensionality reduction techniques can alleviate such difficulty by extracting a few…
This paper is concerned with the Taylor-reduced basis method (Taylor-RBM) for the efficient approximation of eigenspaces of large scale parametric Hermitian matrices. The Taylor-RBM is a local model order reduction method, which constructs…